Gerrymandering-proof social welfare
functions
June 10, 2005
Author 1 : Juan Perote-Peña
Affiliation: Departamento de Análisis Económico, Universidad de Zaragoza.
Running title: Gerrymandering-proofness.
Address for manuscript correspondence:
Juan Perote Peña,
Departamento de Análisis Económico,
Universidad de Zaragoza,
Gran Vía, 2,
50.005
Zaragoza, Spain.
Telephone (Work): +34-976????
Telephone (Mobile): +34-652277537
E-mail address: [email protected]
Fax (Work): +34-976????
1
Abstract
In this paper we deal with a main way of political manipulation of outcomes sometimes called “gerrymandering” in political science. We examine
the possibility of designing constitutions in such a way that the social preferences aggregation is the same regardless of the specific path in which society
has been formed in steps (by merging different coalitions). We conclude
that it is possible for non-trivial environments to design constitutions that
are invariant to the integration process that leads to the formation of the
whole society. We argue that this property is important to avoid conflict and
strategic behavior in regional integration processes like the European Union.
Keywords: gerrymandering-proofness, political manipulation.
JEL classification numbers: D78.
2
1
Introduction
The design of constitutions and the specific properties that should hold when
societies decide the rules to be used to make decisions is not a new topic in
social choice theory. Although the seminal books by Buchanan and Tullock
(1962) and Arrow (1963) stressed the importance of the topic, only recent
works like those of Koray (2000) and Barberà and Jackson (2004a, 2004b)
have developed appropriate theoretical approaches to the problem of constitutional design. Moreover, the economic importance of constitutions has
also been recently found empirically in papers like Persson (2002), Persson
and Tabellini (2000) and Aghion et al. (2002). Our interest in constitutions comes from avoiding their dependency of the society formation path
when there is a regional integration process that can be economic, like the
European Union or the NAFTA or political like the German Re-unification.
Therefore, in this paper we are dealing with a special kind of political manipulation problems that emerge when different institutions within a given
society merge.
Our framework departs from the standard Social Choice Theory in that
we are interested in finding preference aggregation procedures such that always produce the same outcome (social preferences) regardless of how society
has been formed. Imagine the simplest society composed by only three individuals: 1, 2 and 3 and three alternatives: a, b and c. Society could have
been formed by the three individuals joining together in a single step or by a
two-step merging process in which, for example, individuals 1 and 2 formed a
previous institution (possibly in the past) under which a previous preference
aggregation took place and in a second step the institution formed by 1 and
2 merged with individual 3 (itself can be considered a single-agent institution). We claim that this second step aggregation would reflect somehow the
previous agreement made by the institution composed by 1 and 2 in that
the objectives of the institutions do not necessarily coincide with those of
the individuals within it, so that the second step aggregation would not be
equivalent to the social preferences that would surge from a one-step merging
process of the three individuals at the same time. This effect or influence
of previous institutions on the final preference aggregation may take very
different forms and can be justified on very different grounds.
A first possibility is essentially instrumental in that we might well consider
the second merging process as an independent preference aggregation between two new agents (the institutions themselves endowed with preferences
defined by the first step aggregation). The likely final aggregation outcome
would therefore depend on the “bargaining power” of each one. One could
expect that normally the bigger institutions in the first step (those composed
3
by a larger number of individuals) would presumably have a bigger impact on
the final second-step social preferences, but sometimes not only the number
is important, but the names of the individuals within each institution might
have a different effect on the final outcome (assuming that everybody has the
same individual influence on society does not seem a very realistic assumption). Therefore, we should allow for the second step aggregation to depend
on both institutions aggregated preferences and on who are the individuals
contained in each institution. This could reasonably be the same of considering the final outcome of a single step aggregation in which the preferences of
the individuals belonging to each institution in the first step are replaced by
the first step aggregation preferences of their own institution. An example
that can illustrate our approach is Tony Blair’s initial reluctance in the EU
Nice Summit to allow voluntary further-integration processes within countries in the EU. A first consideration for the UK could have been the fact
that further harmonized policies limited to some integrationist members of
the EU could acquire some future non-irreversibility status and would have
more chances to be “imposed” over the rest in the future, but this amounts
virtually to higher bargaining power for these countries in a future integration
step, which is basically our approach.
