Chapter 2
Social Choice Theory
2.1
Collective Choice Environments
A collective choice environment is de¯ned by four things: a set of social states, which
represent information that we want to treat as variable; a set of individual decisionmakers; a
set of alternatives from which they must make a collective choice; and individual preferences
over alternatives.
£
set of social states µ, Á, etc.
N (µ)
¯nite set of individuals i; j, etc.
X (µ)
set of alternatives x; y; z, etc.
jX(µ)j ¸ 2
(Pi (µ); Ri(µ))
i's preferences in state µ, satisfying
Axioms 1-3.
By Axioms 1 and 2, (Pi (µ); Ri(µ)) is a dual pair. We also impose Axioms 3 on individual
preferences, so (Pi (µ); Ri(µ)) is actually an ordering, i.e., Ri(µ) is a weak order and Pi (µ) is
a strict order. As usual, Ii(µ) will denote i's indi®erence relation in state µ.
Recall that Ri (µ) and Pi(µ) contain the same information: xRi(µ)y if and only if not yPi (µ)x,
and xPi(µ)y if and only if not yRi(µ)x. Whether we discuss weak or strict preference is
entirely a matter of convenience.
The dependence of N, X, and (Pi ; Ri ) on µ indicates that, conceivably, these sets could be
treated as variables. It is customary, however, to restrict the analysis to collective choice
problems in which the sets of individuals and alternatives are ¯xed.1 We therefore drop the
1
This is certainly not universal. There is a nice version of Arrow's impossibility theorem (and others)
that is stated in terms of a variable set of feasible alternatives. And Peyton Young, William Thomson, and
60
dependence on µ and write simply N and X. We assume the number of individuals is n,
and we enumerate them 1; : : : ; n.
Paralleling our earlier notation, let
Ri(xjµ) = fy 2 X j yRi(µ)xg
Pi(xjµ) = fy 2 X j yPi (µ)xg
denote the upper sections of individual i's weak and strict preferences in state µ, and denote
¡1
lower sections by R¡1
i (xjµ) and Pi (xjµ).
Denote the pro¯les of individual preferences at µ by
P R(µ) = ((P1(µ); R1(µ)); : : : ; (Pn (µ); Rn(µ)))
To save space, I may sometimes write just (R1; : : : ; Rn ) for a pro¯le of preferences.
The set PR(£) = fPR(µ) j µ 2 £g is the \domain of preferences," the set of possible
preference pro¯les, depending on the state. I assume the existence of a free pair, i.e., we
assume there exist x; y 2 X such that, for every partition fI; J g of N, there exists µ 2 £
such that
I = fi 2 N j xPi(µ)yg
and
J = fi 2 N j yPi(µ)xg:
In words, individual strict preferences over x and y are unrestricted. This is a very mild
assumption satis¯ed in most all environments.
Some extra notation:
R(x; yjµ) = fi 2 N j xRi (µ)yg
P(x; yjµ) = fi 2 N j xPi(µ)yg
r(x; yjµ) = jR(x; yjµ)j
p(x; yjµ) = jP(x; yjµ)j:
We will later consider di®erent restrictions on the domain of preferences: in many applications there are some a priori restrictions that can be imposed on individual preferences by
the analyst. We'll survey some examples in the next section.
We are interested in how individuals overcome possible con°icts of interest to collectively
choose an alternative. Following the standard approach to the analysis of individual decision
making, we view the group as making this decision on the basis of a \social preference." For
others have introduced several axioms of interest when the set of individuals may vary.
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now, we suppose social preferences are determined arbitrarily by the social state according
to the mapping F :
F
µ7
! (PF (µ); RF (µ)):
We call F a Preference Aggregation Rule, or simply PAR.
Here, RF (µ) is the weak social preference relation at µ and PF (µ) is the strict social preference relation at µ. We maintain Axioms 1 and 2, so (PF (µ); RF (µ)) is a dual pair. We use
IF (µ) to denote the corresponding social indi®erence relation at µ.
Thus, strict preference PF (µ) and weak preference RF (µ) contain the same information:
xPF (µ)y , not yRF (µ)x
and
xRF (µ)y , not yPF (µ)x:
Thus, we can use RF (µ) and PF (µ) interchangeably, depending on which is more convenient.
If the PAR F is understood, it may be suppressed. Thus, we may write simply R(µ) and
P(µ).
Paralleling our earlier notation, let
RF (xjµ) = fy 2 X j yRF (µ)xg
PF (xjµ) = fy 2 X j yPF (µ)xg
IF (xjµ) = fy 2 X j xIF (µ)yg
denote the upper sections of weak and strict social preference and social indi®erence, and
¡1
denote lower sections by R¡1
F (xjµ) and P F (xjµ). (Because I F (µ) is symmetric, we don't
need notation for its lower sections.)
In the social choice theory approach, we apply the same theory of choice to groups that we
do to individuals. Thus, given a PAR F , a state µ, and social preferences (PF (µ); RF (µ)),
we posit that the plausible collective choices are just the socially maximal alternatives, if
any. We refer to this set of alternatives as the core of F at µ and denote it by CF (µ).
Formally, it is de¯ned as
CF (µ) = M(RF (µ)) = M(PF (µ)):
When the PAR F is understood, we write simply C(µ) for the core. Whether this set is
nonempty will depend on continuity and transitivity properties of social preferences.
2.2
Examples of Environments
Example 1: The Discrete Choice Model. A classic example of a collective choice
environment from political science is that of a group N of voters who must choose from
a ¯nite set X of candidates to ¯ll a political o±ce. Each voter has preferences over the
62
candidates, and states simply index the set of possible preference pro¯les. Alternatively, a
voting body (e.g., an electorate, a city council, a board of directors) must choose from a
¯nite set of projects to undertake. Or a department faculty must choose from a ¯nite set
of job candidates to ¯ll a position. In these kinds of environments, it is often di±cult to
impose a priori restrictions on individual preferences, and it is natural to allow for every
pro¯le of weak orders. Let
U = f((P1 ; R1); : : : ; (Pn; Rn)) j 8i 2 N : (Pi ; Ri ) is an order of X g
denote the set of all pro¯les of orders of X. We refer to the assumption that P R(£) = U
as unrestricted domain.
Sometimes, especially when the set of alternatives is small, we may wish to consider the possibility that individuals can always discern a strict preference between distinct candidates.
Let
L = f((P1; R1 ); : : : ; (Pn ; Rn )) j 8i 2 N : (Pi ; Ri ) is a linear order of Xg:
We refer to the assumption that PR(£) = L as linear domain.
There are special ¯nite choice environments in which extra structure suggests intuitive
preference restrictions. For example, suppose an electorate has several political positions
to ¯ll, as in multi-member district systems. Suppose there are k candidates and m · k
slots to ¯ll. In this case, an alternative x would represent an assignment of m candidates
to the m di®erent slots, y would represent another assignment. We then take as primitive
voter preferences over assignments of candidates to the slots | not actually over candidates
themselves. In this way, the collective choice problem of ¯lling multiple political o±ces is
isomorphic to the problem of ¯lling one.
Note, however, that preferences over assignments of candidates may implicitly be generated
from (unmodelled) preferences over the candidates themselves. If so, we may be able to
take advantage of this extra structure in formulating reasonable preference restrictions.
For example, suppose there are three o±ces to ¯ll; let x = (a; b; c) denote the assignment
of candidate a to the ¯rst o±ce, b to the second, c to the third; let y = (a; b; d); let
x0 = (a0; b0 ; c); and let y 0 = (a0 ; b 0; d). If xPiy, then we might infer that voter i prefers having
c in the third o±ce over d. Since x0 and y 0 di®er only in that way, it may be reasonable to
infer x0 Pi y0 . This is a restriction on preferences that is not possible in the original model.
It is not without loss of generality, however, for implicit in it is an idea of \separability,"
i.e., that i's preferences over the third slot are independent of which candidates ¯ll the ¯rst
two.
Similarly, we might suppose that a voting body, say a city council, has a ¯xed budget from
which it can fund any number of projects, subject to budget balance. In this case, x would
denote a list of projects funded, and we assume council members have preferences over lists
63
of projects. Or in the example of an academic department, there may be several slots to
¯ll, in which case an alternative x would be a list of hired job candidates.
Example 2: The One-Dimensional Spatial Model. A second classic example of a
collective choice environment is that of a society that must choose a single proportional
income tax rate between 0 and 1. Or that of a town that must choose the location of
a library on its main street. Or, suppose that public policies can be ordered according
to a single summary statistic that contains all information relevant for voter preferences.
The standard interpretation is that policies corresponding to smaller values of the statistic
are more \liberal," while those corresponding to greater values of the statistic are more
\conservative." Thus, under this interpretation, we reduce policies to their ideological
content, placing them in a one-dimensional space to represent their place in the liberalconservative spectrum. If ideological content is indeed all that matters to voters, we can
then model public policy as the choice of a point on the real line. The common element of
these examples is the one-dimensional structure inherent in the set of alternatives.
This structure suggests an intuitive restriction on the domain of preferences. The idea is that
each individual will have some ideal alternative (e.g., a favorite tax rate, location of a library,
ideological mix); and that, if we move to the left or right from that ideal point, then the
alternatives grow worse for the individual. Formally, we say a pro¯le ((P1 ; R1 ); : : : ; (Pn; Rn))
of individual preferences is single-peaked if there exists a weak linear ordering, say ¹, of X
such that, for each i 2 N, there exists x
~ i 2 X satisfying the following conditions:
² for all y 6
=x
~i, x
~ iPiy;
² for all y; z 2 X, if z Á y Á x
~i , then yPi z;
² for all y; z 2 X, if x
~ i Á y Á z, then yPi z,
where Á is the asymmetric part of ¹.2 (Because ¹ is complete, x Á y if and only if not
y ¹ x.) In this case, we say ((P1; R1); : : : ; (Pn ; Rn )) is single-peaked with respect to ¹. Let
S = f((P1 ; R1 ); : : : ; (Pn ; Rn )) j ((P1; R1 ); : : : ; (Pn ; Rn )) is single-peaked g:
We refer to the assumption that PR(£) = S as single-peaked domain.
Given a weak order Ri for individual i, we call x
~i 2 X the individual's ideal point if, for all
i
other x 2 X, x
~ Pi x. Other terminology is \bliss point" or \satiation point."
For an example of single-peakedness, suppose that n = 3, that X = fx; y; zg, and that
2
Many of our results for this model can be adapted to the more general setting in which we allow
x~ 2 f1; ¡1g, representing \extremists" who prefer alternatives as far as possible in one direction or the
other.
i
64
individual preferences are as follows.
R 1 R2 R3
x y
z
y
x y
z
z
x
That (R1 ; R2; R3) is single-peaked with respect to x Á y Á z (with x
~1 = x; x
~ 2 = y; x
~3 = z)
can be easily seen by drawing utility representations of these preferences. See Figure 2.1.
Note that these representations really do look \single-peaked."
Figure 2.1: A Single-peaked pro¯le
It's important to note that, in the de¯nition of single-peakedness, the same weak linear
ordering ¹ of X must be used for all individual preferences. But di®erent ¹'s can be used
for di®erent pro¯les. For example, consider the following preferences.
R 1 R2 R3
y
x z
x z
x
z
y
y
This pro¯le is single-peaked with respect to y Á x Á z, but it is not single-peaked with
respect to the ordering used above, x Á y Á z.
Note that the domain of single-peaked preference pro¯les is not generally \rectangular,"
i.e., there is no set D of weak orders such that the domain of single-peaked pro¯les is just
D £ ¢ ¢ ¢ £ D (n times). To see this, note that
R 01 R02 R03
z
z
z
y
y
y
x x x
is clearly single-peaked with respect to x Á z Á y. But, using R1 and R2 from the previous
example, (R 1; R2 ; R03) is not single-peaked with respect to any weak linear ordering of
fx; y; zg. (Do you see why?)
Note that (R01; R02; R03) is also single-peaked with respect to x Á y Á z, so the weak linear
ordering on X in the de¯nition of single-peakedness is not unique, even up to inversions.
To elaborate on non-rectangularity: single-peakedness is not a property of a weak order
considered in isolation, but rather it is a property of a pro¯le of weak orders taken together.
65
Now that I've emphasized that point, I'll backtrack a bit. In some models, there is a
particular weak linear order of X that suggests itself: when X is a subset of the real line,
for example, we will usually want to consider pro¯les that are single-peaked with respect
to the usual less-than-or-equal-to relation, i.e., ¹ = ·. When the weak linear order ¹ is
¯xed, I may want to consider only preferences single-peaked with respect to ¹. You can
check that the set of pro¯les single-peaked with respect to a given ¹ is, indeed, rectangular.
Thus, single-peakedness with respect to ¹ can be considered a property of weak orders by
themselves.
This observation is borne out in the following proposition, which characterizes the pro¯les
of weak orders single-peaked with respect to the usual · relation on R. Note that the
two-part condition characterizing single-peakedness is stated for each Ri separately.
Proposition 2.1 Let X µ R be convex. The pro¯le ((P1; R1 ); : : : ; (Pn ; Rn )) is singlepeaked with respect to · if and only if, for all i 2 N , (i) (Pi ; Ri ) is strictly convex, and (ii)
Ri has a maximal element, i.e., there exists x
~i 2 X such that, for all y 2 X , we have x
~ iRi y.
Proof: Assume ((P1 ; R1); : : : ; (Pn; Rn)) is single-peaked with respect to ·, and take any
i 2 N. Then i's ideal point is maximal. Take x 2 X and distinct y; z 2 Ri (x). Let
® 2 (0; 1), and de¯ne w = ®y + (1 ¡ ®)z. Assume without loss of generality that y < z. If
w<x
~i, then y < w < ~xi, and then single-peakedness implies wPiy. By transitivity, wPix.
If w > ~xi, then x
~i < w < z, and then single-peakedness implies wPi z. By transitivity, wPix.
i
If w = x
~ , then wPi y, and transitivity implies wPix. Thus, each (Pi ; Ri ) is strictly convex.
I'll leave the other direction to you.
Note that, if X µ R is convex and (Pi ; Ri ) is strictly quasi-concave, then the dual pair has
a utility representation. An implication of Propositions 1.20 and 2.1 is a characterization
of single-peakedness in terms of utility representations.
Corollary 2.1 Let X µ R be convex. The pro¯le ((P1; R1); : : : ; (Pn; Rn)) is single-peaked
with respect to · if and only if, for all i 2 N , (Pi; Ri) has a utility representation ui
satisfying (i) ui is strictly quasi-concave, and (ii) ui has a maximizer.
I follow Austen-Smith and Banks in de¯ning a weakening of the usual single-peakedness
condition. We say a pro¯le ((P1 ; R1); : : : ; (Pn; Rn)) is weakly single-peaked if there exists a
weak linear ordering, say ¹, of X such that, for each i 2 N , there exists ~xi 2 X satisfying
the following conditions:
² for all y 2 X, x
~i Riy;
² for all y; z 2 X, if z Á y Á x
~i , then yRiz;
66
² for all y; z 2 X, if x
~ i Á y Á z, then yRiz.
In this case, we say ((P1 ; R1); : : : ; (Pn; Rn)) is weakly single-peaked with respect to ¹. Let
W = f((P1; R1 ); : : : ; (Pn ; Rn )) j ((P1; R 1); : : : ; (Pn ; Rn )) is weakly single-peaked g:
We refer to the assumption that PR(£) = W as weakly single-peaked domain.
Thus, if we move to the left or right from x
~i , alternatives grow weakly worse for the indvidual. Obviously, weak single-peakedness allows indi®erences that are ruled out by singlepeakedness. As the next variant of Proposition 2.1 shows, it amounts to dropping the
\strict" from strict convexity.
Proposition 2.2 Let X µ R be convex. The pro¯le ((P1; R1); : : : ; (Pn ; Rn )) of weak orders
is weakly single-peaked with respect to · if and only if, for all i 2 N, (i) (Pi; Ri) is convex,
and (ii) Ri has a maximal element, i.e., there exists x
~i 2 X such that, for all y 2 X, we
i
have x
~ Ri y.
For a ¯nal example of an environment with single-peaked preferences, suppose, for example,
that a society must decide on some level of a public good to be provided, assumed to be
a quantity in [0; x]. Individuals also receive utility from \income," which we allow to be
positive or negative, for simplicity. Suppose that i's endowment of money is y i, and that
the cost of x units of public good is C(x) (where C(¢) is strictly convex and continuous) to
be divided equally among the individuals. Thus, an alternative is (x; y1; : : : ; yn ) such that
x ¸ 0 and yi = yi ¡ C(x)=n for each i. Suppose that each individual i has preferences over
(x; y1 ; : : : ; y n) vectors with a utility representation ui satisfying
ui (x; y1; y2; : : : ; yn ) = vi (x) + yi ;
where vi is continuous and strictly concave. Since a public good level determines incomes
uniquely, we can reduce the set of alternatives to X = [0; x]. Then each i has \induced
preferences" on X with utility representation u¤i such that
u¤i (x) = vi (x) ¡
C(x)
+ y i;
n
which is continuous and strictly concave. Therefore, u¤i has a unique maximizer and is
strictly quasi-concave, which implies that each (Pi; Ri) is strictly convex. By Proposition
2.1, therefore, induced preferences are single-peaked with respect to ·.
Example 3: The Multi-dimensional Spatial Model. Now suppose a society must
choose from a set of public policy vectors or from a set of feasible allocations of public
goods. In that case, if there are d policy dimensions or public goods, then we have X µ Rd .
An alternative is x = (x1; : : : ; xd ), where xk is the location of public policy on the kth
67
dimension or the amount of kth public good provided. Usually, we either suppose there
are no restrictions on the set of alternatives, i.e., X = Rd or X = Rd+ , or we suppose
there is some convex, continuous technology for the production of public goods or for the
determination of public policy, implying that X is convex, closed, and possibly compact.
Typically, we assume that X has nonempty interior in Rd, implying that X is in¯nite, in
which case the speci¯cation of X suggests extra structure that we are often willing to impose
in applications.
A classic special case of the multidimensional spatial model has played an important role in
the theoretical political science literature. Given X µ Rd, recall that (Pi ; Ri ) is Euclidean
if i has a unique maximal alternative in X, the individual's \ideal point," denoted x
~i , and,
for all x; y 2 X , xR iy if and only if jjx ¡ x
~i jj · jjy ¡ x
~i jj. In words, Ri is Euclidean if
i's indi®erence curves are concentric circles (more generally, spheres or hyperspheres), with
alternatives closer to i's ideal point being better for i. Let
E = f((P1; R1 ); : : : ; (Pn ; Rn )) j 8i 2 N : (Pi ; Ri ) is Euclidean g:
We refer to the assumption that PR(£) = E as Euclidean domain. Almost all of our results
hold under much more general assumptions about individual preferences.
A minimal condition is continuity of preferences, i.e., each (Pi; Ri) is continuous: for every
¡1 (x) and P (x) are open.
x 2 X, the sets Ri(x) and R¡1
i (x) are closed, or equivalently, P
This assumption, which formalizes the idea that alternatives that are very close to each other
should seem similar to the individual, is a fundamental one in the analysis of in¯nite models.
Note that, if X is assumed to be only closed or compact, then X could be ¯nite. In that
case, individual preferences are automatically continuous. Indeed, many of the results below
assume only compactness of X and continuity of individual preferences, and they therefore
apply to all ¯nite models: given any model with a ¯nite number of alternatives, x1; : : : ; xk ,
and arbitrary individual preferences over these alternatives, we can imbed x1 ; : : : ; xk in Rd
to get a ¯nite set X µ Rd and de¯ne preferences over X as originally speci¯ed | this gives
us a special case of the multi-dimensional spatial model with a compact set of alternatives
and continuous individual preferences!
Another, often reasonable, restriction is convexity of preferences, i.e., each (Pi; Ri) is convex:
for all x 2 X, Ri(x) is convex. This is a formalization of the idea that, if we start from one
alternative and move in the direction of a preferred alternative, then that change will be an
improvement. This assumption does not in itself rule out the possibility that an alternative,
x, contains private good components, e.g., we might have x = (x1; : : : ; xn ; xn+1 ), where,
for i = 1; : : : ; n, xi is an amount of a private good consumed by individual i, and xn+1 is
the amount of a public good consumed by all. Thus, these assumptions are quite general.
A strengthening of convexity is that, for each individual i, (Pi; Ri) is strictly convex: for all
x 2 X, all y; z 2 Ri (x) with y 6
= z, and all ® 2 (0; 1), we have ®y + (1 ¡ ®)z 2 Pi(x). This
assumption rules out the above-mentioned possibility of private good components of x. In
68
the above example, individual i might be indi®erent between alternatives y and z, where y
and z di®er only in the private good consumption levels of individual j 6
= i, but then strict
convexity would require w = (1=2)y + (1=2)zPiy. The problem is that w also di®ers from y
only in j's private good consumption, so i should be indi®erent between all three bundles.
We say the pro¯le ((P1; R1); : : : ; (Pn ; Rn )) satis¯es limited shared weak preferences (LSWP)
if, for all x 2 X and all G µ N,
\
\
\
j
Ri (x)j ¸ 2 )
Ri(x) µ clos
Pi (x):
i2G
i2G
i2G
In other words, if all members of a group weakly prefer some alternative y 6
= x to x, then
there are alternatives arbitrarily close to y that all members of a group strictly prefer to x.
Note that LSWP rules out \locally maximal" elements of (Pi ; Ri ), unless they are global, and
in that case there can be at most one. The condition is clearly implied by strict convexity.
(Why?) To see that it is strictly weaker than strict convexity, note that it is satis¯ed
whenever linear domain holds, as it can in the discrete model. Unlike strict convexity,
LSWP is consistent with the existence of private goods and is, in fact, satis¯ed in a very
large class of such models. We will see that many of our results for the multidimensional
model may use this weak assumption rather than strict convexity, and so they actually
carry over to the discrete choice model.
I have suggested two interpretations of the multidimensional spatial model. One is that
dimensions represent di®erent aspects of public policy: for example, some dimensions might
indicate public policy on economic issues, such as the level of central bank independence
or the restrictiveness of regulations, while others indicate how liberal or conservative public
policy is on various social issues, such as freedom of speech, abortion rights, etc. In this
case, we typically assume that individual preferences are continuous and convex (possibly
strictly), and that each individual has a unique maximal alternative, an \ideal point,"
denoted x
~i , re°ecting that individual's ideal combination of positions on all policy issues.
The other is that dimensions represent levels of di®erent public goods. In this case, a
common type of preference restriction is that each Ri is monotonic, i.e., for all x; y 2 X,
if x À y (so x is greater than y in every component), then xPi y. This, of course, merely
formalizes the idea that public goods are indeed \goods," or at least not bads, because more
of all of them is unambiguously good for the individuals.
Something for you to think about: monotonicity seems like a reasonable assumption, but
in the one-dimensional public good problem with single-peaked preferences, above, it is
violated. How do you reconcile these observations?
A strengthening of monotonicity is the restriction that, for each individual i, (Pi ; Ri ) is
strictly monotonic, i.e., for all x; y 2 X, if x > y (so x ¸ y, greater in at least one
component), then xPiy. Like strict convexity, this restriction also rules out private good
components. (Do you see why?)
69
There is actually a close connection between these two interpretations. Suppose that the
group has a ¯xed amount m of money to purchase ` public goods at ¯xed, positive prices
p. All vectors in R`+ of provision levels are conceivable, and each individual has continuous,
strictly convex, strictly monotonic preferences over R`+. Because of the budget constraint,
however, the set of alternatives is the simplex X = fx 2 R`+ j p ¢ x · mg in R`+ . Indeed, by
monotonicity, all vectors x such that p ¢ x < m are Pareto dominated, so it is customary to
treat the set of alternatives as the outer face of this simplex, where the budget is exhausted:
X = fx 2 R`+ j p¢ x = mg in R`+. Individual preferences will be given by indi®erence curves
on this face: they are continuous and strictly convex, as before. An important feature of
preferences induced on this simplex is that each individual will have an ideal point ~xi, a
vector of provision levels strictly preferred to every other. Thus, we are back to the model
of public policy described above.
Example 4: Private Good Exchange Economies. A group N of individuals is endowed with amounts of k resources that must be allocated. Let w 2 Rk+ denote the social
endowment. Thus, w = (w1; : : : ; wk ), where wj is the total endowment of the jth resource.
In some examples of private good allocation environments, individual property rights over
these resources may be de¯ned, in which case wj is the total amount of privately and publicly owned units of the jth resource. An alternative x 2 Rnk
+ is an allocation of resources to
the individuals, i.e., x = (x1; : : : ; xn ), where xi 2 Rk+ isPa vector indicating i's consumption
n
i
levels. The set of alternatives is then X = fx 2 Rnk
+ j
i=1 x = wg.
Because the set of alternatives is a subset of multidimensional Euclidean space, private
good exchange economies are related to the spatial model. In fact, the familiar restriction
of continuous preferences is always imposed, and convexity and monotonicity are typically
assumed. Because the set of feasible allocations in an exchange economy has empty interior,
however, some of our results for the multidimensional spatial model | even though they
are very general with respect to preferences | do not specialize to private good exchange
economies.
A fundamental preference restriction, not seen in the spatial model, is that an individual
i's preferences depend only on i's consumption: for all x; y 2 X, if xi = y i, then xIiy.
This restriction, which can only be formulated when some form of private consumption is
assumed, is always assumed in private good exchange environments. It leads to a convexity
condition between convexity and strict convexity. We say (Pi ; Ri ) is strictly convex in own
consumption if, for all x 2 X, all y; z 2 R(x) with y i 6
= z i, and for all ® 2 (0; 1), we have
w = ®y +(1¡®)zPi x. The crucial assumption here is, of course, that y and z give i distinct
consumption bundles, so that the mixture w allows i a non-trivial improvement.
Exchange environments also suggest a monotonicity assumption between monotonicity and
strict monotonicity. We say (Pi ; Ri ) is strictly monotonic in own consumption if, for all
x; y 2 X, xi > y i implies xPi y.
70
Perhaps the simplest example of a private good exchange economy is the situation where n
individuals must allocate a single ¯xed resource, often called a \dollar," a \cake," or a \pie."
Thus, the set of alternatives is just the outer face of a simplex in Rn . Letting w
0 denote
P>
n
n
the total amount available, an alternative is x = (x1; : : : ; xn ) 2 R+ , where i=1 xi = w
and xi denotes i's non-negative consumption of the resource. Here, monotonicity and the
assumption of no externalities immediately imply the following restriction on preferences:
xPi y if and only if xi > yi . I will adhere to the least culinary of the standard terminologies
by referring to such a problem as a divide the dollar environment.
2.3
Preference Aggregation Rules
An Assortment of PARs
There are numerous ways of aggregating preferences of individuals. Here are some examples,
de¯ned, for simplicity, in terms of strict preference.
simple majority, FSM
xPSM (µ)y if and only if p(x; yjµ) > n=2:
relative majority, FRM
xPRM (µ)y if and only if p(x; yjµ) > p(y; xjµ):
simple Pareto, FSP
xPSP (µ)y if and only if p(x; yjµ) = n:
relative Pareto, FRP
xPRP (µ)y if and only if r(x; yjµ) = n and p(x; yjµ) > 0:
Here I have de¯ned strict social preference. Note: In doing so, I must be careful to ensure
that PF (µ) is asymmetric in every state. (You can check that this holds.)