A second justification of our treatment of institutions comes from the
Public Choice approach to mainly political institutions: because of the very
nature of institutions, their leaders (managers, politicians, bureaucrats, etc.)
are playing an active non-neutral role when aggregating the individuals’ preferences. These leaders have their own interests and tend to identify them (at
least partially) with those of the institutions, their own status being dependent on the survival or influence of the institution itself.
Finally, a third more bizarre way of justifying the way we model institutional influence on the individuals behind them is provided by evidence
from Sociology. Social Scientists have no problem in assuming that some
institutions are social objects that can exert a tremendous influence or bias
on the real preferences of the individuals, by investing on maintaining a good
image, ideology and propaganda and sometimes by the less subtle means of
terror, coaction or brain-washing. We feel that specific references about this
literature are not really needed for our purpose.
But what kind of questions are we trying to answer? We are interested in
finding step social aggregation procedures that are invariant to the specific
way the whole society has been formed. Let us think about the regional
integration example, and in particular, let us consider the European Union
enlargement historical process: we would like to find step aggregation procedures (or “EU supra-national institutions”) such that regardless of which
countries entered the union in previous steps, the final EU preferences would
4
be the same.
Why are we interested in such questions? A first answer is to check
whether our current institutions are such that we could reasonably expect a
somehow specific long-term social preferences or a different one. The answer
is therefore very relevant in the case of judging the long-term world process of
globalization and regional economic integration (for example, to the recent
debate about regionalism versus multilateralism). A second answer comes
from a strategic point of view: if we cannot find desirable institutions that
are somehow immune to the specific path taken to their merging process,
the whole process of integration could be manipulated by some individuals
that can guarantee better outcomes for themselves under a specific merging
path. Manipulation of constituencies (known as “gerrymandering” in Political Science) in the case of regional merging can be also examined from this
point of view. Moreover, although most people consider the overall process
of world social and economic globalization as unstoppable, there exists an
implicit consensus of the process as leading to just one possible outcome or
ideal “politico-economic equilibrium”, but the groups that argue that under
the present rules the concrete step process of integration under which globalization occurs may affect the final outcome very differently may still have a
case. The design of non-manipulable integration paths in this sense is therefore a relevant problem to address to avoid future social conflict on merging
societies (for instance, the Northern Ireland Peace process) and even in more
specific cases as firm’s mergers regulation, etc.
In this paper we prove that there exists gerrymandering-proof social welfare functions satisfying Pareto-optimality and ex-ante strategy-proofness
that require different issues to be assigned to voters with a fixed priority
order.
2
The model and the results
Let N be a society composed by n agents, individuals or citizens denoted
by i, j, k ∈ {1, ..., n} . Society chooses alternatives from a fixed finite set of
alternatives A. Alternatives are denoted by x, y, z, v and w and let A denote
the set of all ordered pairs xy ∈ A. Let < ⊆ A be the set of complete,
reflexive and transitive binary relations on the set of alternatives and =
the set of all logically possible strict orderings among alternatives. Given a
subset ℘ ⊆ <, each individual i ∈ N is endowed with a preference relation
Ri ∈ <, where Pi and Ii denote the corresponding asymmetric and symmetric
parts of Ri . We use the following notation: xy ∈ Ri means that alternative
x ∈ A is as least as preferred by agent i ∈ N as alternative y ∈ A. Let
5
Q
R = (R1 , ..., Rn ) ∈ ℘ ⊆ <n ≡ ni=1 < denote a preference profile for society.
Let us denote as I the preferences that are always indifferent between any pair
of alternatives, i.e., ∀x, y ∈ A, xIy and yIx. We will also use the following
partitioned notation R = (RS , RN\S ) for any sub-society (or coalition) S ⊆
N. When a given preference R ∈ < has been defined, the preference profile
R0 = (RS , RN\S ) denotes the preference profile such that every agent in
S ∈ N has identical preferences R, i.e., Ri = R ∀i ∈ S. Given any subset
Q ⊆ A and any preference profile R ∈ <, we denote as CQ (R) ⊆ A to
the following alternatives: CQ (R) = {x ∈ Q | xRy for all y ∈ Q} . Given a
preference profile R = (Rj )nj=1 ∈ <n , r(x, y, R) = {i ∈ N | xRi y} is the set
of individuals who weakly prefer alternative x to y. A constitutional social
welfare function (CSWF ) is a function F : 2N ×℘n → ℘ with the restrictions:
∀R ∈ ℘n , F (∅, R) = I. This function is intended to aggregate the individual
preferences Ri of the individuals belonging to a sub-society S ⊆ N, i.e.,
F (S, RS , RN\S ) is the social preference relation that a sub-society S ⊆ N
has when the preferences of the members of sub- S are RS and any other
preferences are RN \S .
Definition 1 A CSWF F is independent of irrelevant inputs (or institub N\S ∈ ℘#{N\S} , F (S, RS , RN \S ) =
tions, III) if ∀S ⊆ N, ∀RS ∈ ℘#S , ∀RN\S , R
b N\S ).
F (S, RS , R
III amounts to the following: social preferences for a sub-society S ⊆ N
should be decentralized, i.e., independent of the preferences of individuals
outside the sub-society, and this whatever their preferences might be.
Definition 2 A SWF F is Pareto optimal (PO) or unanimous if (i) and
(ii) hold:
∀xy ∈ A, ∀R ∈ ℘n , ∀S ⊆ N, S ⊆ r(xy, R) → xy ∈ F (S, R). (i)
∀xy ∈ A, ∀R ∈ ℘n , ∀S ⊆ N, S ⊆ r(xy, R) and ∃i ∈ S such that
q(yRi x) → yx ∈
/ F (S, R). (ii)
PO means that for any pair of alternatives x, y ∈ A, any preference profile
R ∈ ℘n and any sub-society S ⊆ N, x should be socially strictly preferred
to y whenever all individuals on S weakly prefer x to y, and there exists at
least one individual strongly prefering alternative x to alternative y. This
property embodies no more than the traditional (strong) Pareto principle
applied to any sub-society..
Definition 3 A CSWF F is independent of institutions formation (IIF) or
“gerrymandering-proof” if III holds and ∀S ⊆ N, ∀S 0 ⊆ S, ∀R ∈ ℘n
F (S, R) = F (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 , RN\S ).
6
IIF stands for the following: the social preference aggregation for the
society as a whole (N) -or any sub-society- should be independent of the
ways in which society has been formed. We assume that the whole society
could have been formed by merging any possible pair of sub-societies, and
when two sub-societies merge, a new preference aggregation occur in which
the individual preferences of each sub-society are no longer their own individual preferences but her own sub-society preferences (aggregated by the
sub-society in a previous step). This is not an unreasonable assumption, since
the institutions that embody the preferences of the sub-society are imposed
to its members and are normally the relevant preferences to take into account
in any merger. Notice that this does not necessarily imply a substitution of
individual preferences by new aggregated preferences induced by education
and other social processes, although it could be understood in that way. We
prefer the interpretation of the merging process as one of two sub-societies
bargaining the new final preferences and weighting somehow their bargaining
power by the number of members in each sub-society. This is accomplished
by using their own (society-dependent) aggregation rule to the total society
composed of two kinds of single-minded citizens: the individuals belonging
to the first sub-society that have identical preferences to those of their own
sub-society and the citizens belonging to the second sub-society that have
identical preferences to their own sub-society’s preferences. Finally, a third
way to understand this property is as a way to model the absence of possibilities of strategic manipulation of constituencies known as gerrymandering
in the political science. To illustrate this, let us consider two alternatives, x
and y and a society composed by nine citizens, five of them “blacks”, who
would rather prefer x to y and four “whites”, who prefer y to x. Two constituencies have to be formed with the total population to fill two seats in
Parliament. If the majority rule is applied both inside the constituencies and
in the Parliament with both elected members, the final preference aggregation depends on the design of the constituencies: Two constituencies formed
by three blacks and two whites each will mean that a black is elected in each
and the final preferences in parliament will impose x over y. Nevertheless,
two constituencies such that one groups five blacks and the second is composed by the four whites plus one black will allow the whites to gain a seat
in Parliament and the final preferences are supposed to be x is considered
indifferent to y. The majority rule is therefore vulnerable to strategic manipulation of the constituencies design. By imposing IIF we are making the
final social aggregation rule invariant whatever the two merging sub-societies
(constituencies) are.
Definition 4 A CSWF F is dictatorial if ∃j ∈ N such that ∀S ⊆ N, ∀R ∈
7
℘n , j ∈ S → F (S, R) = Rj .