Weak social preference is obtained in the usual way. For example, xRSM (µ)y if and only
if not yPSM (µ)x, which holds if and only if not p(y; xjµ) > n=2, which is equivalent to
r(x; y) ¸ n=2. Weak social preferences for other PARs are derived similarly | I leave that
to you. Note that, as long as PF (µ), de¯ned above, is asymmetric, weak social preference
will be complete, as required.
See Figure 2.2 for an examples of each of these PARs when X is ¯nite.
71
Figure 2.2: Social preferences, ¯nite X
Note the nestings: PSM (µ) µ PRM (µ) and PSP (µ) µ PRP (µ). Under linear domain,
PSM (µ) = PRM (µ) and PSP (µ) = PRP (µ).
What do social preferences corresponding to these PARs look like when X is in¯nite? See
Figure 2.3 for upper sections of strict and weak social preference for the simple majority
and simple Pareto PARs.
Figure 2.3: Social preferences, in¯nite X
The following class of PARs generalizes FSM and FSP by establishing a numerical quota, q,
needed for a strict social preference. Let q be an integer, and, in order to ensure asymmetry
of strict social preference, assume q > n=2.
q-rule, Fq
xPq (µ)y if and only if p(x; yjµ) ¸ q:
n
Simple majority is the case q = n+1
2 when n is odd, or q = 2 + 1 when n is even. Generally,
n+1
simple majority is q = d 2 e, where d¢e denotes \smallest integer greater than or equal to."
And simple Pareto is the case q = n.
The next rule generalizes simple Pareto by designating an arbitrary group G µ N that can
impose a common strict preference among its members on society. Assume G 6
= ;.
G-rule, FG
xPG (µ)y if and only if G µ P(x; yjµ);
\
i.e., PG(µ) =
Pi (µ):
i2G
Equivalently, given G 6
= ;,
RG(µ) =
[
R i(µ)
i2G
for all µ 2 £. That is, x is weakly preferred to y under G-rule if some member of G weakly
prefers x to y.
72
Simple Pareto is the special case where G = N. Note that the classes of G-rules and q-rules
are non-nested: there are G-rules that are not q-rules (e.g., whenever G is a proper subset
of N ), and there q-rules that are not G-rules (e.g., simple majority).
We write Fi for the special case of G-rule with G = fig for some i 2 N. In this case, i is a
\dictator," for the social preference relation is simply equal to i's.
A class of PARs that generalize many of the ones previously de¯ned is the
P \weighted" qrules, where each individual i is given a weight wi ¸ 0 and q satis¯es q > ( i2N wi )=2. We
write w = (w1 ; : : : ; wn) for a vector of such weights.
weighted q-rule, Fw;q
xPw;q (µ)y if and only if
X
i2P (x;yjµ)
wi ¸ q:
Simple majority rule, for example, is obtained by setting wi = 1 for each i 2 N and
q = d n+1
2 e. We obtain G-rule by setting wi = 1 for each i 2 G, wi = 0 otherwise, and
q = jGj. To obtain \weighted majority rule," where a group can impose its preferences on
society if and only if it has more than half of the total weight, de¯ne
X
X
X
q = minf
wi j G µ N;
wi > (
wi)=2g:
i2G
i2G
i2N
P
In fact, any q strictly between this number and ( i2N wi)=2 would work as well.
Here is a PAR de¯ned using the idea of transitive closure of weak preference, which we saw
in the analysis of the weak top cycle. At this point, we de¯ne the transitive closure PAR
using simple majority preference. Later, we will de¯ne the weak top cycle derived from
general PARs. Note that it is now much more convenient to de¯ne social preferences in
terms of weak preference.
simple majority transitive closure, FT
xRT (µ)y if and only if xTRSM (µ) y:
Thus, x is weakly socially preferred to y here if we can get from x to y in a ¯nite number
of weak simple majority preference steps.
Another PAR is de¯ned using the idea of covering from our analysis of the weak uncovered
set. At this point, we de¯ne covering using simple majority preferences. Later, we will
de¯ne the weak uncovered set derived from general PARs.
73
simple majority covering, FC
xPC (µ)y if and only if RSM (xjµ) µ PSM (yjµ):
Thus, PC (µ) is just CRSM (µ) . It follows that xRC (µ)y if and only if there exists z 2 X such
that xRSM (µ)zRSM (µ)y. When xPC (µ)y, we say that x \covers" y.
We could also de¯ne the covering PAR that generates the strong uncovered set, i.e., xPSM (µ)y
and PSM (xjµ) µ RSM (yjµ), but we'll just stick with the above.
A yet di®erent type of PAR is de¯ned for ¯nite X . It was invented by Jean-Charles de
Borda and bears his name. First, let
Bxi(µ) = jfy 2 X j xPi (µ)ygj ¡ jfy 2 X j yPi (µ)xgj
n
X
Bx(µ) =
Bix(µ):
i=1
In words, Bix(µ) registers the net number of strict preferences for x over other alternatives
in i's preference ordering, and B x(µ) sums these across individuals.
Borda rule, FB
xPB (µ)y if and only if B x(µ) > B y (µ):3
One interpretation of Borda rule is that it uses the number of alternatives between, say,
x and y in an individual's ranking to measure \intensity" of preference. That is okay,
if that is how you de¯ne intensity. Since we de¯ne preferences as simply collections of
binary comparisons, however, there is no guarantee that this measure will correspond to
the ordinary language meaning.
Decisive and Blocking Coalitions
Given a PAR F , we say G is decisive if, for all distinct x; y 2 X and all µ 2 £,
G µ P(x; yjµ) ) xPF (µ)y:
When the members of G strictly prefer any x to any y, then x is strictly socially preferred
to y. Let D(F) denote the collection of decisive coalitions of F.
Note that we must have G \ G0 6
= ; for all G; G0 2 D(F ), for we assume the existence of a
0
free pair: if G; G 2 D(F ) are disjoint, then there exists µ 2 £ such that G µ P (x; yjµ) and
G0 µ P (y; xjµ), but then xPF (µ)y and yPF (µ)x, contradicting asymmetry.
We say G is blocking if, for all distinct x; y 2 X and all µ 2 £,
G µ P(x; yjµ) ) xRF (µ)y:
74
Thus, a blocking coalition has less power than a decisive coalition, in that it can enforce
only a weak social preference. Let B(F) denote the collection of blocking coalitions.
Clearly, D(F ) µ B(F). We next develop some slightly deeper (but not much!) results.
Let G denote a collection of groups. We say G is : : :
² proper if, for all G; G0 2 G, we have G \ G0 6
= ;,
² strong if, for all G µ N , there exists G0 2 G such that G0 µ G or G0 µ N n G,
² monotonic if, for all G 2 G and all G0 µ N, G µ G0 implies G0 2 G.
Note that, for proper G, we must have G 6
= ; for all G 2 G. We do not rule out the
possibility, however, that G = ;. Note also that, if G is monotonic, then the following
condition is equivalent to G being strong: for all G µ N, G 2
= G implies N n G 2 G.
We say F is strong if D(F) is strong.
Proposition 2.3
1. D(F ) and B(F) are monotonic.
2. For all G 2 D(F) and all G0 2 B(F ), G \ G0 6
= ;.
3. F is strong if and only if, for all x; y 2 X and all µ 2 £, xRF (µ)y implies R(x; yjµ) 2
D(F ).
4. If F is strong, then D(F ) = B(F ).
Proof: To prove part 2, let G 2 D(F ) and G0 2 B(F), and suppose G \ G0 = ;. Let
fx; yg be a free pair, and take any µ 2 £ such that G µ P(x; yjµ) and G0 µ P (y; xjµ).
Then xPF (µ)y and yRF (µ)x, a contradiction. To prove part 3, assume F is strong. Pick
x; y 2 X and µ 2 £ such that xRF (µ)y. Then P(y; xjµ) 2
= D(F ). Since D(F ) is strong,
R(x; yjµ) 2 D(F ), proving one direction. Now assume F is not strong, so there exists
G µ N such that G 2
= D(F ) and N n G 2
= D(F). Let fx; yg be any free pair, and let
µ 2 £ satisfy P(x; yjµ) = G and P (y; xjµ) = N n G. By completeness, either xRF (µ)y and
R(x; yjµ) 2
= D(F ), or yRF (µ)x and R(y; xjµ) 2
= D(F), proving the converse. For part 4,
take any G 2 B(F). Let fx; yg be a free pair, and let µ 2 £ satisfy P(x; yjµ) = G and
P(y; xjµ) = N n G. Then xRF (µ)y, which, by part 3, implies G 2 D(F).
Note that part 2 of Proposition 2.3 is actually stronger than the condition that D(F) is
proper. (Why?) Is the converse of part 4 true?
You should try to ¯gure out the decisive and blocking coalitions for the PARs de¯ned above.
(Borda rule and the weak covering PAR are a little tricky : : : )
75
The condition that D(F ) is strong is fairly restrictive. The prototypical example of such a
PAR is simple majority rule with n odd. You can check, however, that
P weighted
Pmajority
rule
is
strong
if
and
only
if
there
is
no
partition
I;
J
of
N
such
that
w
=
i2I i
i2J wi =
P
( i2N wi)=2. In one sense, this is true for \most" assignments of weights.
For technical reasons, we de¯ne weaker notions of decisive and blocking coalitions. We say
G is semi-decisive for x over y if, for all µ 2 £,
[G = P(x; yjµ) and N n G = P (y; xjµ)] ) xPF (µ)y:
The group G is simply semi-decisive if it semi-decisive for every x over every y. Thus,
a semi-decisive group is weaker than a decisive one in that it can enforce a strict social
preference only when opposed by all non-members.
We say G is semi-blocking for x over y if, for all µ 2 £,
[G = P (x; yjµ) and N n G = P(y; xjµ)] ) xR F (µ)y:
The group G is simply semi-blocking if it semi-blocking for every x over every y.
Clearly, if a group is semi-decisive for some x over some y, then it is also semi-blocking for
x over y.
Conditions on PARs
Next, I de¯ne several conditions on PARs.
Pareto For all µ 2 £, PN (µ) µ PF (µ)
In words, if every individual prefers some x to some y, then x is strictly socially preferred
to y. Equivalently, N is decisive.
weak Pareto For all µ 2 £, PN (µ) µ RF (µ)
In words, if every individual prefers some x to some y, then x is socially weakly preferred
to y. Equivalently, N is blocking.
independence of irrelevant alternatives (IIA) For all x; y 2 X and all µ; µ0 2 £,
[P (x; yjµ) = P (x; yjµ0 ) and R(x; yjµ) = R(x; yjµ 0) and xPF (µ)y] ) xPF (µ0 )y:
76
An equivalent de¯nition is this: for all x; y 2 X and all µ; µ0 2 £,
[P(x; yjµ) = P (x; yjµ0 ) and R(x; yjµ) = R(x; yjµ 0) and xRF (µ)y] ) xRF (µ0 )y:
Can you see why?
For yet another equivalent de¯nition, we might say F satis¯es IIA if, for all x; y 2 X and
all µ; µ0 2 £,
[PR(µ)jfx;yg = PR(µ0 )jfx;yg] ) [RF (µ)jfx;yg = RF (µ 0)jfx;yg];
where Ri jY = Ri \(Y £Y ) is the \restriction" of Ri to Y , and (R1 ; : : : ; Rn )jY = (R1 jY ; : : : ;
RnjY ).
Thus, if individual preferences between two alternatives, x and y, are the same in two states,
then the social preference between them should be the same as well. That is, individual
preferences over other alternatives are \irrelevant" in the determination of social preferences
over x and y.
The motivation for this condition is straightforward if we view a social preference between
alternatives x and y as indicating the social choice from the pair fx; yg. In that case, if
literally no other alternatives are feasible, then it is plausible that they would not a®ect the
process through which social decisions are ultimately determined. That would be the case,
for example, if those social decisions are the result of a non-cooperative game played by the
individuals. From this perspective, IIA is an \implementation" constraint.
The next lemma is pretty trivial (right?), but we will use it in our analysis of collective
rationality.
Lemma 2.1 Assume F satis¯es IIA. Let fx; yg be any free pair of alternatives, and let
G µ N be any group. Either G is semi-blocking for x over y, or N n G is semi-decisive for
y over x.
The next three conditions are as de¯ned in Austen-Smith and Banks.
neutrality For all x; y; w; z 2 X and all µ; µ0 2 £,
[P(x; yjµ) = P(w; zjµ0 ) and R(x; yjµ) = R(w; zjµ 0) and xPF (µ)y] ) wPF (µ0 )z:
This condition strengthens IIA and formalizes the idea that no alternatives are \special."
If individual preferences over x and y in one state are the same as individual preferences
over w and z in another, social preferences should be too (with w playing the same role as
x, z playing the same role as y).
77
monotonicity For all x; y 2 X and all µ; µ 0 2 £,
[P (x; yjµ) µ P (x; yjµ0 ) and R(x; yjµ) µ R(x; yjµ 0) and xPF (µ)y] ) xPF (µ0 )y:
This condition strengthens IIA by adding the idea that increased support for some x over
some y should preserve a social preference for x over y.
anonymity For all x; y 2 X and all µ; µ0 2 £
[p(x; yjµ) = p(x; yjµ0 ) and r(x; yjµ) = r(x; yjµ0 ) and xPF (µ)y] ) xPF (µ0 )y:
This condition formalizes the idea that no individuals are special: the social preference
between x and y depends only on the numbers of individuals who prefer (strictly and
weakly) x over y, and not on their identities. Note that it also strengthens IIA.
In all of the above conditions save Pareto, a social preference in one state, µ, may have
implications for social preferences in other states. For that reason, they are referred to as
\inter-pro¯le" (or in our framework, \interstate") conditions.
All of the above PARs satisfy Pareto. All but FT , FC , and FB satisfy IIA (Can you see
why?) All but FT , FC , and FB satisfy neutrality and monotonicity. Most (which?) are
anonymous.
We will use three conditions that di®er from neutrality, monotonicity, and anonymity in
that they are expressed in terms of the decisive coalitions of a PAR. We have already de¯ned
an aggregation rule F to be strong if D(F) is strong. We now de¯ne two more conditions.
decisiveness For all x; y 2 X and all µ 2 £,
xPF (µ)y ) P(x; yjµ) 2 D(F ):
In words, a strict social preference is only possible if the members of a decisive coalition
all prefer one alternative to another. This is satis¯ed by FSM , FSP , Fq , FG, and Fw;q , for
example, but not the other PARs we've seen.
weak decisiveness For all x; y 2 X and all µ 2 £,
xPF (µ)y ) R(x; yjµ) 2 D(F ):
This condition is clearly weaker than decisiveness. Note that, under unrestricted or linear
domain, it is implied by the conjunction of neutrality and monotonicity. (Why?) See
Austen-Smith and Banks for a large class of PARs, called \voting rules," that satisfy weak
decisiveness but not decisiveness.
You can check that, under linear domain, F strong implies F is decisive. Note that, by part
3 of Proposition 2.3, F strong implies F is weakly decisive.
78
2.4
Simple PARs
A PAR F is simple if there exists a proper collection G of groups such that
xPF (µ)y if and only if 9G 2 G : xPG (µ)y;
or equivalently,
PF (µ) =
[ \
Pi(µ);
G2G i2G
for all µ 2 £. In words, x is strictly socially preferred to y if and only if all the members
of some group in G prefer x to y. In this case, we denote the PAR by FG , and we call G a
representation of the PAR.
Equivalently, by DeMorgan's law, F is simple if there exists a proper G such that
\ [
RF (µ) =
R i(µ)
G2G i2G
for all µ 2 £. That is, x is weakly socially preferred to y if and only if, in every G 2 G,
there is some member of G who weakly prefers x to y.
Let's check that FG is well-de¯ned, i.e., that PG (µ) is asymmetric and RG (µ) is complete.
Suppose xPG (µ)y, so there exists G 2 G such that, for all i 2 G, xPi(µ)y. Suppose yPG (µ)x
as well, so there exists G0 2 G such that, for all i 2 G0 , yPi(µ)x. Since G is proper, there
exists i 2 G \ G0 , but then xPi(µ)y and yPi(µ)x, contradicting asymmetry of Pi(µ). Thus,
PG (µ) is asymmetric and completeness of R G (µ) follows.
Associated with every PAR F is a related simple PAR FG , de¯ned by G = D(F ). Thus,
xPD(F ) (µ)y if and only if the members of some group that is decisive for F all prefer x to
y. So, for relative majority rule FRM , we have FD(FRM ) = FSM , and for relative Pareto,
we have FD(FRP ) = FSP . For simplicity, we write CD(F ) (µ) for the core of FD(F) at µ. Note
that, for all PARs F and all µ 2 £, we must have CF (µ) µ CD(F ) (µ). (Right?)
Characterization of Simple PARs
Note that simple PARs satisfy IIA, neutrality, and monotonicity. A simple PAR satis¯es
Pareto if and only if it has a nonempty representation G. You can check that a simple PAR
satis¯es anonymity if and only if it is a q-rule for some q.
The same PAR may have several representations. For example, q-rule may be obtained as
a special case by de¯ning
G = fG µ N j jGj ¸ qg
or
79
G = fG µ N j jGj = qg:
Every representation G determines a unique maximal representation, G", de¯ned as follows:
G" = fG µ N j 9G0 2 G such that G0 µ Gg:
You can check that FG = FG" . Also, note that G is monotonic if and only if G = G".
The PAR q-rule is the simple PAR with G"= fG µ N j jGj ¸ qg; G-rule is the simple PAR
P
with G"= fG0 µ N j G µ G0 g; and Fw;q is the simple PAR with G"= fG µ N j i2G wi ¸
qg.
Similarly, every representation G determines a unique minimal representation, G#, de¯ned
as follows: G 2 G# if and only if G 2 G and there is no proper subset G0 $ G such that
G0 2 G. Again, FG = FG#.
Given a collection G, let
G ¤ = fG µ N j 8G0 2 G : G \ G0 6
= ;g
be the collection of all groups having nonempty intersection with all groups in G. Thus, if
N = f1; 2; 3; 4g, then we might have the following.
G
123
124
134
234
1234
G¤
12; 13; 14
23; 24; 34
123; 124
134; 234
1234
G ¤¤
123
124
134
234
1234
The next lemma shows that the above duality holds quite generally. In fact, it holds
whenever G = G".
Lemma 2.2 Let G be a collection of groups. Then
1. G ¤¤ = G",
2. G ¤ = (G")¤ = (G ¤)",
3. G is proper if and only if G¤ is strong,
4. G is strong if and only if G ¤ is proper,
5. if fBi j i 2 Ng is a collection of binary relations on X, then
\ [
[ \
[ \
\ [
Bi =
Bi and
Bi =
Bi:
G2G i2G
G2G¤ i2G
G2G i2G
80
G2G ¤ i2G
Proof: I'll leave the proof of part 1 to you. Note the consequence that (G¤¤)¤ = (G")¤ .
Now apply part 1 to G ¤ to get (G¤ )¤¤ = (G ¤)". Therefore,
(G")¤ = (G ¤)";
and it is clear that
(G")¤ µ G¤ µ (G ¤)";
which proves part 2. For part 3, suppose G is proper. If G¤ is not strong, then there
exists G µ N such that G0 2 G ¤ for no G0 µ G and no G0 µ N n G. In particular,
G2
= G¤ and N n G 2
= G ¤. The former implies there exists G0 2 G such that G0 µ N n G,
and the latter implies there exists G00 2 G such that G00 µ G. But then G0 \ G00 = ;,
a contradiction. Therefore, G ¤ is strong. I leave the other direction to you. For part 4,
apply part 3 to G¤ . Then G¤ is proper if and only if G ¤¤ = G" is strong, which holds if
and only if G is strong. Finally, I'll prove one direction of the ¯rst half of part 5. Take
any (x; y) 2 X £ X, and suppose that, for every G 2 G, there is some i 2 G such that
xBi y. Letting G = fi 2 N j xBiyg, we have G \ G0 6
= ; for every G0 2 G, so G 2 G ¤, as
required.
The following corollary, which gives us alternative formulations of strict and weak social
preferences, is immediate.
Corollary 2.2 Let F be a simple PAR with representation G. Then
\ [
[ \
PF (µ) =
Pi (µ) and RF (µ) =
Ri (µ);
G2G¤ i2G
G2G¤ i2G
for all µ 2 £.
The connection between decisive and blocking coalitions is quite tight for simple PARs.
Recall part 2 of Proposition 2.3, which essentially says
D(F) µ B(F)¤ and B(F) µ D(F)¤:
The next proposition strengthens this conclusion by establishing a duality between decisive
and blocking coalitions.
Proposition 2.4 Let F be a simple PAR with representation G. Then
1. D(F ) = G" = B(F)¤,
2. B(F) = G¤ = D(F)¤,
81
Proof: I ¯rst consider D(F) = G ". One inclusion, that G "µ D(F), is clear. For the
opposite inclusion, take G µ N and suppose there is no G0 2 G such that G0 µ G. Letting
fx; yg be a free pair, take any state µ 2 £ such that P(x; yjµ) = G and P (y; xjµ) = N n G.
Then yRF (µ)x (why?), and it follows that G 2
= D(F). Therefore, D(F) = G". I'll let you
prove that B(F ) = G¤. Now note that
D(F) = G" = G"¤¤ = (G"¤ )¤ = (G ¤)¤ = B(F )¤ ;
where we use parts 1 and 2 of Lemma 2.2. Finally, B(F ) = D(F)¤ follows from D(F) =
B(F )¤ , from part 1 of Lemma 2.2, and from monotonicity of B(F).
An immediate consequence of Lemma 2.2 and Proposition 2.4 is that B(F) is strong. Furthermore, the complement of any not-decisive group is blocking, and the complement of
any not-blocking group is decisive. Thus, for every G, either G 2 D(F) or N n G 2 B(F).
I'll let you prove the former claim, stated in the following corollary.
Corollary 2.3 Let F be a simple PAR. Then
1. B(F) = fG j N n G 2
= D(F)g,
2. D(F ) = fG j N n G 2
= B(F)g.
We can now de¯ne strict and weak social preference in terms of decisive and blocking
coalitions. Note that, in contrast to the general de¯nition of blocking coalition, for a simple
PAR, a common weak preference among the members of some blocking group implies a
social weak preference. Thus, for simple PARs, blocking groups have considerably more
power than in general.
Proposition 2.5 Let F be a simple PAR. Then
[ \
PF (µ) =
Pi(µ) and RF (µ) =
G2D(F ) i2G
[
\
Ri (µ);
G2B(F ) i2G
for all µ 2 £.
Proof: Let G be a representation of F. Because D(F) = G", we have
[ \
[ \
PF (µ) =
Pi(µ) =
Pi(µ):
G2G" i2G
G2D(F ) i2G
And, by Corollary 2.2 and Proposition 2.4, we have
[ \
[
RF (µ) =
Ri (µ) =
G2G¤ i2G
\
G2B(F ) i2G
as required.
82
Ri(µ);
I'll let you prove that a PAR is simple if and only if it satis¯es decisiveness. Thus, every
simple PAR is weakly decisive.
Continuity of Simple PARs
Though not an issue in ¯nite choice environments, when X is in¯nite we have seen that
existence of maximal elements depends crucially on continuity properties of a preference
relation. Thus, assuming continuous domain, we would like to know conditions on PARs
that ensure continuity of social preferences.
It is straightforward to verify that, assuming each Ri(µ) is continuous, the majority weak
preference relation RSM (µ) will also be continuous. The same holds true for simple Pareto.
The next proposition shows that all simple preference aggregation rules preserve continuity
properties of individual preferences.
Proposition 2.6 Assume X µ Rd . Let F be a simple PAR, and let µ 2 £ be any state.
1. If each (Pi(µ); Ri (µ)) is upper semicontinuous, then (PF (µ); RF (µ)) is upper semicontinuous.
2. If each (Pi(µ); R i(µ)) is lower semicontinuous, then (PF (µ); RF (µ)) is lower semicontinuous.
3. Assume X is connected. If each (Pi (µ); Ri(µ)) is continuous, then RF (µ) is closed and
PF (µ) is open.
Proof: I'll prove part 1. Suppose that, for all i 2 N and all x 2 X, Ri (xjµ) = fy 2 X j
yRi (µ)xg is closed. By Proposition 2.5, we have
[ \
RF (xjµ) =
Ri (xjµ):
G2B(F ) i2G
Since intersections of closed sets are closed, and since ¯nite unions of closed sets are closed,
it follows that RF (xjµ) is closed. Thus, RF (µ) is upper semicontinuous. The proof of part
2 is similar, and part 3 follows because each Ri(µ) is closed.
It is clear that continuity of social preferences does not carry over to weakly decisive PARs
generally, for FRM and FRP can easily generate discontinuities. See Figure 2.4.
Figure 2.4: Discontinuous relative majority preferences
83
The following corollaries are an immediate consequence of Proposition 2.6 and our results
on nonemptiness of choice sets. The ¯rst shows that, under weak background assumptions,
nonemptiness of the core of a simple PAR follows under the weak transitivity property of
Condition F or, therefore, under P -acyclicity.
Corollary 2.4 Assume X µ Rd is compact. Let F be a simple PAR, and let µ 2 £ be
any state such that each (Pi (µ); Ri(µ)) is upper semicontinuous. If (PF (µ); R F (µ)) satis¯es
Condition F, then CF (µ) 6
= ;.
A di®erent continuity issue relates to the properties of the mapping RF : £ ! Rc. We
have given the space of complete, closed preference relations a notion of convergence, and
we can de¯ne a notion on £ in a straightforward manner. Given two states µ; µ0 2 £ with
PR(µ); P R(µ 0) 2 RN
c , de¯ne the distance between the states as
X
½(µ; µ 0) =
½(Ri(µ); Ri(µ0 ));
i2N
where we abuse notation slightly by using ½ for this metric. That is, we simply sum the
distance between an individual's preferences in the two states.
A set £0 µ £ is open if, for every µ 2 £0 , there exists ² > 0 such that the open ball
B²(µ) = fµ0 2 £ j ½(µ; µ0 ) < ²g
is contained in £0. A set £0 µ £ is closed if its complement is open.
We say a sequence fµ mg converges to µ if ½(µm ; µ) ! 0. Equivalently, µm ! µ if, for all
i 2 N, ½(Ri (µ m); Ri (µ)) ! 0. Clearly, £0 µ £ is closed if, for every sequence fµmg in £0
converging to some µ 2 £, we have µ 2 £0 .
Assuming PR(£) µ RN
c and RF (£) µ Rc, we say the mapping RF is continuous at µ if, for
every sequence fµmg such that µm ! µ, we have RF (µ m) ! RF (µ). The next proposition
establishes fairly general conditions under which continuity holds for all simple PARs.
Proposition 2.7 Assume X µ Rd is compact, P R(£) µ RN
c , and RF (£) µ Rc. Let F be
a simple PAR, and let µ 2 £ be such that P R(µ) satis¯es LSWP. Then RF : £ ! Rc is
continuous at µ.