A dictatorial CSWF makes an individual a dictator, i.e., the agent can impose his own preferences over the whole set of alternatives for any sub-society
such that he is a member. Notice that dictatorial CSWFs impose no restriction on the aggregation of preferences for sub-societies that do not contain
the dictator. There are dictatorial CSWFs the satisfy III, PO and IIF simultaneously for every number of alternatives and three agents, like the following
one: individual 1 is a dictator and ∀S such that 1/
∈ S, ∀R ∈ ℘n , F (S, R) is
the Borda rule considering only the preferences of individuals in S. Nevertheless, dictatorial CSWFs are considered undesirable, since they allocate
too much power on just a single citizen provided that he belongs to the subsociety. The preferences of the final society will always be imposed by the
dictator. Although the above properties are the core of minimal requirements to be imposed on CSWFs in our contexts, we are also interested in
two properties related to fairness that are specially interesting in the case
of political elections: anonymity and neutrality. We shall need some additional notation for simplicity. Given any subset S ⊆ N and any permutation on the set of agents σ : N → N, Let Sσ ⊆ N be the set defined as
Sσ = {i ∈ N | i = σ(j), ∀j ∈ S} .
Definition 5 A CSWF F is anonymous (A) if for all possible permutations σ : N → N, ∀S ⊆ N, ∀R = (R1 , ..., Rn )∈ ℘n , it holds that
F (S, R1 , ..., Rn ) = F (Sσ , Rσ(1) , ..., Rσ(n) )
Notice that anonymity implies that the agents’ names are unimportant
in the social aggregation process, so it involves non-dictatorship and imposes a fairness equal-treatment of individuals constraint on the CSWF that
is important in many contexts like elections of representatives, etc. Now,
given any preference relation R ∈ ℘ and any permutation on the set of alσ
ternatives
© σ : A → A, Let R ∈ <ªbe the preference relationσdefined as
σ
R = σ(x)σ(y) ∈ A | xy ∈ A & xRy . Notice that preference R replicates
preference relation R and only changes the names of the alternatives following the re-naming implied by permutation σ. We shall only use this property
with preferences domains ℘ ⊆ < such that for all permutation σ and for all
preference relation R ∈ ℘, it holds that Rσ ∈ ℘, such as the universal domain < or the domain of all strict preference rankings of alternatives (with
no indifference allowed).
Definition 6 A CSWF F is neutral (N) if for all possible permutations
σ : A → A, ∀S ⊆ N, ∀R = (R1 , ...,Rn ) ∈ ℘n , ∀xy ∈ A, it holds that
xF (S, R1 , ..., Rn )y ↔ σ(x)F (S, R1σ , ..., Rnσ )σ(y)
8
A neutral CSWF is such that the names of the alternatives are not taken
into account in the social aggregation process, and therefore does not treat
alternatives differently because of its names or identity. This property is
specially interesting when alternatives are intended to be political representatives to be chosen by society rather than issues to be ranked. Finally,
an interesting additional property that can be interesting in many contexts
requires that the social decision process must be immune to strategic misrepresentation of individual preferences, a well-known property called “strategyproofness”. We shall define an extension of this property to our particular
framework.
Definition 7 A CSWF F is manipulable by individual i ∈ S ⊆ N at
preference profile R ∈ ℘n by means of reported preference Ri0 ∈ ℘ if there exists
a real valued utility function ui : A → R representing ordinal preferences
Ri ∈ ℘ (i.e., such that ∀x, y ∈ A, xRi y ←→ ui (x) = ui (y)) such that the
following expression holds true:
X
¤ X
£
ui CQ (F (S, Ri0 , RN\{i} )) >
ui [CQ (F (S, R))] .
Q⊆A, #Q≥1
Q⊆A, #Q≥1
Finally, a CSWF F that is not manipulable by any individual at any preference profile by means of any reported preferences is called “ex-ante strategyproof” (SP).
Ex-ante strategy-proof CSWFs are immune to individual preferences misrepresentation when individuals do not know the feasible set of alternatives
that will come out to choose alternatives within it.
Proposition 1 If #N = 2, there exist non-dictatorial CSWFs such that IIF
and PO hold true.
Proof. Let us consider N = {1, 2} and the following CSWF: ∀i =
1, 2, F ({i} , R1 , R2 ) = Ri and F (N, R1 , R2 ) be the Borda rule. The resulting CSWF is obviously non-dictatorial, P O and III hold trivially for
S = {1} and S = {2} and also for S = {1, 2} , since the Borda rule is
PO in the traditional sense. III also holds trivially for S = {1, 2} . Finally,
gerrymandering-proofness imposes no restriction at all on the CSWFs when
there are only two individuals, so it holds as well.
Proposition 2 If #A = 2, there exist non-dictatorial CSWFs such that IIF
and PO hold true.
9
Proof. Consider A = {x, y} and the following CSWF: ∀S ⊆ N, ∀R ∈
<n , xF (S, R)y ↔ ∃i ∈ S such that xRi y and yF (S, R)x ↔ ∃i ∈ S such
that yRi x. This is an admissible CSWF when there are only two alternatives (transitivity does not impose any constraint), PO obviously hold
and III and IIF also holds since whenever ∃i ∈ S 0 such that xRi y →
xF (S 0 , R)y → ∃i ∈ N such that xRi y, which imply in turn that xF (S, R)y ↔
xF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )y for any S ⊇ S 0 .
Lemma 1 Let ℘ = <. If #N ≥ 2 and #A ≥ 2, the only CSWFs F such that
IIF and PO hold are such that ∀S ⊆ N, ∀y, x ∈ A, ∀R ∈ <n , yF (S, R)x →
yF (S 0 ∪ (r(y, x, R) ∩ S) , R)x ∀S 0 ⊆ S.
Proof. By contradiction, assume that there exist ∃S ⊆ N, ∃x, y ∈
A, ∃R ∈ <n , ∃S 0 ⊆ S such that yF (S, R)x and not yF (S 0 ∪(r(y, x, R) ∩ S) , R)x.
Therefore, it must be that xF (S 0 ∪(r(y, x, R) ∩ S) , R)y. Since S\r(y, x, R) ⊆
(r(x, y, R) ∩ S) , by PO xF (N\ (S 0 ∪ (r(y, x, R) ∩ S)) , R)y and by IIF and
PO, xF (S, F (S 0 ∪ (r(y, x, R) ∩ S) , R)S 0 ∪(r(y,x,R)∩S) ,
,F (S\ (S 0 ∪ (r(y, x, R) ∩ S)) , R)S\(S 0 ∪(r(y,x,R)∩S)) ))y → xF (S, R)y, entering into contradiction with the initial assumption.
Lemma 2 Let ℘ = <, #N ≥ 2 and #A ≥ 2. Every CSWFs F such that
PO, A and N hold must be such that ∀S, S 0 ⊆ N such that S ∩ S 0 = ∅ and
#S = #S 0 , ∀xy ∈ A and given R = (RS , RN\S ) ∈ <n such that ∀i ∈ S,
xPi yIi zIi ...Ii v and ∀j ∈ S 0 , yPj xIj zIj ...Ij v, it holds that xy, yx ∈ F (S ∪
S 0 , R).
Proof. By contradiction and WLG, let xy ∈ F (S ∪ S 0 , R) and yx ∈
/
0
F (S ∪ S , R). Let us consider the following permutation σ : N → N on the
set of alternatives, defined as: σ(x) = y, σ(y) = x, σ(z) = z ∀z ∈ A\ {x, y} .
By N, it must be true that xF (S ∪ S 0 , RS , RS 0 , RN\{S∪S 0 } )y ↔ σ(x)F (S ∪
S 0 , RS 0 , RS , RN\{S∪S 0 } )σ(y), that is,
yF (S ∪ S 0 , RS 0 , RS , RN\{S∪S 0 } )x
(*)
Now, let us first denote as τ S and τ S 0 the following functions: τ S : S →
{1, ..., #S} such that ∀i, j ∈ S, τ S (i) < τ S (j) ↔ i < j and τ S 0 : S 0 →
{1, ..., #S 0 } such that ∀i, j ∈ S 0 , τ S 0 (i) < τ S 0 (j) ↔ i < j (functions τ S and
τ S 0 only rank the individuals in each set S and S 0 following the number assigned to their names). Now,
it holds that #S = #S 0 ,
¡ since by assumption
¢
consider preference profile RS 0 , RS , RN\{S∪S 0 } ∈ ℘n , set (S ∪ S 0 ) ⊆ N and
the following well-defined permutation on the set of individuals σ : ∀i ∈
10

 j iff i ∈ S, j ∈ S 0 and τ S (i) = τ S 0 (j)
j iff i ∈ S 0 , j ∈ S and τ S 0 (i) = τ S (j) . Now, by A, it must be
N, σ(i) =

i
otherwise
0
true that F (S ∪ S , RS 0 , RS , RN\{S∪S 0 } ) = F (S ∪ S 0 , RS , RS 0 , R
¡ N\{S∪S 0 } ), and
¢
0
by (*), it holds that yF (S∪S , RS , RS 0 , RN\{S∪S 0 } )x and since RS , RS 0 , RN\{S∪S 0 } =
R, we enter into contradiction with the initial assumption yx ∈
/ F (S ∪ S 0 , R).