Proof: Let µm ! µ. For each m, let Rm = RF (µ m), and let R = RF (µ). To prove that
fRm g converges to R, ¯rst take any sequence f(xm ; y m)g converging to (x; y) such that
xm Rm ym for all m. For each m, there exists Gm 2 B(F ) such that xm Ri (µm )ym for all
i 2 Gm . Because N is ¯nite, there exists G 2 G such that G = Gm for in¯nitely many
84
m, and then xRi (µ)y for all i 2 G follows from Lemma 1.3. Thus, xRy. Now take any
x; y 2 X with xRy, so there exists G 2 B(F) such that xRi(µ)y for all i 2 G. If x = y, then,
by re°exivity, xRm y for all m. Suppose x 6
= y. Then, by LSWP, for each k there exists
xk 2 PG(yjµ) such that jjxk ¡ xjj < 1=k. (Why?) Furthermore, there exists mk > k such
that xk 2 PG (yjµmk ), for otherwise we would have yRi(µm )xk for some i 2 G and in¯nitely
many m, implying yRi (µ)xk (why?), a contradiction. Thus, we have a sequence f(xk ; y)g
and a subsequence fRmk g such that (xk ; y) ! (x; y) and xk Rmk y for all k. Therefore,
Rm ! R, by Lemma 1.4.
Transitivity of Simple PARs
Another property important in establishing the existence of maximal elements is transitivity.
The next results set the stage for our general analysis of collective rationality, showing that,
when the domain of individual preferences is large, transitivity of strict and weak social
preference corresponding to a simple PAR is very restrictive.
Proposition 2.8 Assume jXj ¸ 3 and PR(£) ¶ L. Let F be a simple PAR. Then F
satis¯es Pareto and PF (µ) is transitive for all µ 2 £ if and only if F = FG for some
nonempty G µ N.
Proof: I will prove one direction. Assume F satis¯es Pareto and PF (µ) is transitive for all
µ 2 £. Let G be a representation of F, nonempty by Pareto, and let G be a minimal group
in G. Suppose there exists G0 2 G such that G nG0 6
= ;. Take distinct alternatives x, y, and
z and any state µ 2 £ with preferences over these alternatives as follows.
RG\G0 (µ)
x
y
z
RGnG0 (µ)
y
z
x
RNnG(µ)
z
x
y
Because xPG0 (µ)y and yPG (µ)z, we have xPF (µ)yPF (µ)z. Note, however, that P(x; zjµ) µ
G \ G0 $ G. By minimality of G, therefore, P(x; zjµ) 2
= G, so not xPF (µ)z, contradicting
transitivity of PF (µ). Therefore, G µ G0 for all G0 2 G, so F = FG. That G 6
= ; follows
because G is proper.
Examples of PARs satisfying the assumptions of Proposition ?? are FSP and FRP , in which
case G = N . Note, as illustrated in Figure 2.2, that these PARs do not always generate
transitive weak social preference. As the next proposition shows, adding that condition
yields the existence of a \dictator."
85
Proposition 2.9 Assume jXj ¸ 3 and PR(£) ¶ L. Let F be a simple PAR. Then F
satis¯es Pareto and RF (µ) is transitive for all µ 2 £ if and only if F = Fi for some i 2 N.
Proof: I will prove one direction. Assume F satis¯es Pareto and RF (µ) is transitive for all
µ 2 £. Let F = FG for some nonempty G µ N, and suppose that i; j 2 G for distinct i and
j. Take distinct alternatives x, y, and z, and take any state µ 2 £ with preferences over
these alternatives as follows.
Ri(µ)
y
z
x
Rj (µ)
z
x
y
RN nfi;jg(µ)
z
x
y
Because xPj (µ)y and yPi (µ)z, we have xRF (µ)yRF (µ)z. But, by Pareto, zPF (µ)x, contradicting transitivity of R F (µ). Thus, jGj = 1.
You should think about how the conclusions of Propositions 2.8 and 2.9 would change if
Pareto were dropped.
Acyclicity of strict social preference is even more fundamental to nonempty social choice
sets, but we will wait to take up that issue.
2.5
The Discrete Choice Model
In this section, I will examine the issue of collective rationality when individual preferences
are unrestricted, as in the discrete choice model. In following sections I will extend the
analysis to other collective choice environments.
We have seen the importance of continuity, transitivity, and convexity conditions in guaranteeing nonempty maximal sets. In this section, I will consider the following transitivity
conditions on a PAR F.
R-transitivity For all µ 2 £, RF (µ) is transitive.
P-transitivity For all µ 2 £, PF (µ) is transitive.
P-acyclicity For all µ 2 £, PF (µ) is acyclic.
Generally, FSP , FRP , FG , and FC are P-transitive. And FT and FB are R-transitive.
Thus, for all µ 2 £, RSP (µ), RRP (µ), RG (µ), and RC (µ) are weak quasi-orders, while RT (µ)
and RB (µ) are weak orders. Corresponding strict preferences are strict quasi-orders, etc.
We have seen that simple rules satisfy P-transitivity and R-transitivity only under very
restrictive conditions. In fact, RSM (µ), RRM (µ), Rq (µ), and R G don't even have to be weak
86
suborders. Consider the simple example below, named \Condorcet's Paradox" after the
Marquis de Condorcet, a scientist and contemporary of Borda's who is generally credited
with initiating the formal analysis of voting.
Condorcet's Paradox. Let n = 3, X = fx; y; zg, and consider a state with individual
preferences as follows.
R1(µ) R2(µ) R3(µ)
x
y
z
y
z
x
z
x
y
Clearly, these individual preferences generate a majority-rule strict preference cycle,
xPSM (µ)yPSM (µ)zPSM (µ)x;
and, as a consequence, the majority core is empty.
The idea of the paradox can be extended to public good environments. In Figure 2.5, for
example, there are three voters with Euclidean preferences and a majority preference cycle
through x, y, and z. In fact, you can check that the presence of such cycles is critical in
this example, because the majority core is empty.
Figure 2.5: Majority preference cycle
There is a classical theorem, due to McGarvey, to the e®ect that, when X is ¯nite, strict
majority preferences can be arbitrarily ugly, subject to the constraint of asymmetry. In other
words, majority preference aggregation does not entail any restrictions on social preferences
other than asymmetry.
Proposition 2.10 (McGarvey) Given a ¯nite set X and an arbitrary asymmetric relation P on X, there exists a set N of individuals and a pro¯le ((P1; R1); : : : ; (Pn; Rn)) 2 L
of preferences, say at state µ, such that PSM (µ) = P .
I'll let you think about the proof. Note that, in McGarvey's Theorem, the set of individuals
is taken as a variable. That means di®erent P's may require di®erent numbers of individuals.
The theorem is stated only for majority rule. I do not think it has been generalized to
arbitrary PARs. (Indeed, it would not be true for every PAR, so the question is: for which
PARs does a McGarvey result hold?) The assumption of ¯niteness is important in the
theorem { there is presently no such characterization of majority preference on in¯nite sets.
87
The next proposition shows that the logic of Condorcet's Paradox is actually quite general.
It di®ers from McGarvey's Theorem in that it establishes the possibility of cycles for most
every size of the set of individuals. Moreover, the cycle established in the proposition is a
nasty one that runs through the entire set of alternatives. In particular, the majority core
is empty.
Proposition 2.11 Assume that n ¸ 3, that jXj ¸ 3, that n = 4 ) jXj ¸ 4, and that
R(£) ¶ L. There exists µ 2 £ such that PSM (µ) = PRM (µ) violates acyclicity. Furthermore, the cycle is exhaustive: for all x; y 2 X, there exist k; l · 4 and x1; x2; : : : ; xk ; y1 ;
y2; : : : ; yl 2 X such that
xPSM (µ)x1 PSM (µ)x2PSM (µ) ¢ ¢ ¢ xk = yPSM (µ)y 1PSM (µ) ¢ ¢ ¢ yl = x:
Proof: I'll prove the result for the n 6
= 4 case, leaving the n = 4 case to you. Divide N into
disjoint groups G1, G2, and G3 such that, for all j 2 f1; 2; 3g, jN nGj j > n=2. (We can do
this because n = 3 or n ¸ 5.) Let w and z be distinct alternatives. By assumption, there
is some state µ 2 £ with preferences as follows.
RG1 (µ)
R G2 (µ)
RG3 (µ)
w
z
Xnfw; zg
z
Xnfw; zg
w
Xnfw; zg
w
z
In the above pro¯le, X n fw; zg indicates where the alternatives other than w and z are
ranked. Any weak linear order within that subset will do. You can check that the claim of
the proposition holds at this state.
Transitivity of Weak Social Preference
Though the weaker conditions are also of interest, I turn ¯rst to the possibility of transitivity
of social weak preference. The reasons for this are threefold: ¯rst, we know that transitivity
is su±cient for the existence of maximal elements in ¯nite sets (and, with compactness and
continuity, in in¯nite sets); second, there is some philosophical interest in the possibility
that groups of people, when acting collectively, might exhibit the same rationality that
individuals do (which would lend some credence to the idea of a \group consciousness");
third, this is historically the ¯rst and most important approach to the problem.
Our ¯rst lemma derives an implication of R-transitivity without the use of any ancillary
conditions on the PAR F.
Lemma 2.3 Assume that jXj ¸ 3, that PR(£) ¶ L, and that F satis¯es R-transitivity. If
G and G0 are semi-blocking, then, for all distinct x; y 2 X, N n(G\ G0 ) is not semi-decisive
for x over y.
88
Proof: Suppose G and G0 are semi-blocking, and take any two alternatives x; y 2 X . Let z
be any distinct alternative, and let µ 2 £ be a state with individual preferences over these
alternatives as follows.
RGnG0 (µ) RG\G0 (µ) RG0 nG (µ) RNn(G[G0 )
x
y
z
x
y
z
x
z
z
x
y
y
Since G is semi-blocking, we have yRF (µ)z. Since G0 is semi-blocking, we have zRF (µ)x.
By R-transitivity, yRF (µ)x. Thus, N n (G \ G0 ) is not semi-decisive for x over y.
Adding the assumption that F satis¯es IIA, then the conclusion of Lemma 2.3 can be
strengthened to: G \ G0 is semi-blocking. (Why?)
The next lemma uses weak Pareto and IIA to show that, if a coalition is semi-blocking in
one instance (i.e., for one alternative over another), then it is semi-blocking generally.
Lemma 2.4 Assume that jXj ¸ 3, that PR(£) ¶ L, and that F satis¯es weak Pareto,
IIA, and R-transitivity. If G is semi-blocking for some x over some y 6
= x, then it is
semi-blocking.
Proof: Let G be semi-blocking for some x over some y 6
= x. Let z be any distinct alternative, and consider a state µ 2 £ with preferences over x, y, and z as follows.
RG(µ)
x
y
z
RNnG(µ)
y
z
x
Since G is semi-blocking, we have xRF (µ)y. By weak Pareto, we have yRF (µ)z. By Rtransitivity, xRF (µ)z. By IIA, G is semi-blocking for x over z. Since z was arbitrary, this
shows that G is semi-blocking for x over every z 6
= x. A similar argument shows that G
is semi-blocking for every w 6
= y over y. (You should try to prove this step.) Of course, x
and y could be any alternatives such that G is semi-blocking for one over the other in some
state, so this argument can be repeated for any such pair of distinct alternatives. Now take
any two alternatives s; t 2 X. Assume without loss of generality that x 6
= s, and take any w
distinct from x and s. Then G is semi-blocking for x over w. Repeating the above argument
for x and w, G is semi-blocking for s over w. If w = t, then we are done. If not, then, again
by the above argument, G is semi-blocking for s over t. Therefore, G is semi-blocking.
Our main result on R-transitive PARs, next, combines the above observations to deduce
the existence of an individual with veto power.
89
weak dictatorship There exists i 2 N such that, for all µ 2 £, Pi(µ) µ RF (µ).
Equivalently, there is some i 2 N such that, for all µ 2 £, PF (µ) µ Ri (µ). The condition
means that some singleton group is blocking, so there is some individual whose strict preference prohibits a strict social preference in the opposite direction. We call this individual
a \weak dictator," or a \vetoer," or a \blocker." Note that there may be several weak
dictators. If a PAR has a weak dictator, we say it is weakly dictatorial.
Proposition 2.12 Assume jXj ¸ 3 and either P R(£) = U or PR(£) = L. If F satis¯es
weak Pareto, IIA, and R-transitivity, then F is weakly dictatorial.
Proof: By weak Pareto, N is semi-blocking. Let G be a minimal semi-blocking group. If
jGj > 1, then take any i 2 G. Taking distinct x; y 2 X, a free pair by linear domain, Lemma
2.1 implies that either G n fig is semi-blocking for x over y or (N nG) [ fig is semi-blocking
for y over x. If Gnfig is semi-blocking for x over y, then, by Lemma 2.4, it is semi-blocking,
contradicting minimality of G. Likewise, if (N nG) [ fig is semi-blocking for y over x, then
it is semi-blocking. But then, by Lemma 2.3, it follows that fig = G \ [(N n G) [ fig] is
semi-blocking, again contradicting minimality of G. Therefore, jGj = 0 or jGj = 1. In the
¯rst case, RF (µ) = X £ X for all µ 2 £ (why?), and every individual is a weak dictator.
Consider the second case, where G = fig for some i 2 N . Take distinct x; y; z 2 X and a
state µ 2 £ with individual preferences over these alternatives as follows.
Ri (µ)
x
z
y
RNnfig (µ)
z
[x y]
(Here \[x y]" indicates that individual preferences between x and y are arbitrary in this
state.) Since fig is semi-blocking, we have xRF (µ)z. By weak Pareto, we have zRF (µ)y.
By R-transitivity, xRF (µ)y. Because x and y are arbitrary here, IIA implies that fig is
blocking, as desired.
As consequence of Proposition 2.12, merely strengthening weak Pareto to Pareto, we deduce
a striking conclusion.
dictatorship There exists i 2 N such that, for all µ 2 £, Pi (µ) µ PF (µ).
The condition means that some singleton group is decisive. Thus, power is concentrated
in a single individual, in the sense that all strict preferences of the individual are imposed
on society. We call this individual a dictator for F . If a PAR has a dictator, we say it is
dictatorial.
90
Proposition 2.13 (Arrow) Assume jXj ¸ 3 and either PR(£) = U or PR(£) = L. If
F satis¯es Pareto, IIA, and R-transitivity, then F is dictatorial.
Proof: Let i be a weak dictator. Take any distinct alternatives x; y; z 2 X and state µ 2 £
with individual preferences over these alternatives as follows.
Ri (µ)
x
z
y
RNnfig (µ)
z
[x y]
Since i is a weak dictator, we have xR F (µ)z. By Pareto, we have zPF (µ)y. By R-transitivity,
xPF (µ)y. By IIA, i is a dictator.
You should consider the robustness of this \impossibility theorem" to relaxations of various
assumptions. If we relax R-transitivity to P -transitivity, for example, the Pareto rules, FSP
and FRP , satisfy Arrow's assumptions. What if jX j = 2? What if we drop Pareto? What
if we drop IIA?
As Proposition 2.9 shows, the converse of Arrow's theorem does hold for simple PARs. It
also holds when PR(£) = L. The converse does not hold when PR(£) = U, as you should
check: if F is dictatorial, it does not necessarily satisfy IIA or R-transitivity. (Though
dictatorship does imply Pareto.)
Can you see where the proof of Proposition 2.13 uses PR(£) = U or P R(£) = L instead
of merely P R(£) ¶ L? Does the conclusion of the proposition hold if we assume merely
the latter inclusion? What if we assume P R(£) = U instead?
Transitivity of Strict Social Preference
As a consequence of Arrow's theorem, assuming IIA and the minimal condition of Pareto, we
cannot expect collective preferences to exhibit the same regularity properties that we expect
of individual preferences | except in groups with highly asymmetric power structures.
Weakening R-transitivity to P-transitivity, we know that there do exist nondictatorial PARs
satisfying the remainder of Arrow's conditions: FSP and FRP , for example. In fact, given
any G with 1 < jGj · n, FG satis¯es Pareto, IIA, and P-transitivity and is nondictatorial.
We will see, however, that this gives us a nearly complete characterization of such PARs!
Lemma 2.5 Assume that jXj ¸ 3, that PR(£) ¶ L, and that F satis¯es P-transitivity. If
G and G0 are semi-decisive, then, for all distinct x; y 2 X , N n(G\G0 ) is not semi-blocking
for x over y.
91
Proof: Suppose G and G0 are semi-decisive, and take any two alternatives x; y 2 X. Let z
be any distinct alternative, and let µ 2 £ be a state with individual preferences over these
alternatives as follows.
RGnG0 (µ) RG\G0 (µ) RG0 nG (µ) RNn(G[G0 )
x
y
z
x
y
z
x
z
z
x
y
y
Since G is semi-decisive, we have yPF (µ)z. Since G0 is semi-decisive, we have zPF (µ)x. By
P-transitivity, yPF (µ)x. Thus, N n (G \ G0 ) is not semi-blocking for x over y.
Adding the assumption that F satis¯es IIA, then the conclusion of Lemma 2.5 can be
strengthened to: G \ G0 is semi-decisive. (Why?)
Lemma 2.6 Assume that jXj ¸ 3, that PR(£) ¶ L, and that F satis¯es Pareto, IIA, and
P-transitivity. If G is semi-decisive for some x over some y 6
= x, then it is semi-decisive.
Proof: Let G be semi-decisive for some x over some y 6
= x. Let z be any distinct alternative,
and consider a state µ 2 £ with preferences over x, y, and z as follows.
RG(µ)
x
y
z
RNnG(µ)
y
z
x
Since G is semi-decisive, we have xPF (µ)y. By Pareto, we have yPF (µ)z. By P-transitivity,
xPF (µ)z. By IIA, G is semi-decisive for x over z. Since z was arbitrary, this shows that G
is semi-decisive for x over every z 6
= x. A similar argument shows that G is semi-decisive
for every w 6
= y over y. (You should try to prove this step.) Of course, x and y could
be any alternatives such that G is semi-decisive for one over the other in some state, so
this argument can be repeated for any such pair of distinct alternatives. Now take any two
alternatives s; t 2 X. Assume without loss of generality that x 6
= s, and take any w distinct
from x and s. Then G is semi-decisive for x over w. Repeating the above argument for x
and w, G is semi-decisive for s over w. If w = t, then we are done. If not, then, again by
the above argument, G is semi-decisive for s over t. Therefore, G is semi-decisive.
We need one more condition before we can state the main result on P-transitive social
preferences.
oligarchy There is some G µ N such that, for all µ 2 £,
\
\
Pi(µ) µ PF (µ) µ
Ri (µ)
i2G
i2G
92
Note that the second inclusion is equivalent to: for all i 2 G, PF (µ) µ Ri(µ). Thus, each
i 2 G is a weak dictator, or equivalently, fig is a blocking coalition for each i 2 G.
In words, the oligarchy condition means that power is concentrated in the members of a
group, G, in the sense that the members of G can impose a common strict preference on
society; moreover, each member of G has individual power, in the form of a veto over strict
social preferences, i.e., if i 2 G strictly prefers x to y, then society cannot have the opposite
preference. We say G is an oligarchy, and we refer to a member of G as an oligarch. If F
has an oligarchy, we call it oligarchical.
Clearly, if i is a dictator, then fig is an oligarchy, and vice versa.
Proposition 2.14 (Gibbard) Assume jXj ¸ 3 and either PR(£) = U or P R(£) = L.
If F satis¯es Pareto, IIA, and P-transitivity, then F is oligarchical.
Proof: By Pareto, N is semi-decisive. Let G be a minimal semi-decisive group, and note
that G is nonempty. (Why?) To see that G is decisive, take distinct x; y; z 2 X and a state
µ 2 £ with individual preferences over these alternatives as follows.
RG(µ)
x
z
y
RNnG(µ)
z
[x y]
(Here, [x y] indicates that individual preferences between x and y are arbitrary.) Because
G is semi-decisive, we have xPF (µ)z. By Pareto, we have zPF (µ)y. By P-transitivity, we
have
IIA implies that G is decisive. Thus,
T xPF (µ)y. Because x and y are arbitrary here, T
P
(µ)
µ
P
(µ)
for
all
µ.
I
claim
that
P
(µ)
µ
F
F
i2G i
i2G Ri(µ) for all µ as well. Take any
distinct x; y 2 X, a free pair by linear domain, and any i 2 G. If fig is not semi-blocking
for x over y, then, by Lemma 2.1, N n fig is semi-decisive for y over x. By Lemma 2.6,
N n fig is semi-decisive. And by Lemma 2.5, it follows that G n fig = G \ [N n fig] is
semi-decisive, contradicting minimality of G. Therefore, fig is semi-blocking. To see that
fig is blocking, take distinct x; y; z 2 X and a state µ 2 £ with individual preferences over
these alternatives as follows.
Ri(µ)
x
z
y
RNnfig(µ)
z
[x y]
By Pareto, we have zPF (µ)y. If yPF (µ)x, then, by P-transitivity, we have zPF (µ)x. But, because fig is semi-blocking, we have xRF (µ)z, a contradiction. Therefore, xRF (µ)y. Because
x and y are arbitrary, IIA implies that fig is blocking, as desired.
93
Since FSP and FRP satisfy all of Gibbard's conditions, they are oligarchical. Can you see
which G is the oligarchy?
You should check the robustness of Gibbard's theorem to relaxations of the assumptions.
What if jXj = 2? What if we drop Pareto or IIA? You can check that, again, the result
does not hold if we assume merely PR(£) ¶ L.
Note that it is not generally true that oligarchical PARs satisfy IIA or P-transitivity.
(Why?)
Acyclicity of Strict Social Preference
What if we drop any of Gibbard's conditions? I'll let you think about Pareto and IIA.
Suppose we weaken P-transitivity to P-acyclicity. Are there non-oligarchical PARs that
satisfy Pareto, IIA, and this condition? The next example shows there are.
The following PAR, call it FSC , is based on the voting rule of the pre-1965 UN Security
Council. Let n = 11, with the ¯rst ¯ve individuals corresponding to the permanent members
of the Security Council. De¯ne
xPSC (µ)y , [f1; 2; 3; 4; 5g µ R(x; yjµ) and p(x; yjµ) ¸ 7]:
This clearly satis¯es Pareto and IIA. Is it oligarchical? If a group were an oligarchy, it
would necessary include f1; 2; 3; 4; 5g, because each permanent member can veto strict social
preference. But f1; 2; 3; 4; 5g is not decisive: an oligarchy would have to contain some i > 5.
But no such i has a veto, so there is no oligarchy. Note that FSC is P -acyclic. (Can you
see why?)
Because FSC is not oligarchical yet satis¯es Pareto and IIA, it must not satisfy P-transitivity
generally. I'll let you construct an example in which strict social preference for this PAR
violates transitivity.
While this does give us an example of a non-oligarchical PAR satisfying Pareto, IIA, and
P-acyclicity, it is apparent that FSC does concentrate some power in a relatively small
group. We will see this observation formalized and generalized shortly.
We ¯rst need another condition on PARs, one that is satis¯ed in most any democratic
societies, though not by FSC .
virtual unanimity For all x; y 2 X and all µ 2 £, p(x; yjµ) ¸ n ¡ 1 implies xPF (µ)y.
In words, if all individuals or all but one prefer x to y strictly, then so does society. Equivalently, G 2 D(F ) for every group G with jGj ¸ n ¡ 1. Note that virtual unanimity implies
Pareto.
94
The next proposition, proved by Tom Schwartz, indicates that strict social preference cycles
can be a deep problem, even if we do not assume IIA. It assumes, however, a large enough
number of alternatives. (Compare this result to Austen-Smith and Banks's Lemma 2.3.)
Proposition 2.15 (Schwartz) Assume jXj ¸ n and P R(£) ¶ L. If F satis¯es virtual
unanimity, then it violates P-acyclicity.
Proof: Since jXj ¸ n, we can take n distinct alternatives and label them x1; x2; : : : ; xn .
Take µ 2 £ such that individual preferences restricted to fx1; x2; : : : ; xn g are as follows.
R1(µ) R2(µ) ¢ ¢ ¢
x2
x3
x3
x4
..
..
.
.
xn
x1
x1
x2
Ri(µ) ¢ ¢ ¢
xi+1
xi+2
..
.
Rn(µ)
x1
x2
..
.
xi¡1
xi
xn¡1
xn:
(This construction is called a \Latin Square".) Note that G n fig = P(xi ; xi+1 jµ) (letting
xn+1 = x1), for all i, so p(xi ; xi+1jµ) = n ¡ 1. By virtual unanimity, we have
x1 PF (µ)x2 PF (µ)x3 ¢ ¢ ¢ PF (µ)xnPF (µ)x1;
contradicting P -acyclicity.
Note that Borda rule never produces strict social preference cycles, so it cannot satisfy
virtual unanimity when jXj ¸ n. I'll let you verify this. Next is another condition on
PARs.
collegiality
T
D(F) 6
= ;.
Roughly, this means there is some individual who is in
T every decisive coalition. If there is
such an individual, we say F is collegial, and we call D(F ) the collegium of F .
Note, however, the logical convention that the intersection of the empty family of coalitions
is NTitself: let D(F) = ;; take any i 2 N, and note that, for all G 2 D(F ), i 2 G; therefore,
i 2 D(F ). Of course, the universal generalization here is vacuously satis¯ed. Technically,
then, collegiality means that either there are no decisive coalitions or there are, and some
individual is in all of them. The minimal condition of Pareto is equivalent to D(F) 6
= ;, in
which case collegiality reduces to the more intuitive idea.
Example: FSC is collegial, and the collegium is the permanent members.
How does a collegium compare to an oligarchy?
95
1. If G is an oligarchy, then it is a collegium.
2. G may be a collegium yet give no member a veto: FB when jXj ¸ n, because then
D(FB) = fNg.
3. G may be a collegium yet not be decisive: FSC .
4. G may be a collegium yet give no member a veto and not be decisive: let N =
f1; 2; 3; 4g, pick two distinct alternatives a and b, and de¯ne F by
8
< xP1(µ) \ P2(µ) \ (P3(µ) [ P4 (µ))y
xPF (µ)y ,
or x = aP1(µ) \ P3(µ) \ P4 (µ)b = y
:
or x = bP2(µ) \ P3(µ) \ P4(µ)a = y:
T
Here, D(F ) = ff1; 2; 3g; f1; 2; 4g; f1; 2; 3; 4gg, so D(F) = f1; 2g. This group is not
decisive. Moreover, assuming linear domain, neither 1 nor 2 has a veto. Also, F is
P-acyclic and satis¯es IIA.
Note that the examples of FB and F , in (4) above, violate neutrality. As the next proposition
shows, if a collegial PAR satis¯es weak decisiveness, then, indeed, every member of the
collegium will have a veto.
T
Proposition 2.16 Assume F satis¯es weak decisiveness. If i 2 D(F), then, for all
µ 2 £, Pi (µ) µ RF (µ).
T
Proof: Take any i 2 D(F) and any µ 2 £ suchTthat xPi (µ)y. If yPF (µ)x, then, by weak
decisiveness, R(y; xjµ) 2 D(F ), contradicting i 2 D(F). Therefore, xRF (µ)y.
The next result, due to Don Brown, gives us a restriction implied by P -acyclicity and is
in the same line of results by Arrow and Gibbard. It is quite di®erent, however, in that it
does not impose IIA but does impose a restriction on the number of alternatives.
Proposition 2.17 (Brown) Assume jXj ¸ n and P R(£) ¶ L. If F is P-acyclic, then it
is collegial.
Brown's result is actually equivalent to Proposition 2.15 and follows from the earlier result
by observing that F is not collegial if and only if it satis¯es virtual unanimity.
Combining Propositions 2.16 and 2.17, we have the following result: If jXj ¸ n, if all pro¯les
of weak linear orders are possible, and if F satis¯es Pareto, monotonicity, neutrality, and
P-acyclicity, then some individual has a veto. (This is essentially Austen-Smith and Banks's
Theorem 2.5.)
96
Because these results assume jXj ¸ n, they give us only a limited characterization of Pacyclic PAR's.
Given a PAR F , de¯ne the Nakamura number of F , denoted N (F ), as
n
o
\
N (F) = min jGj j G µ D(F) and
G=;
T
if D(F ) = ; (i.e., F is non-collegial); otherwise (i.e., F is collegial), set N (F) = j2X j >
jXj. (We can actually do this even when X is in¯nite, in which case j ¢ j denotes the
\cardinality" of a set.)
In words, N (F ) is the size of the smallest collection of decisive groups having empty intersection.
Proposition 2.18 For every PAR F , N (F) ¸ 3. If F is non-collegial, then N (F) · n.
Proof: If F is collegial, then N (F) = j2X j ¸ 3, since jXj ¸ 2. Otherwise, take any
G; G0 2 D(F). Since D(F) is proper, G \ G0 6
= ;. Therefore, N (F) T¸ 3. If F is noni
collegial, then, for each i 2 N, there exists G 2 D(F) with i 2
= Gi . So fGi j i 2 Ng = ;.
i
Therefore, N (F ) · jfG j i 2 Ngj · n.
The next result, due to Nakamura, shows us how the concept of Nakamura number ¯gures
in a more general analysis of P -acyclicity. Note that it does not use any restriction on the
number of alternatives.
Proposition 2.19 (Nakamura) Assume PR(£) ¶ L. If F is P -acyclic, then N (F) >
jXj.
Proof: Suppose jXj ¸ N (F), which T
implies that F is non-collegial. Let m = N (F), and
1
2
m
j
let fG ; G ; : : : ; G g µ D(F) satisfy m
j=1 G = ;. De¯ne
H1
H2
H3
H4
Hm
=
=
=
=
..
.
=
NnG1
G1nG2
(G1 \ G2)nG3
(G1 \ G2 \ G3 )nG4
(G1 \ G2 \ ¢ ¢ ¢ \ Gm¡1 )nGm :
Note that H j may be empty. See Figure 2.6 for an example of this construction.
These sets satis¯es the following three properties:
97
Figure 2.6: The construction of the H j's
1. N =
Sm
j=1 H
j
(i is in ¯rst H j such that i 2
= Gj )
2. H j \ H k = ; for all distinct j and k
3. Gj µ NnH j.
Since jXj ¸ N (F) = m, we may take distinct alternatives x1 ; x2; : : : ; xm and a state µ 2 £
such that individual preferences restricted to fx1 ; x2; : : : ; xm g are as follows.
RH1 (µ) RH2 (µ) ¢ ¢ ¢
x2
x3
x3
x4
..
..
.
.
xm
x1
x1
x2
RHj (µ) ¢ ¢ ¢
xj +1
xj +2
..
.
xj ¡1
xj
RHm (µ)
x1
x2
..
.
xm¡1
xm:
Since Gj µ N nH j , each NnH j is decisive. Therefore,
x1PF (µ)x2PF (µ)x3 ¢ ¢ ¢ PF (µ)xm PF (µ)x1;
contradicting P -acyclicity.
Note that this result generalizes Brown's theorem: Assume jXj ¸ n, all pro¯les of weak
linear orders are possible, but F is non-collegial. Then N (F ) · n · jXj, and Nakamura's
theorem implies F is not P -acyclic.
The next proposition helps clarify the implications of Nakamura's theorem by calculating
the Nakamura numbers of various PARs.
Proposition 2.20
1. If F is non-collegial and strong, then N (F) = 3.
n
2. If q < n, then N (F ) = d n¡q
e.
3. If n = 4, then N (FSM ) = 4. If n ¸ 3, n 6
= 4, then N (FSM ) = 3.
Proof: 1. Since F is non-collegial, there exist distinct minimal decisive groups G and G0 .
Since D(F) is proper, G \ G0 6
= ;. By minimality, G \ G0 2
= D(F ). Since D(F ) is strong,
0
Nn(G \ G ) 2 D(F). Clearly, G \ G0 \ (Nn(G \ G0)) = ;, so N (F ) · 3. By Proposition
98
2.18,
and take distinct G1; G2; : : : ; Gm 2 D(Fq ) with
Tm N (F ) = 3. 2. Let N (Fq ) = m, S
m
j=1 Gj = ;. By DeMorgan's Law, N = j=1(NnGj ). This implies
n = j
m
[
j=1
(NnGj)j ·
m
X
j=1
jN nGj j:
Since Gj 2 D(Fq ), jGjj ¸ q, so jNnGjj · n ¡ q. Thus,
n ·
m
X
j =1
jNnGj j · m(n ¡ q);
implying
N (Fq ) = m ¸
n
:
n¡q
n e. Set t = d n e. For the opposite inequality,
Since N (F) is an integer, N (F ) ¸ d n¡q
Tt n¡q
n
I will ¯nd distinct G1; G2 ; : : : ; Gt 2 D(Fq ) with j =1 Gj = ;. Since t ¡ 1 < n¡q
, i.e.,
(t ¡ 1)(n ¡ q) < n, we can construct t ¡ 1 pairwise disjoint groups of size n ¡ q, call them
H1; H2; : : : ; Ht¡1, with some individuals leftover.
1; 2; : : : ; n ¡ q; n ¡ q + 1; : : : ; 2(n ¡ q); : : : ; (t ¡ 2)(n ¡ q) + 1; : : : ; (t ¡ 1)(n ¡ q);
|
{z
} |
{z
}
|
{z
}
H1
H2
l
m
Ht¡1
(t ¡ 1)(n ¡ q) + 1; : : : ; n
|
{z
}
³
´
leftover
n
n
Since t(n ¡ q) = n¡q
(n ¡ q) ¸ n¡q
(n ¡ q) = n, the leftover group, call it Ht , contains
no more than n ¡ q members. TThus, for each j = 1; : : : ; t, Gj = N nHj has at least q
members, so Gj 2 D(Fq ). Also, tj=1 Gj = ;, as required. 3. Let's use part 2 for the special
case of simple majority rule: q = d n+1
2 e. If n is odd, then
&
'
»
¼
»
¼
n
n
2n
=
=
· 3
n¡q
n¡1
n ¡ n+1
2
2n · 3, i.e., n ¸ 3. With our earlier proposition, this means that N (F
if and only if n¡1
SM ) =
3, whenever n is odd and ¸ 3. If n is even, then
»
¼
»
¼
»
¼
n
n
2n
=
=
· 3
n¡q
n ¡ n2 ¡ 1
n¡2
if and only if
the claim.
2n
n¡2
· 3, i.e., n ¸ 6. You can check that N (FSM ) = 4 when n = 4, proving
These results yield the following corollary.
99
Corollary 2.5 Assume PR(£) ¶ L.
1. If F is P-acyclic and strong, then jXj = 2.
n
2. If Pq µ PF and F is P-acyclic, then jXj < d n¡q
e.
3. If n 6
= 4, n ¸ 3, PSM µ PF , and F is P -acyclic, then jXj = 2.
4. If n = 4, PSM µ PF , and F is P-acyclic, then jX j · 3.
Nakamura also proved the following su±cient condition for P-acyclicity.
Proposition 2.21 (Nakamura) Let F be simple. If N (F) > jXj, then F is P-acyclic.
Proof: Suppose N (F ) > jXj but F is not P-acyclic. Then there is some µ and x1; x2; : : : ; xk
such that
x1 PF (µ)x2PF (µ) ¢ ¢ ¢ xk¡1PF (µ)xk = x1 :
Without loss of generality, we may assume x1; x2; : : : ; xk¡1 are distinct (why?), so jXj ¸
k ¡ 1. Since F is simple,
fP(x1; x2jµ); P(x2 ; x3 jµ); : : : ; P (xk¡1; xkjµ); P(xk ; x1jµ)g µ D(F):
T
Since N (F) > jXj ¸ k ¡ 1, we must have k¡1
= ;. But then there is some
h=1 P(xh; xh+1 jµ) 6
i 2 N with x1Pi(µ)x2Pi(µ) ¢ ¢ ¢ xk¡1Pi(µ)xk = x1 , a contradiction.
As a corollary, we get the following characterization of P-acyclicity for simple PARs, complementing Propositions 2.8 and 2.9 on transitivity of strict and weak social preference for
simple PARs. It di®ers from these earlier propositions in that the characterization places a
joint restriction on F and the number of alternatives.
Corollary 2.6 Assume P R(£) ¶ L. Let F be a simple PAR. Then F is P-acyclic if and
only if N (F ) > jXj.
2.6
The One-Dimensional Spatial Model
We've examined the possibility of collective rationality under broad domain restrictions
(e.g., linear domain or unrestricted domain), and we've discovered rather severe implications
for the distribution of power among individuals (though less so for the case of P -acyclicity).
We now investigate the domain restriction of single-peakedness in the one-dimensional spatial model, where it is quite intuitive. It turns out that single-peakedness is enough to
100
get excellent transitivity properties for many PARs. The next proposition imposes weak
decisiveness to get transitivity of strict social preferences and the stronger condition that
D(F) is strong to get transitivity of weak social preferences. Note that the conditions of
the proposition are stated for individual preferences at a single state, rather than on the
domain of possible preferences. An implication is that, under single-peaked domain, these
transitivity properties hold at every state.
Proposition 2.22 Assume PR(µ) is single-peaked.
1. If F is weakly decisive, then PF (µ) is transitive.
2. If F is strong, then RF (µ) is transitive.
Proof: Let P R(µ) be single-peaked with respect to Á, and take any x; y; z 2 X such that
xPF (µ)yPF (µ)z. If x = y or y = z, then we have xPF (µ)z. Otherwise, one of the following
cases must obtain:
1: x Á y Á z
2: y Á x Á z
3: y Á z Á x
4: z Á y Á x
5: z Á x Á y
6: x Á z Á y:
Case 1: Since xPF (µ)y and F is weakly decisive, we have R(x; yjµ) 2 D(F ). Since individual
preferences are single-peaked, R(x; yjµ) µ P(x; zjµ) (right?), so P (x; zjµ) 2 D(F), which
implies xPF (µ)z. Case 2: Since yPF (µ)z and F is weakly decisive, we have R(y; zjµ) 2 D(F).
By single-peakedness, R(y; zjµ) µ P(x; zjµ), so xPF (µ)z. Case 3: Since xPF (µ)y and F is
weakly decisive, R(x; yjµ) 2 D(F). By single-peakedness, R(x; yjµ) µ P(z; yjµ), implying
zPF (µ)y, a contradiction. Thus, this case cannot occur. Case 4 is symmetric to 1, case 5 is
symmetric to 2, and case 6 is symmetric to 3. With part 3 of Proposition 2.3, the proof of
part 2 proceeds in the same way.
As the following example illustrates, the assumption of single-peakedness in part 1 of Proposition 2.22 cannot be dropped, not even weakened to weak single-peakedness, even for majority rule with n odd. Let n = 3, X = fx; y; zg, and consider a state µ with preferences as
follows.
R1 (µ)
x
yz
R2(µ)
y
x
z
R3 (µ)
z
y
x
Then yPSM (µ)xPSM (µ)z but zRSM (µ)y, violating P -transitivity.
The next example shows that the conclusion of part 1 of Proposition 2.22 cannot be strengthened to R-transitivity, and that the assumption that F is strong in part 2 cannot be weak-
101
ened to weak decisiveness.
R1(µ)
y
x
z
R2(µ)
z
y
x
This pro¯le is single-peaked with respect to x Á y Á z, but we have xRSM (µ)zRSM (µ)y
and yPSM (µ)x, violating R-transitivity.
Note that n = 2 is even in the previous example. Otherwise, if n were odd, then the second
part of Proposition 2.22 would ensure R-transitivity.
Transitivity of strict social preferences is more than we need for nonemptiness of choice
sets. Austen-Smith and Banks show that weak single-peakedness is actually su±cient for
P-acyclicity of social preferences.
Proposition 2.23 Assume P R(µ) is weakly single-peaked. If F is simple, then PF (µ) is
acyclic.
Proof: Let PR(µ) be weakly single-peaked with respect to ¹. Take any ¯nite number
of alternatives, x1; : : : ; xk , and suppose that x1PF (µ)x2 ¢ ¢ ¢ PF (µ)xk . Assume, without loss
of generality, that xk Á xk¡1 . I claim that, for all h; j = 1; : : : ; k, if xh Á xj ¹ xh+1 ,
then xh Á xj+1. Otherwise, we have xj+1 ¹ xh Á xj ¹ xh+1, and, by weak singlepeakedness, P(xh+1 ; xh jµ) µ R(xj ; xj+1jµ) (why?), implying P (xh+1; xhjµ)\P (xj+1; xjjµ) =
;. Since F is simple, however, it must be that P(xh+1 ; xh jµ) and P(xj+1; xj jµ) are decisive,
a contradiction. The result now follows because xk Á x1 (why?), which implies xk 6
= x 1.
We have seen that the conclusion of Proposition 2.23 cannot be strengthened to P -transitivity
or R-transitivity. One way it di®ers from our earlier propositions is that it assumes F is
simple, rather than just weakly decisive. To see that we cannot assume just weak decisiveness, or even that F is strong, let n = 3, X = fx; y; zg, and de¯ne F as follows. For all
v; w 2 X , say vPF (µ)w if and only if any of three conditions holds:
² vP1(µ)w and vR2 (µ)w,
² vP3(µ)w and vR1 (µ)w,
² vP2(µ)w and vP3(µ)w.
This PAR is weakly decisive. In fact, F , so-de¯ned, is neutral and monotonic, and D(F ) =
D(FSM ) is strong. But it is not decisive. Now consider the following pro¯le.
R1 (µ)
x
yz
R2(µ)
z
xy
102
R3 (µ)
y
z
x
This pro¯le is weakly single-peaked with respect to x Á y Á z. Note, however, that
xPF (µ)yPF (µ)zPF (µ)x, violating P -acyclicity.
Thus, when X is ¯nite and F is weakly decisive, single-peakedness implies that the core,
CF (µ), is nonempty. When X is ¯nite and F is simple, weak single-peakedness implies that
the core is nonempty. When X is in¯nite, nonemptiness of the core is not so immediate:
we have not established upper semicontinuity of the social preference relation, nor have
we imposed compactness on the set of alternatives. (In fact, we have not imposed any
topological structure on the set of alternatives.)
The next task is to derive a sharp characterization of the core under single-peakedness for a
large class of PARs. As a consequence, we will obtain an existence result for in¯nite sets of
alternatives, i.e., the usual conditions of compactness and continuity are unneeded. We ¯rst
note that such a characterization for simple PARs carries over immediately to all weakly
decisive PARs.
Recall that, for all PARs F and all µ 2 £, we must have CF (µ) µ CD(F) (µ). For singlepeaked pro¯les, the converse holds for all weakly decisive PARs.
Proposition 2.24 Assume PR(µ) is single-peaked. If F is weakly decisive, then CF (µ) =
CD(F) (µ).
Proof: Let P R(µ) be single-peaked with respect to ¹. Take x 2 CD(F ) (µ) and suppose x 2
=
CF (µ), i.e., there is some y 2 X such that yPF (µ)x. Assume without loss of generality that
x Á y. By single-peakedness, x Á x
~i for all i 2 R(y; xjµ). Let z = minf~
xi j i 2 R(y; xjµ)g,
where the min operation refers to ¹. By single-peakedness, R(y; xjµ) µ P (z; xjµ). By weak
decisiveness, R(y; xjµ) 2 D(F ), so zPD(F ) (µ)x, contradicting x 2 CD(F ) (µ).
Given a state µ and preference pro¯le PR(µ) single-peaked with respect to ¹, de¯ne
NÁ+(xjµ) = fi 2 N j x Á x
~i g
NÁ¡(xjµ) = fi 2 N j x
~ i Á xg:
De¯ne the groups
G1 = fi 2 N j NÁ+(~
xi jµ) 2
= D(F)g
¡ i
G2 = fi 2 N j NÁ (~
x jµ) 2
= D(F)g:
Note that G1 6
= ;, because, letting i have the \greatest" ideal point according to the ordering
Á, we have x
~i 2 G1. Similarly, G2 6
= ;. Thus, we can take
i1 2 arg minf~
xi j i 2 G1g and i2 2 arg maxf~
xi j i 2 G2g;
103
where the minimum and maximum are with respect to Á. Note that, if x
~ i1 Á x
~i , then
i 2 G1; similarly, if x
~i Á x
~i2 , then i 2 G2.
Furthermore, if F satis¯es Pareto, then G1 and G2 are decisive. This is true of G1, for
example, if G1 = N. If G1 $ N, then let i be a solution to maxi2N nG1 x
~i, and note
+ i
that G1 = NÁ
(~
x jµ) 2 D(F ). (Do you see why?) The same argument establishes that
G2 2 D(F). Thus, because D(F) is proper, we have G1 \ G2 6
= ;, i.e., x
~i1 ¹ ~xi2 .
We say x is an F-median with respect to ¹ at µ if
NÁ+(xjµ) 2
= D(F ) and NÁ¡(xjµ) 2
= D(F):
The next proposition establishes that the set of F-medians is independent of the particular
ordering ¹ in the de¯nition of single-peakedness.
Proposition 2.25 Assume PR(µ) is single-peaked with respect to ¹ and with respect to
¹0 . Then x is an F-median with respect to ¹ at µ if and only if it is an F -median with
respect to ¹0 at µ.
Proof: Suppose x is an F-median with respect to ¹ but not an F-median with respect to
+
¹0 . Without loss of generality, suppose N¹
xk j k 2
0 (xjµ) 2 D(F), and take i 2 arg minf~
N¹+0 (xjµ)g, where the minimum is with respect to ¹0 . By single-peakedness with respect to
¹0 , x
~i Pk (µ)x for all k 2 N¹+0 (xjµ). Since x is an F-median with respect to ¹, we cannot
+
¡
have N¹+0 (xjµ) µ N¹+(xjµ) or N¹
~j ¹ x Á x
~i
0 (xjµ) µ N¹ (xjµ). Thus, we may assume that x
+
for some j 2 N¹0 (xjµ). But then single-peakedness with respect to ¹ implies xPj(µ)~
xi , a
contradiction.
Thus, we use the term \F-median at µ" and write simply MF (µ) for the set of F-medians
at µ.
Proposition 2.26 Assume PR(µ) is single-peaked. If F satis¯es Pareto, then
MF (µ) = fx 2 X j x
~i1 ¹ x ¹ x
~ i2 g 6
= ;:
Proof: If x Á x
~ i1 , then G1 µ NÁ+(xjµ) 2 D(F), so x 2
= MF (µ). Similarly, x
~ i2 Á x implies
+
i1
i2
x2
= MF (µ). Now take x such that x
~ ¹x¹x
~ . By the ¯rst relation, NÁ (xjµ) 2
= D(F),
¡
and, by the second relation, NÁ
(xjµ) 2
= D(F). Thus, x 2 MF (µ). This establishes the
equality in the proposition. That MF (µ) 6
= ; follows because D(F) is proper: there exists
i 2 G1 \ G2, and then ~xi 2 MF (µ).
104
Now that we have characterized the F-medians, we can give a characterization of the core
when individual preferences are single-peaked. With Proposition 2.26, the following proposition establishes the nonemptiness of the core, given any weakly decisive PAR, even for
in¯nite X.
Proposition 2.27 Assume PR(µ) is single-peaked.
1. For every PAR F , CF (µ) µ MF (µ).
2. If F is weakly decisive, then MF (µ) µ CF (µ).
Proof: For part 1, take x 2 CF (µ) and suppose x 2
= MF (µ). Without loss of generality,
say NÁ+(xjµ) 2 D(F ), and let x
~ i be the minimum ideal point above x according to Á. Since
PR(µ) is single-peaked, NÁ+(xjµ) = P(~
xi ; xjµ), so x
~ iPF (µ)x, contradicting x 2 CF (µ). For
part 2, take x 2 MF (µ), and suppose there is some y 2 X such that yPF (µ)x. Without
+
loss of generality, suppose x Á y. Since PR(µ) is single-peaked, R(y; xjµ) µ NÁ
(xjµ).
+
Since F is weakly decisive, yPF (µ)x implies R(y; xjµ) 2 D(F). But then N (xjµ) 2 D(F),
contradicting x 2 MF (µ).
Let's examine the idea of F-median in some special cases. If F is a q-rule, then x 2 MF (µ) if
+
¡
and only if jNÁ
(xjµ)j < q and jNÁ
(xjµ)j < q. Ordering the ideal points ~x1 ¹ ~x2 ¹ ¢ ¢ ¢ ¹ x
~n ,
the medians are as in Figure 2.7.
Figure 2.7: F-medians for q-rules
§
¨
If q = n+1
2 , then we have two cases, depending whether n is even or odd, as in Figure
2.8. When n is odd, there is only one majority rule core point: it is the median of the
distribution of ideal points.
Figure 2.8: F-medians for majority rule
This result provides the logic for the famous \Median Voter Theorem," generally attributed
to Harold Hotelling, Anthony Downs, or Duncan Black. Suppose two candidates compete for the votes of an odd number of voters by simultaneously choosing platforms in a
one-dimensional space. If the voters' preferences are single-peaked, then the unique Nash
equilibrium is for both candidates to locate at the median of the voters' ideal points.
The uniqueness of the majority rule core when n is odd can actually be obtained quite
generally, as the next proposition shows. In fact, as long as the collection of decisive groups
is strong, the core must consist of the ideal point of some voter.
105
Proposition 2.28 Assume PR(µ) is single-peaked. If F is strong, then CF (µ) = f~
xig for
some i 2 N. Furthermore, for all x 2 X n f~
xi g, we have x
~i PF (µ)x.
Proof: Note that F is weakly decisive, so CF (µ) = MF (µ). Furthermore, because D(F)
is strong, we have x
~ i1 = x
~ i2 and MF (µ) = f~
xi1 g. That x
~ i1 is strictly socially preferred to
every other alternative follows from Proposition 2.22. (Do you see why?)
Note that, when F is strong and we assume single-peaked domain, the social weak preference
RF (µ) is complete and transitive, i.e., a weak order. Moreover, by the previous proposition,
there is a unique alternative at the top of that ordering, which we may take to be the ideal
point of individual i1. But we have not explored the nature of the social ordering. Is it
single-peaked? (You can check that it is.) Is it equal to the ordering of i1, i.e., do we have
RF (µ) = Ri1 (µ)? (Well?)
The de¯nition of F -median can be extended to weakly single-peaked pro¯les, in which
case we may consider whether Proposition 2.27 extends as well. The next example, due
to Austen-Smith and Banks, shows that it does not, even when individuals have unique
ideal points (so the de¯nition of F-median is as before) and F is strong. Let n = 3 and
X = fx; y; zg, and consider the following pro¯le.
R1 (µ)
y
z
x
R2(µ)
x
yz
R3 (µ)
z
y
x
This pro¯le is weakly single-peaked with respect to x Á y Á z, and the unique majority
rule median is y, but CSM (µ) = fy; zg. Thus, core points need not be medians.
For another example, go back to our example of a weakly single-peaked pro¯le, a PAR with
F strong, and a strict social preference cycle: there, y is the unique median, but xPF (µ)y.
Thus, medians need not be core points.
A more general construction of the set of F-medians must allow for \intervals" of ideal
points, which must have well-de¯ned suprema and in¯ma. To deal with this, suppose
X µ R and PR(µ) is weakly single-peaked with respect to ·, so that each M(Ri (µ)) is a
nonempty interval. Given G µ N, let
[
~ G(µ) =
X
M(Ri(µ));
i2G
and de¯ne
x(µ) =
~ G (µ) and x(µ) = max inf X
~ G(µ);
min sup X
G2D(F )
G2D(F )
106
where we use the convention that sup Y = 1 if Y is unbounded above and inf Y = ¡1
if Y is unbounded below. Then we de¯ne the F-medians, still denoted MF (µ), to be the
interval from x(µ) to x(µ), with endpoints included depending on whether the supremum and
in¯mum are achieved in the above. Of course, if each (Pi (µ); Ri(µ)) is upper semicontinuous,
then each M(Ri (µ)) is closed (right?), and then MF (µ) = [x(µ); x(µ)].4 You should check
that, when P R(µ) is single-peaked, this de¯nition agrees with our earlier one.
The next result extends one direction of Proposition 2.27 to the weakly single-peaked case.
Note that it gives us nonemptiness of the core. I leave the proof to you.
Proposition 2.29 Assume that X µ R, that PR(µ) is weakly single-peaked with respect
to ·, and that each (Pi(µ); Ri (µ)) is upper semicontinuous. If F is simple, then MF (µ) µ
CF (µ).
That the opposite inclusion does not hold, even when F is simple and strong, follows from
the above example for simple majority rule from Austen-Smith and Banks. To see that the
stated inclusion does not hold when F is just weakly decisive, or even when F is strong, go
back to our example of a weakly single-peaked pro¯le and a strict social preference cycle.
There is a simple procedure for ¯nding the F -medians, and therefore a subset of core points,
for a weakly single-peaked pro¯le. You can check that, under the assumptions of Proposition
2.29, x 2 MF (µ) if there is a selection (~
x1; : : : ; x
~n ) such that x
~i 2 M(Ri(µ)) for all i 2 N
1
n
and x is a median with respect to (~
x ;: :: ; x
~ ).
See Sections 4.5 and 4.6 in Austen-Smith and Banks for a di®erent domain restriction,
\order restricted preferences," that can be helpful in analyzing two-dimensional problems.
2.7
The Multi-dimensional Spatial Model
We now impose the assumption that X is a subset of d-dimensional Euclidean space. In
addition, we often impose further structure on the set of alternatives and on individual
preferences. If d = 1, if X is convex, and if preferences are strictly convex, then we are back
to the one-dimensional model with single-peaked preferences. If X is ¯nite, as is allowed in
some results, then we essentially have the discrete choice model. Our main interest here,
however, will be on the multi-dimensional model, with d ¸ 2. In fact, some of the results
in this section assume that X has nonempty interior in Rd , ruling out ¯nite models and
private good exchange economies.
4
Unless one of these end points is 1 or ¡1. Those are not actually points in R, so they are not actually
alternatives.
107
Arrow's Theorem
We ¯rst consider the issue of social rationality, where we impose IIA and allow individual
preferences to vary. Unlike the discrete choice model, however, we can no longer use the
assumptions of linear or unrestricted domain...