Corollary 1 Let ℘ = =, #N ≥ 2 and #A ≥ 2. There do not exist CSWFs
such that PO, A and N hold.
Proof. Consider sets S = {1} , S 0 = {2} and preference profile R =
(RS , RS 0 RN\{S∪S 0 } ) ∈ <n such that xP1 yI1 zI1 ...I1 v and yP1 xI1 zI1 ...I1 v. By
Lemma 2, it must hold that xy, yx ∈ F (S ∪ S 0 , R), so x must be considered
as indifferent to y by society, which is a social preference not allowed in set
=.
Theorem 1 Let ℘ = <. If #N ≥ 3 and #A ≥ 2, there does not exist
CSWFs such that IIF, PO, N and A hold.
Proof. Let us take any two alternatives x, y ∈ A and consider the following preferences R, R0 and R defined as: xP yP zI...Iv, yP 0 xP 0 zI 0 ...I 0 v and
xIyP zI...Iv (all three preference relations R, R0 and R agree on considering
any alternative z, ..., v as indifferent and all of them strictly worse than both
alternative x and y). Let us consider any three individuals 1, 2, 3 ∈ N and
S = {1, 2, 3} and take the following preference profile R = (RS , RN\S ) =
(R1 , R2 , R3 , RN\S ) = (R, R, R0 , RN\S ) ∈ <n where RN\S ∈ <n−3 are any allowed preferences for individuals in N\ {1, 2, 3} . We now consider any CSWF
F such that IIF, PO, N and A hold and we shall prove that this function
cannot exist. Consider preference profile R ∈ <n and subset S ⊆ N. There
are only three possibilities allowed for F (S, R) that do not violate PO:
Case 1 : F (S, R) = R. Since N holds, considering the following permutation of alternatives σ : σ(x) = y, σ(y) = x, σ(z) = z ∀z ∈ A\ {x, y} , preference profile R ∈ <n and subset S ⊆ N, it must be true that xF (S, R, R, R0 , RN \S )y ↔
σ(x)F (S, R0 , R0 , R, RN\S )σ(y), so yF (S, R0 , R0 , R, RN \S )x holds and by PO,
F (S, R0 , R0 , R, RN\S ) = R0 .
(1)
Now, consider the following permutation of individuals σ
b:σ
b(1) = 3, σ
b(2) =
2, σ
b(3) = 1 and σ
b(i) = i ∀i ∈ N\ {1, 2, 3} . Since A holds, given pref0
erence profile (R , R0 , R, RN \S ) ∈ <n , subset S = {1, 2, 3} and permutation σ
b, it must be true that F (S, R0 , R0 , R, RN \S ) = F (Sσ , Rσ(1) , ..., Rσ(n) ) =
11
F (S, R, R0 , R0 , RN\S ) = R0 , by (1). Now, let us consider set {1, 3} ⊂ S and
F ({1, 3} , R). Since we assume that PO, N and A hold, we can apply Lemma
2 to obtain that
F ({1, 3} , R) = R.
(2)
Now, by (2), (1) and IIF, the following two propositions are also true:
F (S, R) = F (S, F ({1, 3} , R){1,3} , F ({2} , R), RN\S ) = F (S, R, R, R, , RN\S ) = R.
(3)
F (S, R, R0 , R0 , RN\S ) = F (S, F ({1, 3} , R){1,3} , F ({2} , R), RN \S ) = F (S, R, R0 , R, RN\S ) = R0 .
(4)
Now, take (3) and consider preference profile (R, R, R, RN \S ) ∈ <n and
set {2, 3} ⊂ S.