[ More to do here. ]
Characterization of the Core
While the core is nonempty and coincides with the F-medians in the one-dimensional spatial
model, the core can be empty in multi-dimensional collective choice problems | even when
the usual regularity conditions hold.
To see this, refer to Figure 2.5, where we have three individuals with Euclidean preferences
over a two-dimensional space. If x is not on the contract curve for individuals 1 and 2, there
is an alternative, y, that both individuals strictly prefer, i.e., yPSM (µ)x. By that logic, a
core alternative would have to be on all three contract curves, which is impossible. Thus,
the simple majority core is empty. Note that this logic applies whenever the three ideal
points form a triangle | no matter how close they are to collinear. Thus, we see that the
simple majority core is \fragile" as well.
Before continuing with the analysis of this issue, we note a useful connection between
the core of a PAR F and the core of the associated simple PAR FD(F ) . We know that
CF (µ) µ CD(F ) (µ) generally. The next proposition establishes the other direction for all
weakly decisive PARs under weak conditions on individual preferences.
Proposition 2.30 Assume X µ Rd and PR(µ) satis¯es LSWP. If F is weakly decisive,
then CF (µ) = CD(F) (µ).
Proof: Take x 2 CD(F ) (µ) and suppose x 2
= CF (µ), i.e., there is some y 2 X such that
yPF (µ)x. By weak decisiveness, G = R(y; xjµ) 2 D(F). By LSWP, there exists z 2 PG (xjµ).
Thus, G µ P (z; xjµ) 2 D(F), so zPD(F) (µ)x, contradicting x 2 CD(F) (µ).
So the cores of weakly decisive PARs are determined by associated simple PARs. This result
justi¯es the terminology \majority core" (without the \simple" or \relative" modi¯er),
which I may use later. Thus, any characterization for the core of a simple rule will apply
to all weakly decisive PARs with the same collection of decisive groups.
Furthermore, Proposition 1.23 immediately yields the following result.
108
Proposition 2.31 Assume X µ Rd and (PF (µ); RF (µ)) satis¯es thin indi®erence. If
CF (µ) 6
= ;, then there exists x 2 X such that CF (µ) = fxg. Furthermore, for all y 2 X nfxg,
we have xPF (µ)y.
Under fairly standard assumptions in the multi-dimensional spatial model, social preferences
satisfy thin indi®erence. Because that is a nice condition to have, the next proposition is
an important (though straightforward) one.
Proposition 2.32 Assume X µ Rd and PR(µ) satis¯es LSWP. If F is strong, then
(PF (µ); RF (µ)) satis¯es thin indi®erence.
Proof: Take y 2 RF (xjµ), so G = R(y; xjµ) 2 D(F ), for otherwise, since D(F) is strong, we
would have P(x; yjµ) 2 D(F ), a contradiction. Suppose y 6
= x and take any open set Y µ X
such that y 2 Y . By LSWP, there exists z 2 PG(xjµ) \ Y . Thus, G µ P (z; xjµ) 2 D(F),
implying z 2 PF (xjµ). Thus, y 2 closPF (xjµ).
A straightforward implication of Propositions 2.31 and 2.32 is that, if LSWP is satis¯ed, if
F is strong, and if the core is nonempty, then it must be a singleton.
Note that the conclusion of Proposition 2.31 is not that CF (µ) consists of the ideal point of
some individual. In fact, that conclusion does not hold generally under the assumptions of
the corollary. Consider simple majority rule in the environment of Figure 2.9.
Figure 2.9: The core is not an ideal point
If we add the assumption that each individual's preferences has a di®erentiable utility
representation, then we can deduce the stronger result.
Proposition 2.33 Assume that X µ Rd and that each (Pi(µ); Ri (µ)) has a di®erentiable
utility representation ui : X ! R such S
that rui(x) = 0 only if x 2 M(R i(µ)). If F is strong
and CF (µ) 6
= ;, then CF (µ) \ intX = i2N M(Ri(µ)).
I'll leave the proof to you. To see that the restrictions on individual preferences are needed
for the latter result, consider the example in the following ¯gure, where x¤ is a relative
majority core point but is no one's ideal point.
The following proposition may also be of interest.
109
Figure 2.10: Core still not an ideal point
Proposition 2.34 Assume X µ Rd. If (RD(F ) (µ); PD(F ) (µ)) satis¯es thin indi®erence,
then so does (PF (µ); RF (µ)).
Proof: Take any x 2 X and y 2 RF (xjµ). If y = x, then clearly y 2 fxg [ closPF (xjµ).
Otherwise, take any open set Y containing y. Note that yRF (µ)x implies yRD(F ) (µ)x. Since
(RD(F ) (µ); PD(F ) (µ)) satis¯es thin indi®erence, there exists z 2 Y such that zPD(F ) (µ)x.
This implies zPF (µ)x, as required.
[ Do medians in all directions here. ]
Nonemptiness of the Core
Recall that, when all pro¯les of weak linear orders are possible, N (F ) > jXj is necessary
for P-acyclicity of social preferences. The next result shows that, with a su±ciently rich
domain d+1 plays the role of the cardinality of jX j: the condition N (F ) > d+1 is necessary
for nonemptiness of the core to hold generally. Note that the result does not impose any
restrictions on the PAR F.
Proposition 2.35 (Scho¯eld; Strnad) Assume that X µ Rd has nonempty interior
and PR(£) ¶ E. If CF (µ) 6
= ; for all µ 2 £, then N (F ) > d + 1.
Proof: I will be somewhat less formal in thisTproof. Suppose d+1 ¸ N (F). Let m = N (F),
and take G1; G2; : : : ; Gm 2 D(F) such that m
j =1 Gj = ;. Since X contains an open set and
d + 1 ¸ m, we can ¯nd m distinct points, z1; z2; : : : ; zm 2 X that are pointwise equidistant
from each other.
Figure 2.11: Equidistant points for d = 3
.
This is depicted in Figure 2.11 for d = 3. We construct a pro¯le of Euclidean preferences
as follows: for each i 2 N, let °(i) = fj j i 2
= Gj g, and de¯ne
x
~i =
X
1
zj :
j °(i) j
j2°(i)
110
T
Note that m
~ i is well-de¯ned. By assumption,
j=1 Gj = ; implies j°(i)j ¸ 1 for all i, so x
there exists µ 2 £ such that each (Pi(µ); Ri (µ)) is Euclidean with ideal point x
~i . Note that
x
~i lies on the face opposite zj if and only if i 2 Gj. Claim: CF (µ) = ;. Why? Take any
x 2 X. There is some face, say that opposite zj , that x is not on. Let y be the point on
that face closest to x. Since preferences are Euclidean, we have yPi (µ)x for all i 2 Gj. Since
Gj 2 D(F), yPF (µ)x.
We will now use previous results, and one new one, to extend this parallel: in many environments, N (F) > d + 1 is su±cient for nonemptiness of the core of a simple PAR.
Recall, from Proposition 1.27, that compactness of X, upper semi-continuity of (P; R), and
Condition F imply M(P) 6
= ;. Recall, from Proposition 1.28, that convexity of X and
upper semi-continuity and semi-convexity of (P; R) imply Condition F.
Applying these results to social preferences, CF (µ) 6
= ; follows if:
² X is compact and convex,
² (PF (µ); RF (µ)) is upper semicontinuous and semi-convex.
We already have a result, Proposition 2.6, that gives a weak su±cient condition on individual
preferences for upper semi-continuity of social preferences: if F is simple, then we need only
assume that each Ri (µ) is upper semicontinuous.
What about semi-convexity? That is addressed next.
Proposition 2.36 Assume that X µ Rd is convex and that, for all i 2 N, (Pi (µ); Ri (µ))
is convex. Let F be a simple PAR. If d + 1 < N (F), then (RF (µ); PF (µ)) is semi-convex.
Proof: By Caratheodory's Theorem, if Y µ Rd and y 2 convY , then we can ¯nd y 0; y 1; : : : ;
yd 2 Y such that y 2 convfy0 ; y 1; : : : ; yd g. Take any x 2 X and suppose that x 2
convPF (xjµ). By Caratheodory's Theorem, there exist y 0; y 1; : : : ; y d 2 PF (xjµ) such that
x 2 convfy0 ; y 1; : : : ; y dg. Since F is simple, for each j T= 0; 1; : : : ; d, we have Gj =
P(yj ; xjµ) 2 D(F ). Since d + 1 < N (F ), we must have dj=1 Gj 6
= ;, i.e., there exists
j
i such that y Pi(µ)x for all j = 0; 1; : : : ; d. But then x 2 convPi(xjµ), contradicting convexity of (Ri (µ); Pi (µ)).
Combining these observations, we have the following result on nonemptiness of the core of
a simple rule.
111
Proposition 2.37 Assume that X µ Rd is compact and convex and that, for all i 2 N,
(Pi(µ); Ri (µ)) is upper semicontinuous and convex. Let F be a simple PAR. If N (F) > d+1,
then CF (µ) 6
= ;.
Using Proposition 2.30, this gives us the following corollary for all weakly decisive PARs.
Corollary 2.7 Assume that X µ Rd is compact and convex and that, for all i 2 N,
(Pi(µ); Ri (µ)) is upper semicontinuous and strictly convex. Let F be weakly decisive. If
N (F) > d + 1, then CF (µ) 6
= ;.
To see the role of semi-convexity in Proposition 2.37, go back to one of our earlier examples:
three voters with Euclidean preferences over a two-dimensional space, ideal points arranged
in a triangle. We can make this space compact and convex. Clearly, individual preferences
are continuous and strictly convex. If simple majority preferences were semi-convex, Proposition 2.37 would yield nonemptiness of the core, which clearly does not hold. Indeed, as
we have noted, simple majority preferences are not semi-convex in this example. If d + 1
were less than N (F), Proposition 2.36 would yield semi-convexity, but you can see that
condition does not hold: d + 1 = 3 = N (F).
As an application of Proposition 2.37, add a fourth individual. Then the Nakamura number
of simple majority rule is N (F) = 4 > 3 = d + 1, and the result tells us that the majority
core is nonempty | regardless of the preferences of the individuals. If we add another
dimension as well, however, the antecedent condition of the proposition is violated.
Symmetry Conditions for Majority Core Points
Note that the condition N (F ) > d + 1 in Proposition 2.35 is necessary for nonemptiness of
the core as µ varies across £, generating all pro¯les of Euclidean preferences. It is possible,
however, that d + 1 ¸ N (F) and CF (µ) 6
= ; in particular states (though there will be other
states in which the core is empty). This is pictured below in two examples: n = 3, which
we have seen before, and n = 5.
Figure 2.12: Nonempty core with d + 1 ¸ N (F).
Our next goal is to understand, given any state, the conditions under which the majority
core is nonempty. We skip some interesting topics, like \medians in all directions" and
the separating hyperplane characterization of core points. (See Austen-Smith and Banks's
Theorem 5.6).
112
Note that, in the last examples, while the majority core was nonempty, it exhibited two
properties: ¯rst, the core point was the ideal point of one individual; second, individual
gradients at the core point were matched against each other in a very precise way. We will
see that these observations generalize.
Given a state µ in which each (Pi(µ); Ri (µ)) has a di®erentiable utility representation ui ,
given a group G µ N, and given an alternative x 2 intX, we say that radial symmetry
holds at x if, for all p 2 Rd ,
jfi 2 N j 9® > 0 : rui(x) = ®pgj = jfi 2 N j 9® > 0 : rui (x) = ¡®pgj:
This is pictured below.
Figure 2.13: Radial symmetry
The next proposition, due to Charlie Plott, uses these concepts to deduce a strong necessary
condition on majority core points.
Proposition 2.38 (Plott) Assume X µ Rd and:
1. n is odd;
2. each (Pi(µ); Ri (µ)) is represented by some di®erentiable ui ;
3. x¤ 2 intX;
4. for all i; j 2 N, rui (x¤) = ruj(x¤ ) = 0 implies i = j (no shared critical points at
x¤ ).
If x¤ 2 CSM (µ), then (i) there is some i 2 N with rui(x¤ ) = 0, and (ii) radial symmetry
holds at x¤.
Proof: To establish (i), suppose x¤ 2 CSM (µ) but rui (x¤) 6
= 0 for all i 2 N. Then there
is at least one p 6
= 0 such that rui (x¤) ¢ p 6
= 0 for all i. (Why?) Since n is odd, either
fi 2 N j rui(x¤ ) ¢ p > 0g
or
fi 2 N j rui (x¤) ¢ p < 0g
is a majority. Without loss of generality, suppose the former. Since x¤ 2 intX , we can pick
² > 0 small enough that x¤ + ²p 2 X. Furthermore, ui (x¤ +²p) > ui(x¤ ) for all i in the ¯rst
set above. (For this, recall that rui (x¤) ¢ p is the derivative of ui in the direction p:
ui (x¤ + ²p) ¡ ui(x¤)
> 0:)
²!0
²
rui (x¤) ¢ p = lim
113
So for small enough ² > 0, this is true for a majority of individuals, so (x¤ + ²p)PSM (µ)x¤ ,
a contradiction. Therefore, rui (x¤) = 0 for some i 2 N. (Note that, for this conclusion, we
used only that D(F ) was strong.) Now suppose that (ii) is false, i.e., there is some p 2 Rd
such that
jfj 6
= i j 9® > 0 : ruj (x¤) = ®pgj 6
= jfj 6
= i j 9® > 0 : ruj (x¤) = ¡®pgj:
Call the ¯rst group G1 and the second G2 . Without loss of generality, suppose jG1j > jG2 j.
If G1 [ G2 [ fig = N, then de¯ne the vector q = 0. Otherwise, d ¸ 2 and we can de¯ne q
as a non-zero vector such that
fj 2 N j ruj (x¤) ¢ q = 0g = G1 [ G2 [ fig:
(Why?) See the ¯gure below.
Figure 2.14: A helpful picture
Therefore,
fj 2 N j ruj (x¤) ¢ q > 0g [ fj 2 N j ruj (x¤) ¢ q < 0g µ Nn(G1 [ G2 [ fig):
Call the ¯rst group on the lefthand side H1 and the second H2, and without loss of generality
suppose jH1j ¸ jH2j. Note that jG1j + jH1j ¸ n+1
2 (why?), so G1 [ H1 is a majority. If we
can ¯nd some y 2 X such that uj (y) > uj (x¤) for all j 2 G1 [ H 1, we have yPSM (µ)x¤, a
contradiction. First, pick ± > 0 small enough that
ruj (x¤) ¢ (q + ±p) > 0
for all j 2 H1. Also, note that
ruj(x¤ ) ¢ (q + ±p) = ±ruj(x¤ ) ¢ p > 0
for all j 2 G1 . Now we can use an argument similar to that used before. Picking ² > 0
small enough, we have x¤ + ²(q + ±p) 2 X and
uj (x¤ + ²(q + ±p)) > uj (x¤)
for all j 2 G1 [ H1 . Setting y = x¤ + ²(q + ±p), we have yPSM (µ)x¤, a contradiction, and
the proof is complete.
Does Plott's theorem need n odd? Yes. See Figure 2.15.
Note that, in Figure 2.15(a), x¤ satis¯es (i) but violates (ii). In Figure 2.15(b), x¤ satis¯es
(ii) but violates (i). Is it possible to ¯nd an example in which x¤ violates both (i) and (ii)?
As the following proposition makes clear, no.
114
Figure 2.15: n even
Proposition 2.39 Assume X µ Rd and:
1. each (Pi(µ); Ri (µ)) is represented by some di®erentiable ui ;
2. x¤ 2 intX;
3. for all i; j 2 N , rui(x¤) = ruj (x¤) = 0 implies i = j.
If x¤ 2 CSM (µ), then either (i) there is some i 2 N with rui(x¤ ) = 0, or (ii) both rui(x¤ ) 6
=
0 for all i 2 N and radial symmetry holds at x¤.
Proof: I will sketch the proof. Let x¤ 2 CSM (µ), and suppose neither (i) nor (ii) hold.
Add another individual, say n + 1, giving us an odd number, with ideal point at x¤. Thus,
run+1(x¤) = 0. Since (i) does not hold, no other individual has a zero gradient at x¤ .
By Proposition 2.38, then, if x¤ is a majority core point for this augmented society, radial
symmetry must hold. Since (ii) does not hold, x¤ is not a majority core point, i.e., there
is a majority, say G, of individuals who strictly prefer some y 2 X to x¤ . By construction,
G µ N. Returning to the original model, we have yPSM (µ)x¤, a contradiction.
Does Plott's theorem need no shared critical points at x¤? Yes. See Figure 2.16 for an
example.
Figure 2.16: Shared critical points at x¤
Can you see how you could extend Plott's theorem to account for shared critical points?
(We would need to de¯ne a weaker type of radial symmetry.)
Though the ¯rst part of Plott's theorem generalizes to all strong PARs, the second part
clearly does not: consider Fi, the PAR de¯ned as the preference relation of individual i.
Indeed, the second part does not generalize even to non-dictatorial PARs with D(F) strong.
Let n = 5, and let D(F) consist of every group containing individual 1 and some other i 6
=1
and the group f2; 3; 4; 5g. Consider preferences as in Figure 2.17.
Figure 2.17: Non-dictatorial PAR, D(F) strong
Note that, under the conditions of Plott's theorem, x¤ must be a critical point of some
individual's utility function. To see that x¤ need not be anyone's ideal point, consider the
115
example of Figure 2.10, following Proposition 2.33. Note that, as that proposition implies,
some individual has non-convex preferences in the example.
At any rate, we have a strong necessary condition for x to be a majority rule core point
when n is odd. The next result addresses su±ciency. We say ui is pseudo-concave if it is
di®erentiable and, for all x; y 2 intX, ui (y) > ui (x) implies rui(x) ¢ (y ¡ x) > 0. Note that
pseudo-concavity implies di®erentiability and quasi-concavity (why?), but not the converse.
(Think about ui(x) = x3 .)
Proposition 2.40 (Plott) Assume X µ Rd and each (Pi (µ); Ri(µ)) has a pseudo-concave
utility representation ui . Let x 2 intX. If radial symmetry holds at x, then x 2 CSM (µ).
Proof: Suppose radial symmetry holds at x but yPSM (µ)x for some y. Then, by pseudoconcavity,
P(y; xjµ) µ fi 2 N j rui (x) ¢ (y ¡ x) > 0g:
The latter group is a majority. But then fi 2 N j rui (x) ¢ (y ¡ x) < 0g is a minority,
contradicting radial symmetry.
Thus, radial symmetry essentially characterizes the majority rule core points. Note that,
unlike Proposition 2.38, Proposition 2.40 does not assume n is odd. How does this su±ciency
result generalize beyond majority rule? I'll let you think about it.
Majority Core Usually Empty
Plott's Theorem tells us that, when n is odd (etc.), it is very di±cult to ¯nd majority
rule core points. Indeed, not only are core points rare, but, even when they exist, they
are \fragile." That is, a minor perturbation of individual utility functions can remove an
alternative from the core. We will prove formally that, for our metric on £, the set of states
in which the majority core is empty is \generic," de¯ned as follows. A set £0 µ £ is dense
if, for every µ 2 £ and every ² > 0, B²(µ) \ £0 6
= ;. In other words, clos£0 = £. Then we
0
say £ is generic if it contains an open, dense subset of £.
Equivalently, £0 is generic if its complement £ n £0 is contained in a closed set with empty
interior, i.e., clos(£ n £0 ) contains no open set.
We ¯rst need the following result on upper hemicontinuity of the core. The result assumes
continuity of the mapping RF from states to social weak preference relations. Note that,
from Proposition 2.7, we have conditions under which this is satis¯ed by all simple PARs.
116
Proposition 2.41 Assume X µ Rd is compact, PR(£) µ RN
c , and RF (£) µ Rc . Let
RF : £ ! Rc be continuous at µ. Then CF (µ) is closed and CF is upper hemicontinuous at
µ.
Proof: Take any open set U with CF (µ) µ U, and note that CF (µ) = M(RF (µ)). We know
from Proposition 1.49 that M is upper hemicontinuous on Rc, so there exists an open set
W containing RF (µ) such that, for all R 2 W , M(R) µ U. Assuming RF is continuous
at µ, there is an open set V containing µ such that, for all µ0 2 V , RF (µ0 ) 2 W . Thus,
CF (µ0 ) µ U for all µ0 2 V , as required. I'll let you prove that CF (µ) is closed.
The next proposition establishes weak conditions under which the set of states for which
the core is nonempty is closed in the metric ½. Given a PAR F , let £F µ £ be the subset
of states for which CF (µ) 6
= ;.
Proposition 2.42 Assume X µ Rd is compact, PR(£) µ RN
c , and RF (£) µ Rc . Let
RF : £ ! Rc be continuous. Then £F is closed.
Proof: Consider a sequence fµm g of states and a state µ such that µ m 2 £F for all m and
µm ! µ. For each m, let xm be an element of the core of the mth state, i.e., xm 2 CF (µm ).
Since X is compact, the sequence fxm g has a subsequence that converges to some x 2 X.
Without loss of generality, index this subsequence by m. By Proposition 2.41, x 2 CF (µ),
so µ 2 £F .
Of course, I eventually want to argue that the set £n£F is generic. Since £F is closed, this
amounts to showing that £F contains no open set. The next lemma, on nowhere denseness
of unions of closed sets, will be helpful. It is taken from McKelvey (1979).
Lemma 2.7 Let X µ Rd be endowed with the relative topology, and let Y µ X . If Y =
Sn
i=1 Yi, where each Yi µ X is closed and has empty interior, then Y is closed and has
empty interior.
In contrast to Proposition 2.42, the next result will depend on our speci¯cation of the
domain of preferences. Di®erent authors have considered di®erent restrictions. We will use
a subset of those in Plott's theorem. We say ui is strictly pseudo-concave if it is di®erentiable
and, for all x; y 2 intX with x 6
= y, ui (y) ¸ ui (x) implies rui(x) ¢ (y ¡ x) > 0. Note that,
if (Pi; Ri) has a strictly pseudo-concave representation, then it is strictly convex (but not
conversely).
Let Rsp µ RN
c denote the set of preference pro¯les such that each (Pi ; Ri ) is representable by
some strictly pseudo-concave ui : Rd ! R, with no shared critical points. 5 Let £±SM µ RN
c
5
Did you notice that I'm assuming ui is de¯ned over all of Rd ? Can you see where I use this in the next
proof ? This is cheating a little, but what the heck.
117
denote the set of pro¯les with a majority core point interior to X. For simplicity, I will
show that £±SM , rather than £SM , contains no open sets.
Proposition 2.43 Assume n is odd and X µ Rd is compact and convex with d ¸ 2,
±
PR(£) = RN
sp, and RSM (£) µ Rc . Then £ n £SM is dense in £ with respect to the
Hausdor® metric.
Proof: Take any µ 2 £±SM , where ui is a strictly pseudo-concave representation of (Pi(µ);
Ri(µ)). Thus, CSM (µ) contains an element interior to X , say x¤. Note that, with our
assumptions that n is odd and that each ui is strictly pseudo-concave, Proposition 2.33
implies that CSM (µ) = f~
xig for some i 2 N. Thus, x¤ is the unique core point and x¤ = x
~i .
Furthermore, radial symmetry must hold at x¤. Let Cjh µ X be the alternatives that are
\Pareto optimal for j and h," i.e.,
Cjh = fx 2 X j 8y 2 X : xRj (µ)y or xRh(µ)yg;
and note that the set of interior points at which radial symmetry holds is contained in
S
C = j;h6
=i Cjh. Furthermore, each Cjh is closed and, since d ¸ 2, is nowhere dense.
(Why?) By Lemma 2.7, C is nowhere dense. For arbitrary integer m, we can therefore
¯nd an interior point zm within 1=m of x¤ such that z m 2
= C. De¯ne the utility function
m
m
¤
ui (x) = ui (x ¡ z + x ). This is a new strictly pseudo-concave representation such that
m
m
m
m
m
rum
i (z ) = 0, i.e., the unique maximizer of ui is z . Let µ 2 £ be a state such that ui
m
is a representation of Ri (µ ) and the preferences of other individuals are unchanged. The
weak preference Ri(µm ) can be made arbitrarily close to Ri (µ) in by a suitable choice of m.
(Check this.) I claim that, for high enough m, CSM (µ m) \ intX = ;. To see this, suppose
there exists xm 2 CSM (µm ) for in¯nitely many m. By Proposition 2.38, we have xm = x
~j
m
for some j 2 N . By construction, the radial symmetry condition does not hold at z , so
we must have j 6
= i. Since N is ¯nite, we may assume, going to a subsequence if needed,
m
j
that x = x
~ for all m. By Proposition 2.41, CSM has closed graph, so x
~j 2 CSM (µ). But
¤
this contradicts our assumption that CSM (µ) = fx g. Thus, we can perturb µ to obtain an
empty core, as required.
2.8
The Weak Top Cycle
Recall the de¯nition of the weak top cycle set: given a dual pair (P; R), it is T OP (R) =
M(TR). That is, it is the maximal elements of the transitive closure of weak preference.
We have already seen this idea in the context of collective choice with the de¯nition of the
PAR FT , the transitive closure of weak simple majority preference. The core of this PAR
is CT (µ) = TOP(RSM (µ)), where we use the social weak preference RSM (µ) for R.
We can apply this de¯nition more generally to any PAR. Given a PAR F , de¯ne the weak
top cycle of F at µ by W TOPF (µ) = TOP(RF (µ)).
118
Properties of the Weak Top Cycle
Recall some properties of the weak top cycle applied to social preferences, where WF (µ) is
the union of PF (µ)-dominant RF (µ)-cycles.
² W TOPF (µ) = CF (µ) [ WF (µ).
² The weak top cycle is externally PF (µ)-stable and, if Y µ X is externally PF (µ)-stable,
then W T OPF (µ) µ Y .
² Assume CF (µ) 6
= ;. If (PF (µ); RF (µ)) satis¯es thin indi®erence, then WT OPF (µ) =
CF (µ).
² If X is compact and RF (µ) is upper semicontinuous, then W T OPF (µ) 6
= ;.
The last item listed above is signi¯cant: whereas we have seen that CSM (µ) is typically
empty, the weak top cycle for majority rule will be nonempty under very general conditions.
Indeed, this is true for all simple rules.
Proposition 2.44 Assume X µ Rd is compact and each (Pi(µ); Ri (µ)) is upper semicontinuous. Let F be a simple PAR. Then WT OPF (µ) 6
= ;.
The second-to-last item above relies on the property of thin indi®erence of social preferences,
a su±cient condition for which was presented in Proposition 2.32. This yields the following
result.
Proposition 2.45 Assume X µ Rd and PR(µ) satis¯es LSWP. If F is strong and CF (µ) 6
=
;, then W TOPF (µ) = CF (µ).
As a consequence, we have conditions under which the weak top cycle and core (when
nonempty) coincide. In particular, the conditions are ful¯lled, under convexity assumptions,
by the simple majority PAR when n is odd.
The above result on nonemptiness of the weak top cycle is stated for simple PARs, which
have the required continuity properties. But relative majority rule and relative Pareto are
obvious examples of PARs that do not generally generate continuous social preferences.
Such PARs are captured through the following proposition. It shows that, under fairly
weak conditions on individual preferences, weak decisiveness is enough.
Proposition 2.46 Assume that X µ Rd , that each (Pi(µ); Ri (µ)) is continuous, and that
PR(µ) satis¯es LSWP. Let F be weakly decisive. Then WT OPF (µ) 6
= ;.
119
Proof: From Proposition 2.6, it follows that (PD(F ) (µ); RD(F) (µ)) is continuous, and then
Proposition 1.40 implies that the corresponding strong top cycle is nonempty, i.e.,
T OP (PD(F ) (µ)) 6
= ;:
I claim that this strong top cycle is a subset of the weak top cycle of F at µ, i.e.,
T OP(PD(F ) (µ)) µ TOP(RF (µ)):
The claim is clearly true, by Proposition 2.30, if CD(F ) (µ) 6
= ;, so suppose CD(F ) (µ) = ;.
Take any x 2 TOP(PD(F) (µ)). Proposition 1.34 implies that x is an element of some
undominated PD(F ) (µ)-cycle, say Z, and by de¯nition there must exist y; z 2 Z such that
xPD(F ) (µ)yPD(F) (µ)z. To see that x 2 TOP(RF (µ)), take any w 2 X. If w 2 Z, then,
because x 2 Z, we have xTPD(F) (µ) w, which implies xTRF (µ) w. Suppose w 2
= Z, so by
¡1
de¯nition we have yRD(F) (µ)w. Since (PD(F ) (µ); RD(F) (µ)) is continuous, PD(F)
(xjµ) \
¡1
0
PD(F) (zjµ) is an open set containing y. By LSWP, there exists y 2 PD(F ) (xjµ) \ PD(F ) (zjµ)
such that R(y; wjµ) = P (y 0 ; wjµ) and P(w; yjµ) = P(w; y0 jµ). (Do you see why?) We then
have y 0RF (µ)w: otherwise we would have wPF (µ)y0 , and then weak decisiveness would imply
that P (w; y0 jµ) = R(w; y 0 jµ) 2 D(F), but then we would have wPD(F ) (µ)y 0 , implying w 2 Z,
a contradiction. Thus, xPD(F ) (µ)y 0 RF (µ)w, which implies xTRF (µ) w, as required.
We can say more if we add the assumption that F is strong. In fact, it is enough if social
preferences are upper semicontinuous and satisfy thin indi®erence: then the weak top cycles
of F and FD(F) di®er only at points of closure. Note that the proposition actually gives
more: the weak top cycles of these PARs essentially coincide with the strong top cycles.
Proposition 2.47 Assume X µ Rd. If (PD(F) (µ); R D(F ) (µ)) is upper semicontinuous and
satis¯es thin indi®erence, then
TOP(PD(F) (µ)) µ T OP (PF (µ)) µ W TOPF (µ) µ W TOPD(F) (µ) µ closTOP(PD(F ) (µ)):
Proof: The result follows easily from Proposition 2.31 when CD(F ) (µ) 6
= ;. Otherwise,
recall that, by Propositions 1.38 and 1.39, we have
D(TPD(F )(µ) ) = TOP(PD(F) (µ)) µ WT OPD(F ) (µ) µ closTOP(PD(F) (µ)):
Since
D(TPD(F )(µ) ) µ D(TP F (µ) ) µ T OP (PF (µ)) µ W TOPF (µ) µ W TOPD(F ) (µ);
the desired inclusions follow.
120
The weak top cycle will be nonempty even when the core is empty, presenting an alternative
approach to placing bounds on collective choices. As we will see, however, the bounds
imposed by weak top cycle are typically not very restrictive: when the core is nonempty,
W TOPF (µ) can be quite large.
Extent of the Weak Top Cycle
For simplicity, we now assume X = Rd and consider a state µ in which individual preferences
are Euclidean, i.e., for each i 2 N, there exists x
~i 2 X such that xRi(µ)y if and only if
i
i
jjx ¡ x
~ jj · jjy ¡ x
~ jj.
We note that, if (Pi(µ); Ri (µ)) is Euclidean, then the following statements are equivalent to
xPi (µ)y:
jjx ¡ x
~i jj
jjx ¡ x
~ijj 2
(x ¡ x
~i) ¢ (x ¡ x
~ i)
x ¢ x ¡ 2x ¢ x
~i + x
~i ¢ x
~i
x ¢x¡ y ¢y
1
(x + y) ¢ (x ¡ y)
2
<
<
<
<
<
jjy ¡ x
~ ijj
jjy ¡ x
~ ijj2
(y ¡ x
~i ) ¢ (y ¡ x
~i )
y ¢ y ¡ 2y ¢ x
~i + x
~i ¢ x
~i
2~
xi ¢ (x ¡ y)
< x
~i ¢ (x ¡ y):
Geometrically, think of x ¡ y as the gradient of a linear function. The left hand side is the
value of the function at 12 (x + y). The inequality says x
~i must be on the (x ¡ y)-side of
1
the hyperplane with normal x ¡ y through 2 (x + y), which we refer to as the \bisecting
hyperplane" for x and y.
We can use this observation to derive a useful characterization of the core. We call a vector
y 2 Rd a direction if jjyjj = 1. Given a direction y 2 Rd and c 2 R, let
Hy;c = fx 2 Rd j x ¢ y = cg
+
Hy;c
= fx 2 Rd j x ¢ y > cg
+
H y;c = fx 2 Rd j x ¢ y ¸ cg
¡
¡ and H
De¯ne Hy;c
y;c similarly.
Given a PAR F , we say Hy;c is a F -median hyperplane at µ if
+
¡
fi 2 N j x
~ i 2 Hy;c
g2
= D(F) and fi 2 N j x
~i 2 Hy;c
g2
= D(F):
Equivalently, when F is simple, Hy;c is a F -median hyperplane at µ if
¡
+
fi 2 N j x
~i 2 H y;cg 2 B(F) and fi 2 N j x
~i 2 H y;c g 2 B(F ):
121
Let MF (µ) denote the set of F-median hyperplanes at µ. Note that, for all y 2 Rd , there
exists c 2 R such that Hy;c 2 MF (µ). (Why?)
When F is strong, for every direction y 2 Rd , there is exactly one F-median hyperplane with
normal y. (Why?) Generally, assuming F satis¯es Pareto, there will be a compact interval,
[c0 ; c 00], such that Hy;c is a F-median hyperplane with normal y if and only if c 2 [c0 ; c00 ].
Denote this interval by Cy (suppressing dependence on F and µ). Note that Cy = ¡C¡y
(because y ¢ x = c if and only if (¡y) ¢ x = ¡c).
We say x¤ is a total F-median at µ if, for all y 2 Rd , Hy;x¤¢y 2 MF (µ). In other words, x¤
is a total F-median at µ if and only if, for all y 2 Rd, y ¢ x¤ 2 Cy . Let T MF (µ) denote the
set of total F-medians at µ.
The next proposition is well-known and straightforward to prove.
Proposition 2.48 Assume that X µ Rd , that each (Pi (µ); Ri(µ)) is Euclidean, and that
x¤ 2 intX.
1. For every PAR F , x¤ 2 CF (µ) implies x¤ 2 T MF (µ).
2. If F is weakly decisive, then x¤ 2 T MF (µ) implies x¤ 2 CF (µ).
Proof: For part 1, suppose x¤ 2 CF (µ) but is not a total median. Then there exists y 2 Rd
such that Hy;x¤ ¢y 2
= MF (µ). Without loss of generality, suppose the individuals with ideal
+
points in Hy;x¤¢y are decisive. For ² > 0, let z² = x¤ + ²y. Note that, for each i 2 N , z² is
closer than x¤ to x
~ i if and only if
1
(z² + x¤ ) ¢ (z² ¡ x¤) < x
~ i ¢ (z² ¡ x¤ );
2
or equivalently,
²
x¤ ¢ y + y ¢ y < x
~i ¢ y:
2
(2.1)
If i is such that x
~i ¢ y > x¤ ¢ y, then this inequality holds for small enough ². Therefore,
+
choosing ² > 0 small enough that z² 2 X and (2.1) holds for all i with x
~i 2 Hy;x
¤ ¢y , we have
¤
z²PF (µ)x , a contradiction. I'll leave part 2 for you.
Before proceeding, I give a lemma on solutions to systems of weak inequalities: if fyj ¢ x ¸
cj j j = 1; 2; : : : ; kg is a system with no solution, and if, after deleting any one of the
inequalities, the remaining ones admit a solution, then the remaining ones can be solved
with equality.
122
Let K = f1; 2; : : : ; kg, and let J denote a subset of K. For y1 ; y2; : : : ; yk 2 Rd and
c1; c2; : : : ; ck 2 R, let S(J) = fx 2 Rd j 8j 2 J : y j ¢ x ¸ cj g denote the set of solutions to the J inequalities. Let S =(J) = fx 2 Rd j 8j 2 J : yj ¢ x = cj g denote the set of
solutions which solve the inequalities with equality.
Lemma 2.8 Let y1 ; y2 ; : : : ; yk 2 Rd, and let c1; c2; : : : ; ck 2 R. Assume that S(K) = ; and
that, for all j 2 K, S(K n fjg) 6
= ;. Then, for all j 2 K, S =(K n fjg) 6
= ;.
Proof: Suppose that, for some j 2 K, S= (K n fjg) = ;. Note that the maximization
problem
maxx2Rd yj ¢ x
s.t. y` ¢ x ¸ c ` (` 6
= j)
has a solution, say x
^. Furthermore, note that yj ¢ x
^ < c j. (Why?) I claim that y` ¢ x
^ = c`
for all ` 6
= j. If not, some constraint, say h, is not binding at x
^, i.e., yh ¢ x
^ > ch. Then
S(K n fhg) = ;: if x 2 S(K n fhg), then, for small enough ² > 0, x² = (1 ¡ ²)^
x + ²x solves
the constraints in the above maximization problem (why?), and, since yj ¢ x ¸ cj > y j ¢ x
^,
we have
yj ¢ x² = (1 ¡ ²)yj x
^ + ²yj ¢ x = yj ¢ x
^ + ²(yj ¢ x ¡ y j ¢ x
^) > yj ¢ x
^;
a contradiction. But S(K n fhg) 6
= ; by assumption, a contradiction.
The next result was proved by McKelvey (1976) for the special case of simple majority
rule. I give a version of it that covers all simple PARs and other PARs such as relative
majority rule. Referred to as a \chaos" theorem, it shows that the weak top cycle (in
fact, the strong top cycle, if you look at the proof carefully) exhausts the entire space of
alternatives. The result assumes X = Rd and Euclidean preferences. McKelvey (1979)
drops those assumptions, adding, however, the assumption that F is strong.
Proposition 2.49 (McKelvey) Assume X = Rd and each (Pi(µ); Ri (µ)) is Euclidean.
Let F be weakly decisive. If CF (µ) = ;, then W TOPF (µ) = X.
Proof: Assume CF (µ) = ;, and take any v; w 2 X. We will construct a sequence t0; t1; : : : ;
tk 2 X such that vRF (µ)tk RF (µ)tk¡1 ¢ ¢ ¢ RF (µ)t0 = w. Since the core is empty and F is
weakly decisive, F must satisfy Pareto, so Cy is a compact interval for every direction y.
Let cy = min Cy , and de¯ne Hy = Hy;cy . Since the core is empty, Proposition 2.48 implies
that there is no total median, so
\
+
H y = ;:
y 2 Rd
jjyjj = 1
123
+
+
This follows because, for all y 2 Rd, x 2 H y \ H ¡y implies
min Cy = c y · x ¢ y · ¡c¡y = ¡ min C¡y = max Cy :
By Helly's theorem (Rockafellar (1970), Corollary 21.3.2), there are d+1 vectors y0 ; y1; : : : ; yd
with no common solution x to
x ¢ yi ¸ cyi ;
i = 0; 1; : : : ; d. Let ci = cyi . Pick a minimal subset with this property, and, without loss of
generality, assume that subset is y 0; y 1; : : : ; yp. So, for all j = 0; 1; : : : ; p, there is a solution
to
x ¢ y i ¸ ci (i 6
= j):
Dropping the jth inequality, Lemma 2.8 allows us solve the remaining p inequalities with
equality. Let zj solve them, i.e.,
zj ¢ yi = ci (i 6
= j);
(2.2)
Pp
1
and note that zj ¢ yj < c j . Set z = p+1
j=0 zj , and assume without loss of generality that
z = 0. Then c i > 0 for all i = 0; 1; : : : ; p, since
0 = (p + 1)z ¢ yi =
p
X
j=0
zj ¢ yi = pc i + zi ¢ yi < (p + 1)ci ;
where the third equality uses (2.2), and the inequality uses zi ¢ yi < ci . See Figure 2.18.
Figure 2.18: What's going on?
Note that, for all x 2 Rd , there is some i = 0; 1; : : : ; p such that x ¢ yi · 0 (or else we could
solve all p + 1 inequalities). Given any tk , we de¯ne t k+1 as follows: pick any y i such that
tk ¢ yi < 0 (there is at least one), and let
tk+1 = tk + [ci ¡ 2yi ¢ tk ]yi :
Note that the projection of tk onto the subspace Hyi;0 is equal to the projection of t k+1
onto this subspace. Denote it by s. Furthermore, you can check that
tk = s + (yi ¢ tk )yi
t k+1 = s ¡ (yi ¢ tk + ci )y i:
What's going on here? See Figure 2.19.
124
Figure 2.19: What's going on?
That tk+1 PF (µ)tk follows since, letting c 0 = 12 (tk + tk+1 ) ¢ yi, we have
ci
c0 = (t k + ( ¡ yi ¢ tk )y i) ¢ yi
2
ci
=
2
< ci;
which implies c0 2
= Cyi , and so Hyi;c0 2
= MF (µ). Also, by the Pythagorean theorem, we have
jjtk+1jj2 = (ci ¡ tk ¢ yi )2 + jjsjj2
jjtk jj2 = (tk ¢ yi)2 + jjsjj2:
Thus,
jjt k+1 jj2 ¡ jjtk jj2 = c 2i ¡ 2(tk ¢ yi )ci
¸ c 2i :
In particular, the squared norm of tk+1 is larger than that of tk by a discrete amount. By
taking k high enough, we may construct a sequence satisfying
tk PF (µ)tk¡1 ¢ ¢ ¢ PF (µ)t2PF (µ)t 1PF (µ)t0 = w
and such that jjtk jj2, and therefore jjtk jj, is arbitrarily large. Making jjt k jj big enough, we
can ensure vRF (µ)t k (how?), as required.
The following corollary is (nearly) immediate. You should try to prove it.
Corollary 2.8 Assume X = Rd and PR(£) ¶ E. If d + 1 ¸ N (F), then WT OPF is not
upper hemicontinuous.
What about Non-Euclidean Preferences?
McKelvey's later theorem is not proved constructively. There, he takes a strong simple rule
F and an arbitrary alternative x 2 X, and he considers the set of alternatives that are
socially preferred to x directly or indirectly, in a ¯nite sequence of social preference steps.
Letting P = PF (µ), this is the set
TP (x) = fy 2 X j yTP xg:
125
Assuming individual preferences are continuous, PF (zjµ) is open for all z 2 X, and therefore
TP (x) is open. (Why?) Suppose the boundary of TP (x) is nonempty. We can show that,
for each y and z in the boundary of TP (x), it must be that yIF (µ)z. (Same question.)
By a clever argument using a \preference diversity" assumption, McKelvey shows that
there is some individual j such that the boundary (well, \frontier") of TP (x) lies within
an indi®erence class of j. Finally, take i 6
= j and alternatives z and w in the boundary of
TP (x) such that zIiw. Then, under weak conditions, there must be another individual k
such that zIk w.
We conclude that the necessary conditions that must be met by boundary points of TP (x)
are very restrictive. So restrictive that, typically, the boundary of TP (x) will be empty.
This essentially means, assuming X is connected, that either the core is nonempty or the
set TP (x) comprises almost all of X. So, if the core is empty, then W TOPF (µ) is almost
the entire space. In fact, McKelvey shows this for the strong top cycle. This is also referred
to as a \chaos" result.
I will present a version of McKelvey's general chaos result that is stronger than his in the
sense that we do not need F to be strong or even simple | merely weakly decisive | but
weaker in that we prove the expansiveness of the weak top cycle, rather than the strong. In
this latter respect, it is similar to the conjunction of Austen-Smith and Banks's Theorems
6.5 and 6.6. It di®ers from theirs in that they restrict attention to simple PARs (and,
for simplicity, they impose X = Rd and convexity of individual preferences). Results on
expansiveness of the weak top cycle do, however, extend to the strong top cycle when thin
social indi®erence holds: by Proposition 2.47, the strong top cycle contains the weak within
its closure. All of these results rely on a rather complicated restriction on decisive coalitions
related to a certain individual, which I simplify somewhat.
Before proceeding to that result, I will establish a much weaker conclusion without these
ancillary assumptions. In the following, let TF (µ) = TRF (µ) be the transitive closure of social
weak preference at µ, and let TF (xjµ) be the upper section of TF (µ) at x.
Proposition 2.50 Assume X µ Rd and each (Pi(µ); Ri (µ)) is continuous. Let F be weakly
decisive. For all x 2 X and all y; z 2 bdTF (xjµ), there exists i 2 N such that yIi(µ)z.
Furthermore, if Z \ Pi¡1(zjµ) \ TF (xjµ) 6
= ; or (Z \ Pi(zjµ)) n TF (xjµ) 6
= ; for every open
set Z around z, then there exists j 6
= i such that yIj (µ)z.
Proof: First suppose there is no i 2 N such that yIi(µ)z, i.e., P(y; zjµ) [ P(z; yjµ) = N .
By continuity of individual preferences, and since y; z 2 bdTF (xjµ), there exist y0 2 TF (xjµ)
and z0 2 X n TF (xjµ) such that P (y 0 ; z0 jµ) = P (y; zjµ) and P(z0 ; y 0 jµ) = P(z; yjµ). I claim
that z 0 RF (µ)y0 . If not, then y 0PF (µ)z0 , and, by weak decisiveness, R(y0 ; z 0 jµ) = P (y; zjµ) 2
D(F). Now take y^ 2 X n TF (xjµ) and z^ 2 TF (xjµ) such that P (^
y ; z^jµ) = P(y; zjµ) and
P(^
z; y^jµ) = P(z; yjµ). Then P(^
y ; z^jµ) 2 D(F), which implies y^PF (µ)^
z . But then y^ 2
TF (xjµ), a contradiction. Therefore, there exists i 2 N such that yIi(µ)z. Suppose there is
126
no j 6
= i such that yIj (µ)z, i.e., P(y; zjµ) [ P(z; yjµ) = N n fig. By continuity, there exists
an open set V around z such that V µ Pj (yjµ) for all j 2 P(z; yjµ) and V µ Pj¡1 (yjµ) for all
j 2 P (y; zjµ). If (Z \ Pi(zjµ)) n TF (xjµ) 6
= ; for every open set Z around z, then there exists
0
z 2 (V \ Pi(zjµ)) n TF (xjµ). By continuity, since z0 Pi (µ)y and y 2 bdTF (xjµ), there exists
y0 2 TF (xjµ) such that P(y 0; z0 jµ) = P(y; z 0jµ) = P (y; zjµ) and P (z 0; y 0jµ) = P (z 0; yjµ) =
P(z; yjµ) [ fig. See Figure 2.20.
Figure 2.20: What's going on?
I claim that z 0R F (µ)y 0 . Otherwise, we have y 0PF (µ)z0 , and, by weak decisiveness, R(y0 ; z0 jµ) =
P(y; zjµ) 2 D(F). By continuity and y; z 2 bdTF (xjµ), there exist y^ 2 X n TF (xjµ) and
z^ 2 TF (xjµ) such that P (^
y ; ^zjµ) = P (y; zjµ), implying y^PF (µ)^
z . But then y^ 2 TF (xjµ),
0
0
a contradiction. Therefore, z RF (µ)y , as claimed. But then z0 2 TF (xjµ), a contradiction. If Z \ Pi¡1(zjµ) \ TF (xjµ) 6
= ; for every open set Z around z, then there exists
z00 2 V \ Pi¡1 (zjµ) \ TF (xjµ). By continuity, since yPi (µ)z 00 and y 2 bdTF (xjµ), there exists y 00 2 TF (xjµ) such that P(y 00 ; z00 jµ) = P (y; z 00 jµ) = P(y; zjµ) [ fig and P(z00 ; y 00 jµ) =
P(z; yjµ). See Figure 2.20. I claim that y00 RF (µ)z00 . Otherwise, we have z 00 PF (µ)y00, and, by
weak decisiveness, R(z00 ; y00 jµ) = P(z; yjµ) 2 D(F). This leads to a contradiction as above,
establishing the claim. But then y00 2 TF (xjµ), a contradiction.
Thus, if TF (xjµ) has two distinct boundary points, then some individual must be indi®erent
between them. And if that individual's indi®erence curve crosses the boundary of TF (xjµ)
at those points in a \transversal" way, then some other individual's indi®erence curve also
must cross at those two points. This observation can have very restrictive implications when
jbdTF (xjµ)j ¸ 2 for some x 2 X, depicted in Figure 2.21, so we would expect that, for all
x 2 X, jbdTF (xjµ)j = 0 or 1.
Figure 2.21: Restrictive implications
But the condition jbdTF (xjµ)j = 1 is possible, essentially, only if the core is a singleton and
is externally PF (µ)-stable. (There are some other special cases where this can happen. Can
you think of any?) And when X is connected, jbdTF (xjµ)j = 0 is only possible if either
closTF (xjµ) = ; or closTF (xjµ) = ;. The former cannot be, since x 2 TF (xjµ), and the
latter implies TF (xjµ) = X. Thus, as the next proposition states, the weak top cycle can
be quite large.
Proposition 2.51 Assume X µ Rd is connected and bdTF (xjµ) = ; for all x 2 X. Then
W TOPF (µ) = X.
127
The next proposition ampli¯es these conclusions for a special case of the one-dimensional
spatial model. The result applies to simple majority rule with n even, i.e., q = (n=2) + 1,
and it shows that the weak top cycle may be very large, even when the core is nonempty.
Proposition 2.52 Assume X = R, each (Pi(µ); Ri (µ)) is Euclidean, and, for all distinct
i; j 2 N, x
~i 6
=x
~j . Let Fq be a quota rule with q > (n + 1)=2. Then W TOPq (µ) = R.
Proof: By Proposition 2.51, it su±ces to show that jbdTq (xjµ)j = 0 for all x 2 X. Recall
that, by Proposition 2.27, we have Cq (µ) = [~
xi1 ; x
~i2 ] for some i1; i2 2 N. By the assumption
that ideal points are distinct and q > (n + 1)=2, it must be that x
~i1 < ~xi2 . Thus, for each
i1 i2
x 2 X, we see that Tq (xjµ) contains a non-degenerate interval [~
x ;x
~ ]. There are two cases
to rule out: jbdTq (xjµ)j = 1 and jbdTq (xjµ)j ¸ 2. In the ¯rst case, let bdTq (xjµ) = fyg,
and assume without loss of generality that there exists z 2 Tq (xjµ) such that y < z. Pick
w 2 X high enough that z < w and P (y; wjµ) = N. If w 2 Tq (xjµ), then there is an open
set Y around y such that Y µ Pq (wjµ) µ Tq (xjµ), contradicting our assumption that y 2
bdTq (xjµ). Thus, w 2
= Tq (xjµ). But then clos(Tq (xjµ))\[z; w] 6
= ; and clos(Tq (xjµ))\[z; w] 6
=
;. Since [z; w] is connected, we must have bdTq (xjµ) \ [z; w] 6
= ;, implying jbdTq (xjµ)j > 1,
a contradiction. In the second case, take distinct y; z 2 bdTq (xjµ). Assume without loss of
generality that jy ¡ x
~i1 j < jz ¡ x
~ i1 j. It follows that there is some open set Y around y and
some open set Z around z such that this inequality holds for all y 0 2 Y in place of y and
all z0 in place of z. Then our assumption of Euclidean preferences implies that y0 Rq (µ)z0
for all y 0 2 Y and all z0 2 Z. (Why?) Since Z \ Tq (xjµ) 6
= ;, we have y 2 intTq (xjµ), a
contradiction.
The conclusion of Proposition 2.52 obviously does not carry over the the case where F is
strong, in which case the core is a singleton and coincides with the weak top cycle set.
That it does not extend generally to non-Euclidean preference pro¯les can be seen in the
following example.
Figure 2.22: Bounded weak top cycle
We now proceed to a version of McKelvey's general chaos theorem.
To formalize McKelvey's main assumption on individual preferences, de¯ne the frontier of
Y µ X, denoted frY , as
frY
= clos(intY ) \ clos(intY );
where closures and interiors are with respect to the relative topology on X. Note that frY is
a closed subset of the boundary of Y , and that frY = fr(intY ) = fr(closY ). (Right?) Also,
when X is connected, frY 6
= ; if and only if intY and intY are nonempty. (Same question.)
128
We say a pro¯le ((P1; R1); : : : ; (Pn ; Rn )) satis¯es preference diversity if, for every Y µ X,
every x 2 frY , and all distinct i; j 2 N, Ii (x) \ Ij (x) has empty interior in the relative
topology on frY . This assumption guarantees that no two voters have preferences with
indi®erence curves that exactly coincide locally.
We're ready for McKelvey's theorem. It is similar to Proposition 2.50 in that it establishes
precisely matched individual indi®erences across the frontier of TF (xjµ). One di®erence is
that, because it relies on preference diversity, the proposition focuses on the frontier, rather
than the boundary, of this set. This leads to a more important di®erence: McKelvey's
theorem establishes that the frontier of TF (xjµ) is entirely contained within an indi®erence
class of a single individual, signi¯cantly pinning down the way in which those individual
indi®erences need to be matched.
Proposition 2.53 (McKelvey) Assume that X µ Rd , that each (Pi (µ); Ri(µ)) is continuous, and that PR(µ) satis¯es preference diversity. Let F be weakly decisive. Take any
x 2 X.
1. There is some j 2 N such that, for all y 2 frTF (xjµ), frTF (xjµ) µ Ij (yjµ).
Furthermore, take any y; z 2 frTF (xjµ) and any i 6
= j such that yIi (µ)z.
2. If P(y; zjµ) [ fjg 2
= D(F ) and (Z \ Pi (zjµ)) nTF (xjµ) 6
= ; for every open set Z around
z, then there exists k 2 N , with k 6
= i; j, such that yIk (µ)z.
3. If P(z; yjµ)[ fjg 2
= D(F ) and Z \Pi¡1 (zjµ) \TF (xjµ) 6
= ; for every open set Z around
z, then there exists k 2 N , with k 6
= i; j, such that yIk (µ)z.
Proof: From Proposition 2.50 and the fact that frTF (xjµ) µ bdTF (xjµ), it follows that, for
all y; z 2 frTF (xjµ), there exists i 2 N such that yIi (µ)z. Therefore, given any y 2 frTF (xjµ),
S
we have frTF (xjµ) µ i2N [Ii (yjµ) \frTF (xjµ)]. Since each Ii(yjµ)\frTF (xjµ) is closed in the
relative topology on frTF (xjµ), and since frTF (xjµ) is nonempty by supposition, Lemma 2.7
implies that some Ij (yjµ) \ frTF (xjµ) contains a nonempty open set in the relative topology
on frTF (xjµ). I claim that frTF (xjµ) µ Ij (yjµ). Suppose there exists z 2 fr(TF (xjµ))nIj(yjµ).