Let us consider now preference profile (R, R, R0 , RN\S ) ∈ <n . Notice that
by PO, it must hold that F ({1, 2} , R, R, R0 , RN\S ) = R and F ({1, 3} , R, R, R0 , RN \S ) =
R0 . But IIF then implies that
F (S, R, R, R0 , RN\S ) = F (S, F ({1, 2} , R, R, R0 , RN\S ){1,2} , R0 , RN \S ) = F (S, R, R, R0 , RN\S ) = R
(5)
and
F (S, R, R, R0 , RN\S ) = F (S, F ({1, 3} , R, R, R0 , RN\S ){1,3} , R, RN\S ) = F (S, R0 , R, R0 , RN\S )
(6)
But take permutation of agents σ
e:σ
e(1) = 2, σ
e(2) = 1, σ
e(3) = 3 and σ
e(i) =
i ∀i ∈ N\ {1, 2, 3} . Since A holds, F (S, R0 , R, R0 , RN\S ) = F (S, R0 , R0 , R, RN\S ),
and by (1) and (6), we conclude that F (S, R, R, R0 , RN \S ) = R0 , entering into
contradiction with (5).
Case 2 : F (S, R) = R0 .We can apply Lemma 1 or alternatively notice that
by Lemma 2, (2) holds and using IIF, F (S, R) = F (S, F ({1, 3} , R){1,3} , R2 , RN\S ) =
F (S, R, R, R, , RN \S ) = R by PO, entering into contradiction with the assumption in Case 2.
Case 3 : F (S, R) = R.Identical to Case 2. By Lemma 2, PO and IIF,
F (S, R) = R 6= R.
Proof. We prove the theorem by demonstrating that IIF, PO, N and A
are mutually incompatible for the case of a society with #N = 3 and #A = 3
for simplicity of notation, so from now on we assume N = {1, 2, 3} and
A = {x, y, z} . At the end of the proof we show the way to extend the proof
easily to the more general case of #N > 3 and #A > 3. Let us consider any
SWF F such that IIF, PO, N and A hold and take the following admissible
preference profile R ∈ ℘3 : xR1 yR1 z, yR2 zR2 x, zR3 xR3 y. Notice that this
12
profile is a member of the family of profiles known as the Condorcet triple.
Therefore, it must be true that there exists at least a pair of alternatives
vw ∈ A such that vF (N, R)w and there is a unique individual i ∈ N such
that vRi w, while the other two agents’ preferences are such that wRj v for
all j 6= i (this follows trivially for any Condocet triple profile: the majority
rule between pairs do not generate a transitive social ranking). Let xy ∈ A
WLG be that pair such that yF (N, R)x and only one individual, 2 ∈ N in
this case, prefers y to x with preferences R2 , while agents 1 and 3 prefer x
to y with their corresponding preferences in profile R ∈ ℘3 . Now, we are
in the conditions to apply Lemma 1, so it holds that yF ({2, 3} , R)x and
yF ({2, 3} , R)x are true, together with the fact that zRj x for j = 2, 3, yRj z
for j = 1, 2 and since PO holds, there are only two possibilities open for
social preferences F ({2, 3} , R): either
zF ({2, 3} , R)yF ({2, 3} , R)x
(A23)
yF ({2, 3} , R)zF ({2, 3} , R)x
(B23)
or
And there exist only two possibilities open for social preferences F ({1, 2} , R):
either
yF ({1, 2} , R)zF ({1, 2} , R)x
(A12)
yF ({1, 2} , R)xF ({1, 2} , R)z
(B12)
or
Let us first assume that (B23) is true. Let us consider the following
permutation of alternatives σ : σ(y) = x, σ(z) = y, σ(x) = z. Since (B23) is
true, by neutrality (N), it must also be true that
σ(y)F ({2, 3} , R1σ , R2σ , R3σ )σ(z)F ({2, 3} , R1σ , R2σ , R3σ )σ(x)
or written differently:
xF ({2, 3} , R1σ , R2σ , R3σ )yF ({2, 3} , R1σ , R2σ , R3σ )z
(1)
where xR2σ yR2σ z and yR3σ zR3σ x. But consider now the following permutation