Note that, by preference diversity, for all i 6
= j, Ii(zjµ) \ Ij (yjµ) has empty interior in
fr(TF (xjµ)). Therefore, by Lemma 2.7, there exists w 2 Ij(yjµ) \ fr(TF (xjµ)) such that
wIi(µ)z for no i 6
= j, a contradiction. This establishes the claim and proves part 1 of the
proposition. Now take any y; z 2 frTF (xjµ) and any i 6
= j such that yIi(µ)z and such that
there is no k 6
= i; j for which yIk (µ)z. By continuity, there is an open set V around z such
that V µ Pk (yjµ) for all k 2 P(z; yjµ) and V µ Pk¡1 (yjµ) for all k 2 P(y; zjµ). Suppose
that P(y; zjµ) [ fjg 2
= D(F) and (Z \ Pi(zjµ)) n TF (xjµ) 6
= ; for every open set Z around
0
z. Take z 2 (V \ Pi(zjµ)) n TF (xjµ). By continuity, since z 0Pi(µ)y and y 2 frTF (xjµ), there
exists y0 2 TF (xjµ) such that R(y 0; z0 jµ) µ P(y; zjµ) [ fjg, which is not decisive. Since F
is weakly decisive, therefore, we have z0 RF (µ)y0 , implying z0 2 TF (xjµ), a contradiction.
This proves part 2 of the proposition. Suppose that P(z; yjµ) [ fjg 2
= D(F) and Z \
Pi¡1(zjµ) \ TF (xjµ) 6
= ; for every open set Z around z. Take z 00 2 V \ Pi¡1 (zjµ) \ TF (xjµ).
129
By continuity, since yPi(µ)z00 and y 2 frTF (xjµ), there exists y00 2 X n TF (xjµ) such that
R(z00 ; y 00 jµ) µ P(z; yjµ) [ fjg, which is not decisive. Since F is weakly decisive, therefore,
we have y 00 RF (µ)z00 , implying y 00 2 TF (xjµ), a contradiction. Therefore, there exists k 2 N ,
with k 6
= i; j, such that yIk (µ)z, proving part 3 of the proposition.
How restrictive are the assumptions in parts 2 and 3 of Proposition 2.53? To gauge this,
suppose that the frontier of TF (xjµ) is contained in an indi®erence class of individual j,
and take two alternatives, y and z, in the frontier, and let i be indi®erent between them.
Moreover, let i's indi®erence curve through y and z cut the frontier in a \transversal" way,
so (Z \Pi(zjµ))nTF (xjµ) 6
= ; and Z \Pi¡1(zjµ)\TF (xjµ) 6
= ; for every open set Z around z.
Beyond these assumptions, the existence of another individual who is indi®erent between y
and z relies on the following:
P(y; zjµ) [ fjg 2
= D(F)
or
P (z; yjµ) [ fjg 2
= D(F ):
Note that, if F is anonymous, then this condition is automatically satis¯ed: if P (y; zjµ)[fjg
is decisive, then, by anonymity, P(y; zjµ) [ fig must be decisive, so P (z; yjµ) [ fjg cannot
be. Thus, it should be considered a weak condition.
McKelvey argues that the conclusions of Proposition 2.53 are very restrictive, especially
when d ¸ 3, so that we would expect them to hold only in the trivial case of frTF (xjµ) = ;.
When X is connected, this is only possible if intTF (xjµ) = ; or intTF (xjµ) = ;. The former
is only possible if intRF (xjµ) = ;, which is only possible if x is in the core and the core has
empty interior. Thus, for all x 2 X, we may very well have intTF (xjµ) = ;, i.e., TF (xjµ) is
dense in X .
The next proposition, which parallels Proposition 2.51, shows that the weak top cycle can
be quite large as a consequence. It di®ers from the earlier result in that its main assumption
concerns the frontier of TF (xjµ) rather than the boundary. To compensate, we must add
the extra assumption that the core is empty and accept a slightly weaker conclusion.
Proposition 2.54 Assume X µ Rd is connected, (PF (µ); RF (µ)) is continuous, frTF (xjµ) =
; for all x 2 X , and CF (µ) = ;. For all x 2 X, if PF¡1(xjµ) 6
= ;, then x 2 W TOPF (µ).
Proof: Fix an arbitrary x 2 X. Since X is connected, frTF (xjµ) = ; implies that either
intTF (xjµ) = ; or intTF (xjµ) = ;. Suppose the former. Because PF (xjµ) is open and
PF (xjµ) µ intTF (xjµ), it must be that PF (xjµ) = ;, i.e., x 2 CF (µ), a contradiction. Thus,
intTF (xjµ) = ;, or equivalently, clos(TF (xjµ)) = X. To prove the proposition, assume
PF¡1(xjµ) 6
= ;, and take any y 2 X. By the above argument, clos(TF (yjµ)) = X, so
PF¡1(xjµ) \ clos(TF (yjµ)) 6
= ;, so PF¡1(xjµ) \ TF (yjµ) 6
= ;. This implies that x 2 TF (yjµ),
and we conclude that x 2 W TOPF (µ).
130
Proposition 2.53 is stated di®erently than McKelvey's theorem, in that after part 1 he
assumes that i is either \not a dummy voter" or is \as strong as j." I will argue that, if
yIi(µ)z and yIj (µ)z as in the proposition, then it is weaker to assume
P(y; zjµ) [ fjg 2
= D(F)
or
P (z; yjµ) [ fjg 2
= D(F );
meaning my formulation of the assumptions of the proposition is somewhat weaker than
his. I will use the phrase \j is not a swing voter" to denote this condition.
We say i is a dummy voter for y over z at µ if, for every G µ N n fig with
P (y; zjµ) n fig µ G µ R(y; zjµ);
we have
G [ fig 2
= D(F) or G 2 D(F):
So i is not a dummy voter if there exists G µ N n fig with P (y; zjµ) n fig µ G µ R(y; zjµ)
such that G [ fig 2 D(F ) and G 2
= D(F).
We say i is as strong as j for y over z at µ if, for every G µ N n fig with
P(y; zjµ) n fi; jg µ G µ R(y; zjµ);
we have
G [ fjg 2
= D(F) or G [ fig 2 D(F ):
So, no matter how we divide up indi®erent voters, if j makes the group voting for y over z
decisive, then so does i.
To compare the formulations, assume yIi(µ)z and yIj (µ)z, and suppose i is not a dummy
voter for y over z. Thus, there exists G µ N n fig with P(y; zjµ) µ G µ R(y; zjµ) such that
G [ fig 2 D(F) and G 2
= D(F). If j is contained in G, then P(y; zjµ) [ fjg 2
= D(F). If j is
not contained in G, then G[fig µ R(y; zjµ)nfjg 2 D(F ), and this implies P (z; yjµ)[ fjg 2
=
D(F). Thus, j is not a swing voter, as claimed.
Now suppose i is as strong as j 6
= i for y over z. Suppose that xIi (µ)y and xIj (µ)y, and
take G as in the de¯nition such that G [ fjg 2
= D(F) or G [ fig 2 D(F ). In the ¯rst case,
we have P(y; zjµ) [ fjg 2
= D(F). In the second case, we have P(z; yjµ) [ fjg 2
= D(F ). Thus,
again, j is not a swing voter.
Lastly, recall that McKelvey assumed F was strong and proved the expansiveness of the
strong top cycle set. With strictly convex individual preferences, that would give us thin
social indi®erence. McKelvey doesn't assume strict convexity, but his other assumption do,
indeed, deliver thin social indi®erence. Then the conclusion from Proposition 2.54 that the
131
weak top cycle can be quite large carries over, by Proposition 2.47, to the strong top cycle
set. In fact, it carries over to the strong top cycle of any weakly decisive PAR F such that
FD(F) generates thin social indi®erence at µ.
In sum, we have seen that the weak top cycle set is of little use in bounding collective
choices in the presence of an empty core: while nonempty quite generally, the top cycle set
is much too big, typically encompassing the entire space of alternatives.
We are left in something of a quandary. The majority core is rarely nonempty in multiple
dimensions. When it is, arbitrarily small perturbations of individual preferences can annihilate it. In such cases, the weak top cycle gives no indication that collective choices need
have any relation to the core, i.e., predictions based on the core may fail to be robust to
even the smallest misspeci¯cations of individual preferences.
2.9
The Weak Uncovered Set
Recall the de¯nition of the weak uncovered set: given a dual pair (P; R) it is UC(R) =
M(CR), where the covering relation of R, CR, is de¯ned as
xCRy , R(x) µ P(y):
We can apply this de¯nition when, in particular, R and P are weak and strict social preferences. Given a collective choice environment and PAR F , de¯ne the weak uncovered set
of F at µ by W UCF (µ) = U C(RF (µ)).
Properties of the Weak Uncovered Set
Recall some properties of the weak uncovered set applied to social preferences.
² x 2 W U CF (µ) if and only if, for all y 2 X , there exists z 2 X such that xRF (µ)zRF (µ)y.
(Two-step principle)
² Assume CF (µ) 6
= ;. If (PF (µ); RF (µ)) satis¯es thin indi®erence, then W UCF (µ) =
CF (µ).
² If X is compact and (PF (µ); RF (µ)) is upper semicontinuous, then W U CF (µ) 6
= ;.
² CF (µ) µ W UCF (µ) µ W TOPF (µ).
Thus, the weak uncovered set shares some of the properties of the weak top cycle. In
particular, the fourth item listed above establishes nonemptiness of the weak uncovered set
under the same minimal continuity condition used for the weak top cycle. Thus, we have
the following proposition, which actually implies our nonemptiness result for the weak top
cycle.
132
Proposition 2.55 Assume X µ Rd is compact and each (Pi(µ); Ri (µ)) is upper semicontinuous. Let F be a simple PAR. Then WU CF (µ) 6
= ;.
The third item above yields the following result, also paralleling the results for the weak
top cycle.
Proposition 2.56 Assume X µ Rd and PR(µ) satis¯es LSWP. If F is strong and CF (µ) 6
=
;, then W UCF (µ) = CF (µ).
I want to digress brie°y on the conditions under which we get equivalence of the core (when
nonempty) and the weak uncovered set: F is strong and LSWP is satis¯ed. If either of
those assumptions is dropped, then the equivalence need not hold. For example, assume
n = 3, di®erentiable, concave utility representations, as in Figure 2.23.
Figure 2.23: Core 6
= weak uncovered set
Here, ru2(z) = 0 for all z on the straight line segment above. The Plott conditions are
satis¯ed at x¤ , and imagine that it is the only such point. Therefore, the unique majority
core alternative is x¤. But note that y is weakly majority preferred to x¤, so y 2 W U CSM (µ).
Why do we need F to be strong? Consider an environment with n = 2, X = R, Euclidean
preferences with ~x1 = ¡² and x
~2 = ². Here, the simple majority core is [¡²; ²], and it can
be checked that the weak uncovered set is [¡2²; 2²]. This is in stark contrast to the weak
top cycle, which, by Proposition 2.52, is equal to the entire real line! Note also that, as ²
goes to zero, the weak uncovered set collapses to f0g in a continuous way.
Nonemptiness of the weak uncovered set for PARs such as relative majority rule is a di±cult
issue. The next proposition gives some su±cient conditions for nonemptiness, including a
new one on social preferences. We say a dual pair (P; R) is rich if, for all x; y 2 X,
R(x) = R(y) implies x = y. This would appear to be a very weak condition, but I do not
know of any general su±cient conditions for it.
Proposition 2.57 Assume that X µ Rd is compact, that (PD(F ) (µ); RD(F ) (µ)) is upper
semicontinuous and satis¯es thin indi®erence, and that either CD(F ) (µ) 6
= ; or (PD(F ) (µ);
RD(F ) (µ)) is rich. Then
; 6
= W UCF (µ) µ W U CD(F ) (µ):
133
Proof: The result is trivial, by Proposition 2.31, if CD(F ) (µ) 6
= ;, so suppose otherwise.
De¯ne xBy if and only if xRD(F ) (µ)y and PD(F ) (xjµ) µ RD(F ) (yjµ). Note that, by thin
indi®erence and upper semi-continuity, xBy implies RD(F ) (xjµ) µ RD(F ) (yjµ). In the proof
of Proposition 1.48, we showed that there exists a B-maximal element, say x¤ . Suppose
that, for some x 2 X, we have xCRF (µ) x¤ , i.e., RF (xjµ) µ PF (x¤jµ). A consequence of
PD(F) (xjµ) µ RF (xjµ) and PF (x¤ jµ) µ RD(F ) (x¤jµ) is, then, that xBx¤. Since x¤ is Bmaximal, we therefore have x¤Bx. This implies RD(F ) (xjµ) = RD(F ) (x¤jµ), and then, by
richness, we have x = x¤, contradicting xPF (µ)x¤. Therefore, there is no such x, and
we conclude that x¤ 2 W UCF (µ). Now take x 2 W UCF (µ) and y 2 X, and suppose
that yCRD(F )(µ) x, i.e., RD(F ) (yjµ) µ PD(F ) (µ). This implies that RF (yjµ) µ PF (xjµ), or
equivalently, yCRF (µ) x. But then x is not in the weak uncovered set W U CF (µ), a contradiction. Therefore, there is no such y, and we conclude that x is in the weak uncovered set
W UCD(F ) (µ), as required.
If you're interested in continuity of the strong uncovered set, there is no general result.
You can show, however, that, assuming that F is strong and LSWP is satis¯ed, the strong
uncovered set is upper hemicontinuous at any state with a nonempty core. This follows because, at such a state, the core is an externally PF (µ)-stable singleton and will coincide with
the weak and strong uncovered sets. Then upper hemicontinuity of the strong uncovered
set follows from Proposition 2.59.
The above example and the last property of the weak uncovered set listed above suggest
that it may be a more useful bound on social choices than the weak top cycle. The next
proposition immediately tells us somewhat more generally that the uncovered set won't
su®er from the above \chaos" results for the weak top cycle: it says that, under \reasonable"
conditions, the elements of the weak uncovered set are always Pareto optimal. Note that
the result does not rely on any structure of the set of alternatives or individual preferences.
Proposition 2.58 Assume that X µ Rd, that (PF (µ); RF (µ)) satis¯es thin indi®erence,
and that each (Pi (µ); Ri (µ)) is upper semicontinuous. Let F be weakly decisive and satisfy
Pareto. Then W U CF (µ) µ CSP (µ).
Proof: Take x; y 2 X and µ 2 £ such that xPSP (µ)y, i.e., P (x; yjµ) = N. Letting R =
RF (µ), I claim that xCRy, i.e., R F (xjµ) µ PF (yjµ). Take z 2 RF (xjµ). If z = x, then
zPF (µ)y by Pareto. If z 6
= x, then, by thin indi®erence, there is a sequence fzmg in PF (xjµ)
converging to z. By weak decisiveness, Gm = R(zm ; xjµ) 2 D(F) for all m. Since N is ¯nite,
we must have G = Gm for some decisive G and in¯nitely many m. By upper semicontinuity,
we have G µ R(z; xjµ). By transitivity of individual preferences, R(z; xjµ) µ P(z; yjµ), so
zPF (µ)y, as required. Therefore, y 2
= W UCF (µ).
The requirement of thin social indi®erence in the above proposition is not usually imposed,
because most analyses consider a notion of covering somewhere between CR and CP . This
134
gives us a smaller uncovered set than W UCF (µ), one that is always contained in the Pareto
optimals. You can check that the strong uncovered set is contained in the Pareto optimals
generally. When considering the weak uncovered set, we need the additional assumption.
Let n = 4, X = fx; y; zg, and consider a state with individual preferences as follows.
R1(µ) R2 (µ) R3 (µ) R4(µ)
x
x
z
z
y
y
x
x
z
z
y
y
Here, xPSP (µ)y but y 2 W UCSM (µ).
Like the nonemptiness of the weak uncovered set, the upper bound of the Pareto optimals,
in Proposition 2.58, relies on continuity properties of social preferences. It is easily checked,
given an arbitrary PAR F , that W U CF (µ) µ WU CD(F ) (µ). Thus, as long as FD(F ) generates social preferences satisfying thin indi®erence at µ, as with relative majority rule when
n is odd, at least Proposition 2.58 holds more generally.
Though Proposition 2.58 gives us an upper bound on the weak uncovered set, the set of
Pareto optimal alternatives can be quite large and does not give us very tight bounds on
collective choices | even when the core is \close" to being nonempty (in the sense of the
Hausdor® metric de¯ned above.) The next result, on continuity of the weak uncovered
set correspondence solves that problem. The result assumes continuity of the mapping
RF : £ ! Rc . Note that, from Proposition 2.7, we have conditions under which this is
satis¯ed by all simple PARs.
Proposition 2.59 Assume X µ Rd is compact, PR(£) µ RN
c , and RF (£) µ Rc . Let
RF : £ ! Rc be continuous at µ. Then W U CF (µ) is closed and WU CF is upper hemicontinuous at µ.
Proof: Take any open set U with W U CF (µ) µ U , and note that WU CF (µ) = U C(RF (µ)).
We know from Proposition 1.50 that U C is upper hemicontinuous on Rc , so there exists
an open set W containing RF (µ) such that, for all R 2 W , UC(R) µ U. Assuming RF is
continuous at µ, there is an open set V containing µ such that, for all µ0 2 V , RF (µ0 ) 2 W .
Thus, WU CF (µ 0) µ U for all µ0 2 V , as required. I'll let you prove that W UCF (µ) is
closed.
Proposition 2.59 tells us that, when we move slightly from a state with a nonempty core, the
weak uncovered set does not \blow up" discontinuously. We next give more global bounds
for the weak uncovered set under the preference restriction of Euclidean domain.
135
The Yolk
We can tighten bounds on the weak uncovered set, if we focus on a strong PAR F and on
Euclidean preferences. Let BF (µ) consist of the closed balls B µ Rd that intersect F -median
hyperplanes with all normals: that is, B 2 BF (µ) if and only if
8y 2 Rd; 8c 2 R : Hy;c 2 MF (µ) implies B \ Hy;c 6
= ;:
Let BF (µ) denote the unique element of BF (µ) with smallest radius. We call BF (µ) the
yolk at µ. Let it have center y ¤ and radius r. Thus, BF (µ) = Br (y ¤). See Figure 2.24.
(Uniqueness of this ball is not obvious. It seems to be widely accepted, though I do not
know of a published or circulated proof. I had a proof sketched out last year, but I got too
busy to write it up. So I think the claim is true.)
Figure 2.24: The yolk
To see that the yolk is generally well-de¯ned only for strong rules, consider Figure 2.25. Here
there are four individuals with Euclidean preferences, ideal points arranged in a rectangle.
Drawn are two elements of BF (µ), assuming simple majority rule, of minimal radius.
Figure 2.25: No yolk
It turns out that uniqueness of the yolk is not such a critical issue. Proposition 2.61, below,
will hold for any ball intersecting F-median hyperplanes in all directions. (Check this.)
Clearly the yolk can be a relatively small set. In fact, it becomes arbitrarily small as
we move ideal points closer to radial symmetry. In the following propositions, I assume
Euclidean domain, so we can identify each state with a pro¯le (~
x1; ~x2; : : : ; x
~n ) of ideal
points. Convergence in the Hausdor® metric is just convergence of ideal points in the usual
Euclidean metric.
Proposition 2.60 Assume that X µ Rd convex, that P R(£) µ E, and that F is strong.
If CF (µ) 6
= ;, then BF (µ) = CF (µ) and BF is upper hemicontinuous at µ.
Proof: Take a state µ = (~
x1; x
~2; : : : ; x
~n ) in which the core is nonempty, so, under the
conditions of the proposition, it consists of a single alternative, say x¤, that is strictly
socially preferred to every other alternative. Thus, BF (µ) = fx¤ g. Take a sequence of
states fµm g, where µm = (~
xm
~m
~m
1 ;x
2 ;: :: ; x
n ), converging to µ. To simplify notation, let
136
B = BF (µ), and let Bm = BF (µm ). I will show that, for all ² > 0, there exists m0 such that,
for all m ¸ m0 , Bm µ B²(x¤). If not, then we can ¯nd ² > 0 and a subsequence fwmg, also
indexed by m for convenience, such that wm 2 Bm nB² (x¤) for all m. I claim that, for each
m, there exists a F-median hyperplane Hm , relative to the ideal points (~
xm
~m
~m
1 ;x
2 ; :: : ;x
n ),
¤
¤
such that Hm \ B²=3(x ) = ;. To prove this, suppose not. Then B²=3(x ) intersects all Fmedian hyperplanes, so Bm must have radius less than ²=3. Then the F -median hyperplane
with normal wm ¡ x¤ does not intersect B²=3(x¤ ) (why?), a contradiction. This establishes
the claim. Denote the mth hyperplane by Hm , and let ym and c m satisfy Hm = Hym ;cm ,
jjym jj = 1, and cm ¡ ym ¢ x¤ ¸ ²=3. Note that f(ym ; c m)g has a convergent subsequence
(why?), also indexed by m. Denote the limit by (y; c). Furthermore, note that c¡y¢x¤ ¸ ²=3,
i.e., y ¢ x¤ · c ¡ ²=3. Since N is ¯nite and each Hm is a F -median hyperplane, there is a
blocking group G whose ideal points are on the ym -side of in¯nitely many hyperplanes Hm .
That is,
ym ¢ x
~m
¸ cm
i
for each i 2 G. Therefore, y ¢ x
~i ¸ c for all i 2 G. De¯ne z = x¤ + (c ¡ y ¢ x¤ )y, i.e., z is the
vector in H y;c closest to x¤ in the y-direction. In particular, z 6
= x¤. For each i 2 G,
1 ¤
1 ¤
(x + z) ¢ y =
(x ¢ y + z ¢ y)
2
2
1 ¤
=
(x ¢ y + c)
2
· c
· x
~i ¢ y;
where the ¯rst inequality uses x¤ ¢ y · c ¡ ²=3 and the second uses y ¢ x
~ i ¸ c. Therefore,
¤
zRF (µ)x , a contradiction.
More generally, the yolk will be some \centrally" located subset of alternatives. To establish bounds on the weak uncovered set using the yolk, we ¯rst establish a result on the
alternatives weakly socially preferred to y ¤.
Lemma 2.9 Assume that X µ Rd convex, that each (Pi(µ); Ri (µ)) is Euclidean, and that
F is strong. Then RF (y¤ jµ) µ B2r (y ¤).
Proof: Take any x 2
= B2r (y¤ ) and note that the hyperplane through (1=2)(x + y¤ ) with
normal x ¡ y¤ does not intersect BF (µ). Therefore, it is not a F -median hyperplane (nor is
there one closer to x), so y¤ PF (µ)x, establishing the claim.
And now another result on the strict upper sections of alternatives far from y ¤.
137
Lemma 2.10 Assume that X µ Rd convex, that each (Pi (µ); Ri (µ)) is Euclidean, and that
F is strong. For every x 2
= B4r (y¤ ), B2r (y ¤) µ PF (xjµ).
Proof: Take any x
^2
= B4r (y ¤), and take any y^ 2 B2r (y ¤). See Figure 2.26, below. To show
y^ is strictly socially preferred to x
^, I will show that the hyperplane through (1=2)(^
x + y^)
with normal x
^ ¡ y^ does not intersect B F (µ) and is, therefore, not a F -median hyperplane.
Without loss of generality, let y¤ = 0. The closest point in BF (µ) to that hyperplane is
r
®(^
x ¡ y^), where jj®(^
x ¡ y^)jj = r, i.e., ® = jj^x¡^
yjj . Thus, I need to show
1
(^
x + y^) ¢ (^
x ¡ y^) > ®(^
x ¡ y^) ¢ (^
x ¡ y^) = rjj^
x ¡ y^jj:
2
Let y0 and ¯ 0 solve
min y ¢ (^
x ¡ y^)
s.t. y = y^ + ¯(^
x ¡ y^)
¯ ·0
y 2 B2r (y¤ )
and let x0 and ° 0 solve
min x ¢ (^
x ¡ y^)
s.t. x = y^ + °(^
x ¡ y^)
°¸0
±
x2
= B4r
(y ¤);
±
where B4r
(y ¤) is the open ball of radius 4r around y¤ . See Figure 2.26.
Figure 2.26: What's going on?
± (y¤ ), we have °0 > 0; second, since x
Observe three things: ¯rst, since y^ 2 B4r
^2
= B4r (y ¤),
0
0
0
we have x ¢ (^
x ¡ y^) < x
^ ¢ (^
x ¡ y^); third, jjy jj = 2r and kx k = 4r. From
x0 = y^ + °0 (^
x ¡ y^)
0
0
y = y^ + ¯ (^
x ¡ y^);
we deduce
^x ¡ y^ = c(x0 ¡ y 0);
where, from the ¯rst observation above, c =
1
° 0¡¯ 0
> 0. Thus, I need to show
1
(^x + y^) ¢ c(x0 ¡ y 0 ) > rjjc(x0 ¡ y 0)jj;
2
138
or equivalently,
1
(^
x + y^) ¢ (x0 ¡ y 0) > rjjx0 ¡ y0 jj:
2
From the construction of y 0 and x0 , and from the second observation above, this would be
implied by
1 0
(x + y0 ) ¢ (x0 ¡ y0 ) ¸ rjjx0 ¡ y 0jj;
2
or equivalently,
¢
1¡ 0 2
jjx jj ¡ jjy 0 jj2 ¸ rjjx0 ¡ y0 jj:
2
From the third observation above, namely, jjx0jj = 4r and jjy 0 jj = 2r, this is
6r ¸ jjx0 ¡ y 0jj;
which follows since, by the triangle inequality, jjx0 ¡ y0 jj · jjx0 jj + jjy0 jj = 6r.
The preceding lemmas immediately yield the following result.
Proposition 2.61 Assume X µ Rd convex and each (Pi(µ); Ri (µ)) is Euclidean. Then
W UCF (µ) µ B4r (y ¤).
Note the implication, already proved for the general case, that the weak uncovered set is
upper hemicontinuous at states with a nonempty core. That is, suppose x¤ is the unique
core point when ideal points are x
~1; x
~2; : : : ; x
~n . If we perturb one ideal point slightly, so
that the core becomes empty, the weak uncovered set will not \blow up." It will get bigger,
but continuously. Contrast this with the weak top cycle: if we perturb one ideal point and
the core becomes empty, the weak top cycle set blows up to the entire set of alternatives.
2.10
Variable Feasible Sets
[ More to do here. ]
2.11
Exercises
2.1.
Prove that a pro¯le ((P1; R1); : : : ; (Pn; Rn)) is single-peaked only if the following
holds: for every triple fx; y; zg of distinct alternatives,
139
(_) There is some alternative that no individual ranks last among the three, i.e., there
exists w 2 fx; y; zg such that, for every i 2 N , there exists v 2 fx; y; zg such that
wPi v.
2.2.
We say a pro¯le ((P1; R1); : : : ; (Pn; Rn)) is relatively single-peaked if every triple
fx; y; zg of distinct alternatives can be indexed, as in fx1; x2 ; x3 g, so that the following
holds:
(^) For each i 2 N, x1 Rix2 implies x2 Pix3, and x3Rix2 implies x2Pi x1.
Prove that the conditions (_) and (^) are equivalent.
2.3. Explain how relative single-peakedness is weaker than single-peakedness. (I can think
of three ways.) What about weak single-peakedness?
2.4. Show that (R1; R2; R03) preceding Proposition 2.1 is not single-peaked.
2.5. Prove the remaining direction of Proposition 2.1.
2.6. Prove Corollary 2.1.
2.7. In the one-dimensional public good provision problem from Example 2, monotonicity
is violated. Does this mean that monotonicity is an unreasonable condition in public good
economies? Does this mean that the example is unreasonable?
2.8. In multiple public good provision problems, why does strictly monotonic domain rule
out the possibility of private good components in the de¯nition of an alternative x?
2.9. Consider a multiple public good provision problem in which a society has a ¯xed
budget m to spend and each good k is priced at some price pk > 0. As in Example 3, let
X = fx 2 R`+ j p ¢ x = mg. Individual preferences on X are continuous and strictly convex,
and each individual has an ideal point. Are individual preferences monotonic? Strictly so?
2.10. Prove that, if X µ Rd is convex and each (Pi(µ); Ri(µ)) is strictly convex, then
LSWP holds.
2.11. Prove that LSWP holds in the divide-the-dollar environment. (What about private
good exchange economies?)
2.12. Calculate weak social preferences for the PARs de¯ned in Section 2.3.
2.13.
Check that the PARs de¯ned in Section 2.3 are well-de¯ned, i.e., strict social
preference is asymmetric and weak preference is complete.
140
2.14. I claimed in Section 2.3 that, whenever G is a proper subset of N, FG is not a q-rule;
and I claimed that simple majority is not a G-rule. Why?
2.15. Let n = 5, let X = fa; b; c; dg, let q = :55, and let w1 = :25, w2 = :24, w3 = w4 =
w5 = :17. Suppose individual preferences in µ are as follows:
R 1(µ) R2(µ) R3(µ) R4(µ) R5 (µ)
a
dc
b
b
cd
d
a
d
c
a
c
b
ca
a
b
b
d
Graph the strict and weak social preference relations Pw;q (µ) and Rw;q (µ) of the corresponding weighted quota rule.
2.16. Let n = 5, and consider the PAR F de¯ned as follows: xPF (µ)y if and only if either
f1; 2; 3g µ P(x; yjµ) or f1; 2; 4g µ P(x; yjµ) or f3; 4; 5g µ P(x; yjµ):
Prove that F is not a weighted quota rule, i.e., there do not exist weights w and a quota q
such that F = Fw;q .
2.17. Assume X is ¯nite. Prove that, for all µ 2 £,
P
x2X
Bx (µ) = 0.
2.18.
Assume that B y (µ) =
6 0 for some y 2 X. Prove that, if x 2 CRM (µ), then
x
B (µ) > miny2X By (µ). Give an example showing that we may have CSM (µ) \ CB(µ) = ;,
even if CSM (µ) 6
= ;. Could we weaken the assumption of this problem to merely that
PRM (µ) 6
= ;?
2.19. Suppose that x is a \relative majority Condorcet loser," i.e., for all y 6
= x, yPRM (µ)x.
Prove that x 2
= CB (µ).
2.20. Why does part 2 of Proposition 2.3 imply that D(F) is proper? Is the converse of
part 4 true?
2.21. Prove that
P weightedPmajority rule is strong if and only if there is no partition I; J
of N such that i2I wi = i2J wi = n=2.
2.22. What are D(FB ), B(FB), D(FT ), B(FT ), D(FC ), and B(FC )?
2.23. Prove that a PAR F satis¯es IIA if and only if, for all x; y 2 X and all µ; µ0 2 £,
[P(x; yjµ) = P (x; yjµ0 ) and R(x; yjµ) = R(x; yjµ 0) and xRF (µ)y] ) xRF (µ0 )y:
141
2.24. Prove that a PAR F satis¯es IIA if and only if, for all x; y 2 X and all µ; µ0 2 £,
[PR(µ)jfx;yg = PR(µ0 )jfx;yg] ) [RF (µ)jfx;yg = RF (µ 0)jfx;yg];
where Ri jY = Ri \(Y £Y ) is the \restriction" of Ri to Y , and (R1 ; : : : ; Rn )jY = (R1 jY ; : : : ;
RnjY ).
2.25. Prove Lemma 2.1.
2.26. Give an example showing that FB , FT , and FC do not satisfy IIA generally. What
if jXj = 2?
2.27. Which PARs in Section 2.3 aren't anonymous? Why not?
2.28. Assume there is a \very free pair," fx; yg, such that, for every pair of disjoint groups
G; G0 µ N, there exists a state µ 2 £ such that G = P(x; yjµ) and G0 = P (y; xjµ). Prove
that, if a PAR F is neutral and monotonic, then it is weakly decisive. Why is a free pair
not enough?
2.29. Prove that, under linear domain, F strong implies F is decisive.
2.30. Assume unrestricted domain. Prove that if F is anonymous and strong, then n
is odd and F satis¯es the weak Condorcet principle: for all µ 2 £ and all x; y 2 X,
xPSM (µ)y ) xPF (µ)y.
2.31. Give an example of an anonymous and strong PAR F that is not FSM or FRM .
2.32. Show by example that F strong does not imply F is decisive, and that F decisive
does not imply F strong.
2.33. Prove that a PAR is simple if and only if it is decisive.
2.34. Is FB simple when jXj = 2? Why or why not? How does your answer depend on
the domain of preferences?
2.35. Prove that a PAR is anonymous and simple if and only if it is q-rule for some q.
2.36. Prove that, given a simple PAR F, G " and G # are unique maximal and minimal
(respectively) representations.
2.37. For Lemma 2.2, prove part 1, prove the remaining direction of part 3, and prove the
rest of part 5.
2.38. Prove Corollary 2.2.
142
2.39. Prove Corollary 2.3.
2.40. Prove parts 2 and 3 of Proposition 2.6.
2.41.
Assume that X µ Rd, and that each (Pi(µ); Ri (µ)) is continuous. Let F be a
simple PAR. Prove that the weak upper contour correspondence RF (¢jµ): X ¶ X is upper
hemicontinuous.
2.42. Now add the assumption that PR(µ) satis¯es LSWP, and prove that R F (¢jµ) is lower
hemicontinuous.
2.43. In the proof of Proposition 2.7, why does LSWP imply that, for each k, there exists
xk 2 PG(yjµ) such that jjxk ¡ xjj < 1=k? In the following line, why does yRi(µm )xk for
in¯nitely many m imply yR i(µ)xk ?
2.44. Prove the other direction in Proposition 2.8.
2.45. Prove the other direction in Proposition 2.9.
2.46. We say a PAR F satis¯es positive responsiveness if, for all µ; µ0 2 £ and all x; y 2 X,
the following holds: Suppose xRF (µ)y, R(x; yjµ) µ R(x; yjµ0 ), and P (x; yjµ) µ P(x; yjµ0 )
with at least one inclusion strict; then xPF (µ0 )y. Assume unrestricted domain.
(a) Prove that FRM is the only PAR satisfying anonymity, neutrality, and positive responsiveness.
(b) Give an example showing that FRP violates positive responsiveness.
(c) Prove that, assuming n ¸ 2, no simple PAR satis¯es positive responsiveness.
2.47. Prove McGarvey's Theorem, Proposition 2.10.
2.48. Verify that the claim in Proposition 2.11 holds for the preference pro¯le constructed
in the proof.
2.49. Prove the n = 4 case of Proposition 2.11.
2.50. Prove that, if we add IIA to the assumptions of Lemma 2.3, then we can deduce
that G \ G0 is semi-blocking.
2.51. In the proof of Lemma 2.4, ¯ll in the omitted step: If G is semi-decisive for x over y
at µ, then G is semi-decisive for every w 6
= y over y.
2.52. In the proof of Proposition 2.12, why does RF (µ) = X £ X in the ¯rst case?
143
2.53. Does Arrow's theorem still hold if jX j = 2? What if we drop Pareto? IIA?
2.54. Do Propositions 2.12 and 2.13 hold if we merely assume L µ PR(£) µ U?
2.55. Show by example that the existence of a dictator does not generally imply that F
either is R-transitive or satis¯es IIA.
2.56. Assume P R(£) = L.
(i) Characterize the anonymous and neutral PARs in terms of a set Q µ f0; 1; : : : ; ng of
\quotas." What conditions must Q satisfy?
(ii) Assume jXj ¸ 3. Using your answer to (i), what are the PARs satisfying anonymity,
neutrality, and P -transitivity?
(iii) Now strengthen P-transitivity to R-transitivity. Are there any PARs satisfying the
three conditions? What if we also add Pareto?
(iv) Repeat parts (ii) and (iii) after replacing neutrality by Pareto.
(v) How would your answers change if P R(£) = U?
2.57.
Assume jX j ¸ 3, PR(£) ¶ L, and F is neutral and monotonic. Also assume
that F inherits the resoluteness of individuals: for all x; y 2 X and all µ 2 £ such that
N = P(x; yjµ) [ P (y; xjµ), either xPF (µ)y or yPF (µ)x. (i) Prove that F is strong. (ii)
Demonstrate that F need not be simple, i.e., give an example of X, N , and a PAR F
satisfying these conditions that is not simple. (iii) What can you conclude about F if it is
P-acyclic?
2.58. We say F satis¯es strong Pareto if, for all x; y 2 X and all µ 2 £, xPRP (µ)y implies
xPF (µ)y. It satis¯es Pareto indi®erence if xIi (µ)y for all i 2 N implies xIF (µ)y. We say
F is a lexicographic dictatorship if there is some ordering, say (i1; i2 ; : : : ; in ), of individuals
such that: for all x; y 2 X and all µ 2 £, xPF (µ)y if and only if there exists some j such that
xPij (µ)y and, for all k < j, xIik (µ)y. That is, the PAR ¯rst checks i1 's preferences over x
and y; if i1 prefers x to y, then so does society; if i1 is indi®erent, it checks i2's preferences,
and so on. Assume jXj ¸ 3 and unrestricted domain. Use Arrow's theorem to prove the
following: if F satis¯es strong Pareto, Pareto indi®erence, IIA, and R-transitivity, then it
is a lexicographic dictatorship. Give examples showing that this conclusion does not hold
if either strong Pareto or Pareto indi®erence are dropped.
2.59. Prove that, if we add IIA to the assumptions of Lemma 2.5, then we can deduce
that G \ G0 is semi-decisive.
2.60. Fill in the omitted step in the proof of Lemma 2.6.
144
2.61. In the proof of Proposition 2.14, why is G nonempty?
2.62. Assuming unrestricted domain, identify the oligarchies for FSP and FRP .
2.63. Does Gibbard's theorem, Proposition 2.14, still hold if we drop Pareto or IIA?
2.64. Does Gibbard's theorem hold if we assume merely L µ P R(£) µ U?
2.65. Show that an oligarchical PAR need not satisfy IIA or P-transitivity.
2.66. Assume linear domain, and let F satisfy Pareto, IIA, and P-transitivity. Prove that
F is neutral.
2.67. Prove that the Security Council PAR, FSC , is P-acyclic. What if the total number
of individual strict preferences needed were six instead of seven?
2.68. Construct an example in which FSC violates P -transitivity.
2.69. Prove that every oligarchy is a collegium.
2.70. Consider weighted q-rule with n = 5 and weights w1 = w2 = :35 and w3 = w4 =
w5 = :1. Letting q take values in (:5; 1], identify the ranges of q for which the PAR is
oligarchic, collegial, or strong.
2.71. Assume unrestricted domain and F is simple. Prove that, if F is not P -acyclic, then
there exists µ 2 £ at which social preferences contain a \universal" cycle: for all x; y 2 X,
there exist k and x1; x2; : : : ; xk 2 X such that
xPF (µ)x1PF (µ)x2 ¢ ¢ ¢ PF (µ)xk¡1PF (µ)xk = y:
2.72. Consider the example in item 4 preceding Proposition 2.16. Show that F is P-acyclic.
2.73. Prove that a PAR is not collegial if and only if it satis¯es virtual unanimity.
2.74. Prove the last step of part 3 of Proposition 2.20, namely, that N (FSM ) = 4 when
n = 4.
2.75. Prove Corollary 2.18.
2.76. In the proof of Proposition 2.21, why may we assume without loss of generality that
the alternatives x1; x2; : : : ; xk¡1 are distinct?
145
2.77. The ¯fteen member European Union recently adopted the following simple PAR: a
group of states is decisive if and only if it consists of at least eight members and the total
population of its members exceeds half the total population of all ¯fteen states. Formally,
let wi denote the population P
of state i as a fraction of the total. Then a group G is decisive
if and only if jGj ¸ 8 and i2G wi > 1=2. Assume wi > 0 for all i. Without loss of
generality, let the states be indexed in order of population size, so state 1 is the largest, 2
next largest, and so on. Clearly, G = f1; 2; 3; 4; 5; 6; 7; 8g is decisive. Assume it is possible
to partition G into two groups of four, say H and I, such that
X
X
1
1
wi <
and
wi < :
2
2
i2H
i2I
That is, we can partition the top eight states into two groups of four states, each group
with less than half of the total population. What is the Nakamura number of this voting
rule?
2.78. Referring to the previous question, what is the Nakamura number of the European
Union voting rule for the general case, assuming only positive weights?
2.79. Assume unrestricted domain, and let F satisfy Pareto. We say a collection G1 ; : : : ; Gk
of groups is a decisive chain for a collection x1 ; : : : ; xk ; xk+1 of distinct alternatives if, for
all h = 1; : : : ; k,
Gh is decisive for xh over xh+1 ;
in the sense that, for all µ 2 £, Gh µ P (xh; xh+1) implies xhPF (µ)xh+1. In this de¯nition,
k is the
If there is no decisive chain with empty intersection,
Tk length of the decisive chain.
¤
i.e., h=1 Gh = ;, then de¯ne N = 2jXj . Otherwise, de¯ne N ¤ to be the length of the
smallest decisive chain. Prove that, if jX j ¸ N ¤ +1, then F violates P-acyclicity. Compare
this result to Nakamura's theorem.
2.80. Assume jX j ¸ 4 and unrestricted domain. Let F satisfy Pareto and the following
condition. There are distinct individuals, i and j, with \positive rights" over distinct pairs
of alternatives: there exist distinct x; y; z; w 2 X such that, for all µ 2 £, xPi (µ)y implies
xPF (µ)y and zPj(µ)w implies zPF (µ)w. Prove that F violates P-acyclicity. (Hint: You
might use the result from the previous exercise.)
2.81. Prove that, for every PAR F and every state µ, we must have CF (µ) µ CD(F ) (µ).
2.82. Prove part 2 of Proposition 2.22.
2.83. In the proof of Proposition 2.23, why does weak single-peakedness imply P(xh+1; xhjµ) µ
R(xj ; xj+1 jµ)? How do we conclude that xk Á x1?
2.84. In the discussion before Proposition 2.27, why must we have G1 = NÁ+(~
xi jµ) 2 D(F)?
146
2.85. In the proof of Proposition 2.28, how do we know x
~i1 = x
~ i2 ? How is Proposition
2.22 used in the proof?
2.86. Assume F is strong and P R(F) is single-peaked. Prove that RF (µ) is single-peaked.
Is it true that RF (µ) = Ri1 (µ) (in Proposition 2.26)? Prove or provide a counterexample.
2.87. Assuming X µ R, prove that, if (P; R) is upper semicontinuous, then M(R) is closed.
Use this to show that, if R(µ) is single-peaked with respect to ·, then MF (µ) = [x(µ); x(µ)].
2.88.
Check that, if X µ R and PR(µ) is single-peaked with respect to ·, then
[x(µ); x(µ)] = M F (µ) (as originally de¯ned).
2.89. Prove Proposition 2.29.
2.90.
Prove that, assuming PR(µ) is weakly single-peaked, x 2 MF (µ) if there is a
selection (~
x1 ; : : : ; x
~ n) such that x
~i 2 M(Ri(µ)) for all i 2 N and x is a median with respect
1
n
to (~
x ;: :: ; x
~ ).
2.91. Let X = R, let U : R ! R be a strictly quasi-concave function with maximum at
zero, and suppose
(?) each individual i has a preference relation represented by ui (x) = U (x ¡ x
~ i).
Thus, in the above, x
~ i 2 R is i's ideal point. Suppose F is strong, and let individual k
k
satisfy CF (µ) = f~
x g.
(a) Prove that, if F is simple, then the social preference relation is represented by uk (i.e.,
the core voter is \representative").
(b) Redo part (a), replacing the assumption that F is simple with the assumptions that
F is weakly decisive and that, for all i 6
= j, x
~i 6
=x
~j (distinct ideal points). Why is
the assumption of distinct ideal points needed?
(c) Again assume F is simple. Show that the core voter k is not generally representative if
(?) is weakened to the assumption merely that individual preferences are single-peaked
(with respect to ·).
(c) Now assume that X = Rd , that U : Rd ! R, with x
~i 2 Rd , and that (?) continues to
hold. Is the social preference relation necessarily represented by uk ?
2.92. We say a preference pro¯le is order restricted if the individuals can be numbered so
that, for all x; y 2 X,
fi j xPi yg < fi j xIiyg < fi j yPi xg
147
or
fi j yPixg < fi j xIiyg < fi j xPiyg;
where \<" here indicates that all elements of the lefthand set are less than all elements of
the righthand set. Austen-Smith and Banks give an example showing that a pro¯le may
be order restricted but not single-peaked, and they give an example of the other direction:
there are single-peaked pro¯les that are not order restricted. But consider a weakening of
order restriction. We say a pro¯le is relatively order restricted if, for every triple a; b; c 2 X,
the individuals can be numbered so that, for all x; y 2 fa; b; cg, either
fi j xPi yg < fi j xIiyg < fi j yPi xg
or
fi j yPixg < fi j xIiyg < fi j xPiyg:
Prove that, if a pro¯le is single-peaked, then it is relatively order restricted. What if, instead
of single-peakedness, (_) holds for every triple of distinct alternatives?
2.93.
Austen-Smith and Banks prove the following result: if a preference aggregation
rule, F , is neutral and monotonic, and if a pro¯le PR(µ) is order restricted, then PF (µ) is
transitive.
(a) How does this result bear on part 1 of Proposition 2.22?
(b) Let F be a strong simple PAR, and let P R(µ) be order restricted. Prove that some
individual is a representative, i.e., there is some i 2 N such that PF (µ) = Pi(µ).
(c) How is the result from (b) related to the above exercise on representative voters?
(d) Now drop the assumption that F is strong from (b) and assume Pareto. Prove that
there is some group G µ N such that PF (µ) = PG(µ).
(e) How are the conclusions of parts (b) and (d) altered if we drop the assumption that
F is simple and add monotonicity and neutrality?
2.94. Assume X µ R and that every pro¯le ((P1 ; R1 ); : : : ; (Pn; Rn )) 2 P R(£) is singlepeaked with respect to ·. We have shown that the correspondence CSM : £ ¶ X is
nonempty-valued, and we have characterized CSM (µ) for each state. We know from Proposition 1.49 that CSM is upper hemicontinuous. Prove that it is actually continuous. (Hint:
We had an exercise in Chapter 1 on convergence of Euclidean preferences that may be
helpful.)
2.95. Consider a divide-the-dollar environment, where X is the simplex in Rn and where
each Ri (µ) has the representation ui(x) = xi . Prove that, if F is non-collegial, then CF (µ) =
;.
148
2.96. Let X µ Rd be convex, let F be a simple PAR, and let R = F (µ) and P = PF (µ).
Prove that, if F is strong and each (Pi(µ); Ri (µ)) is strictly convex, then (P; R) is \strictly
star-shaped," meaning: for all x 2 X, all y 2 R(x), and all ® 2 (0; 1), ®y + (1 ¡ ®)xPx.
Use this to prove that (P; R) satis¯es thin indi®erence.
2.97. Assuming n = 5, consider the weighted q-rule with w1 = w2 = :35 and w3 = w4 =
w5 = :1. Assume X µ Rd is convex, and suppose that each Ri(µ) is Euclidean with ideal
points arranged in a pentagon. Assume that individuals 1 and 2 are not adjacent. Identify
the core of the weighted q-rule for all values of q 2 (:5; 1].
2.98. Prove Proposition 2.33.
2.99. Assume that X = Rd , that each (Pi(µ); Ri (µ)) is continuous and strictly convex. Let
F be a strong simple PAR. Prove that fxg [ PF (xjµ) = fxg [ intRF (xjµ).
2.100. Consider the proof of Plott's theorem, Proposition 2.38.
(i) Why does there exist p 6
= 0 such that, for all i 2 N, p ¢ rui(x¤) 6
= 0?
(ii) Prove that, if G1 [ G2 [ fig ( N, then there exists a non-zero q 2 Rd such that
fj 2 N j ruj(x¤ ) ¢ q = 0g = G1 [ G2 [ fig:
(iii) How do we know jG1 j + jH1j ¸ (n + 1)=2?
2.101. Extend part (i) of Proposition 2.38 to every strong PAR. How is this result related
to Proposition 2.33?
2.102. In stating Proposition 2.38, we assume that rui(x) = 0 for at most one individual.
I gave an example showing the theorem isn't generally true if we drop that quali¯cation.
Suppose we just assume rui (x) = 0 for at most two individuals. How would we weaken
the de¯nition of \radial symmetry at x" so as to maintain the truthfulness of the theorem?
What if more individuals could have a zero gradient at the same alternative?
2.103. Assume X µ Rd and each (Pi(µ); Ri (µ)) is represented by di®erentiable ui such that
there are no shared critical points. Let x¤ 2 intX \ CSM (µ). Prove that, if n is even, then
there exists y 2 X n fx¤g such that yR SM (µ)x¤.
2.104. Assume X µ Rd and each (Pi(µ); Ri (µ)) is represented by di®erentiable ui . Let
x 2 intX, let G µ N, and suppose that zero is not in the semi-positive span of the
gradients
of members of G, i.e., there do not exist f®i 2 R+ n f0g j i 2 Gg such that
P
®
ru
i(x) = 0. Prove that there exists y 2 X such that yPG(µ)x. (In answering
i2G i
this, you may use the following result, which is known as a \theorem of the alternative."
149
Let A denote an m £ m matrix. Either the equation xA = 0 has a semi-positive solution
x 2 Rm n f0g or the inequality Ay > 0 has a solution.)
2.105. Prove that, if ui is a pseudo-concave representation of (P; R), then (P; R) is convex.
2.106. Let X µ Rd . When preferences do not have di®erentiable utility representations,
gradients are not well-de¯ned. But given x 2 X , say p 2 Rd supports Ri (x) at x if, for
all y 2 Ri (x), we have p ¢ (y ¡ x) ¸ 0. In the absence of di®erentiability, Ri (x) could be
supported by many di®erent vectors. Now let x¤ 2 intX and assume each (Pi(µ); Ri (µ)) is
strictly convex. Prove that x¤ 2 CSM (µ) if:
There exist p1 ; p 2; : : : ; pn 2 Rd such that each pi supports Ri (x¤ ) at x¤ and, for
all y 2 Rd , jfi 2 N j pi ¢ y > 0gj < n=2.
Give an example showing that the converse is not generally true.
2.107. Prove that, if ui is a strictly pseudo-concave, then it is strictly quasi-concave. Can
you ¯nd a counter-example to the converse direction?
2.108. Consider the proof of Proposition 2.43.
(i) Where do I use the assumption that the domain of ui is Rd?
(ii) Give the argument for the claim that, if d ¸ 2, then each Cj h is closed and nowhere
dense. What if d = 1?
(iii) Verify the claim that Rim ! Ri in the Hausdor® metric.
2.109. Assume that X = Rd and that PR(£) consists of all pro¯les of Euclidean preferences satisfying CSM (µ) 6
= ;. Give an example demonstrating that, when n is even, the
correspondence CSM is not necessarily lower hemicontinuous. (Hint: Think about n = 4
and d = 2.) What if n is odd?
2.110. Explain how strict convexity is used in the proof of Proposition 2.46.
2.111. Assume X = Rd and individual preferences are Euclidean. Prove the following
claims: for all y 2 Rd, there exists c 2 R such that Hy;c 2 M; when F is strong, there exists
exactly one c for which this is true; the set of such c's is a compact interval.
2.112. Prove part 2 of Proposition 2.48.
2.113. In the proof of Lemma 2.8, how do we know that yj ¢ x
^ < c j? Why does x² solve
the constraints of the maximization problem for small enough ²?
150
2.114.
In the proof of Proposition 2.49, verify that t k = s + (yi ¢ tk )yi and tk+1 =
s ¡ (yi ¢ tk + ci )y i. How do we ensure, at the end of the proof, that vRF (µ)tk ?
2.115. Prove Corollary 2.8.
2.116. How would you adapt Proposition 2.49 (and proof) for the strong top cycle?
2.117. In the discussion following the proof of Proposition 2.49, why is it that, assuming
individual preferences are continuous and F is simple, TP (x) is open? Also, why is it that
y; z 2 bdTP (x) implies yIF (µ)z?
2.118. Prove that, for every Y µ X µ Rd ,
(i) frY is a closed subset of the boundary of Y
(ii) frY = fr(intY ) = fr(closY )
(iii) frY 6
= ; if and only if intY and intY are nonempty.
2.119. Check that, in the example following Proposition 2.56, the weak uncovered set for
simple majority rule is [¡2²; 2²].
2.120. Complete the proof of Proposition 2.59 by showing that W UCF (µ) is closed.
2.121. Assume that n = 3, that X = R2 , and that each (Pi (µ); Ri(µ)) is Euclidean, with
ideal points arranged in an equilateral triangle. Let x
^ be the point at the center of the
triangle. Prove that x
^ 2 W U CSM (µ).
2.122. Consider a divide-the-dollar environment, where X is the unit simplex in Rn and
where each (Pi (µ); Ri(µ)) has the representation ui (x) = xi . What is W UCSM (µ)? (You
can assume n = 3 if it helps.)
2.123.
Let F be a weakly decisive PAR satisfying Pareto. Prove that, regardless of
whether social preferences satisfy thin indi®erence, the strong uncovered set is contained in
the Pareto optimal alternatives: U C(PF (µ)) µ CSP (µ).
2.124. In the proof of Proposition 2.60, how do we know that the median hyperplane with
normal wm ¡ x¤ does not intersect B²=3(x¤ )?
2.125. Give the argument for the claim, in the proof of Proposition 2.60, that f(ym; cm )g
has a convergent subsequence.
2.126. Check that Proposition 2.61 holds for any ball intersecting the F-median hyperplanes in all directions, even if the yolk is not uniquely de¯ned.
151
Chapter 3
Implementation Theory
3.1
The Implementation Problem
3.2
Dominant Strategy Equilibrium
3.3
Nash Equilibrium
3.4
Nash Re¯nements
3.4.1
Strong Nash Equilibrium
3.4.2
Undominated Nash Equilibrium
3.4.3
Subgame Perfect Equilibrium
3.5
Bayesian Equilibrium
152