of individuals σ : σ(2) = 1, σ(3) = 2, σ(1) = 3. By anonymity (A) and (1),
the following is true:
σ
σ
σ
σ
σ
σ
xF ({σ(2), σ(3)} , Rσ(1)
, Rσ(2)
, Rσ(3)
)yF ({σ(2), σ(3)} , Rσ(1)
, Rσ(2)
, Rσ(3)
)z
or written in other words:
xF ({1, 2} , R3σ , R1σ , R2σ )yF ({1, 2} , R3σ , R1σ , R2σ )z
13
where xR1σ yR1σ z and yR2σ zR2σ x, which clearly contradicts both (A12) and
(B12), that must hold as well, so our assumption that (B23) is true is false
and therefore (A23) must hold. Now, consider (A23) and permutation σ. By
N, it holds that
σ(z)F ({2, 3} , R1σ , R2σ , R3σ )σ(y)F ({2, 3} , R1σ , R2σ , R3σ )σ(x)
or in other words:
yF ({2, 3} , R1σ , R2σ , R3σ )xF ({2, 3} , R1σ , R2σ , R3σ )z
(2)
where xR2σ yR2σ z and yR3σ zR3σ x. Now take again permutation of individuals
σ and by (A) and (2), it holds that
σ
σ
σ
σ
σ
σ
, Rσ(2)
, Rσ(3)
)xF ({σ(2), σ(3)} , Rσ(1)
, Rσ(2)
, Rσ(3)
)z
yF ({σ(2), σ(3)} , Rσ(1)
or written in other words:
yF ({1, 2} , R3σ , R1σ , R2σ )xF ({1, 2} , R3σ , R1σ , R2σ )z
where xR1σ yR1σ z and yR2σ zR2σ x, so (A12) must be false and (B12) is the only
remaining possibility open for SWF F . Now, take again (A23) and consider
the following permutation of alternatives σ 0 : σ 0 (y) = z, σ 0 (z) = x, σ 0 (x) = y.
By N, it is true that σ 0 (z)F ({2, 3} , R1σ , R2σ , R3σ )σ 0 (y)F ({2, 3} , R1σ , R2σ , R3σ )σ 0 (x)
where zR2σ xR2σ y and xR3σ yR3σ z and together with the following permutation
of individuals σ 0 : σ 0 (2) = 3, σ 0 (3) = 1, σ 0 (1) = 2, by A, it holds true that
σ 0 (z)F ({σ 0 (2), σ 0 (3)} , Rσσ0 (1) , Rσσ0 (2) , Rσσ0 (3) )σ 0 (y)F ({σ 0 (2), σ 0 (3)} , Rσσ0 (1) , Rσσ0 (2) , Rσσ0 (3) )σ 0 (x)
so the following statement must also be true:
xF ({1, 3} , R)zF ({1, 3} , R)y
(A13)
Notice that by PO, xF ({1, 3} , R)y. Now, notice that by initial assumption,
yF (N, R)x and by GP, the following three statements (C12, C23 and C13
below) must be true:
F (N, F ({1, 2} , R){1,2} , R3 ) = F (N, R)
(C12)
F (N, F ({2, 3} , R){2,3} , R1 ) = F (N, R)
(C23)
F (N, F ({1, 3} , R){1,3} , R2 ) = F (N, R)
(C13)
Let us take now the permutation of alternatives σ 0 and the permutation
of
¡
¢
agents σ 0 defined above and apply them to profile of preferences F ({1, 2} , R){1,2} , R3 .
14
We obtain that since yF (N, F ({1, 2} , R){1,2} , R3 )x, by N and A applied sequentially for permutations σ 0 and σ 0 ,
zF (N, F ({2, 3} , R){2,3} , R1 )y
(3)
must also be true. But since by initial assumption yF (N, R)x and by (C12)
yF (N, F ({1, 2} , R){1,2} , R3 )x, transitivity of social preferences and (3) imply
that the following is true:
zF (N, F ({2, 3} , R){2,3} , R1 )x
(4)
Now, we know that yF (N, F ({1, 3} , R){1,3} , R2 )x hold by initial assumption
yF (N, R)x and (C23). Let us take the permutation of alternatives σ and the
permutation
of agents σ defined
above and apply them to profile of prefer¡
¢
ences F ({1, 3} , R){1,3} , R2 . We obtain easily that xF (N, F ({2, 3} , R){2,3} , R1 )z
is true, which clearly enters into contradiction with statement (4) or with the
assumption of transitivity of social preferences, so therev are no more possibilities open, the assumptions are not compatible and the theorem is proved.
Finally, notice that extending the reasoning above to the case of #N =
n > 3 and #A > 3 is straightforward since all the steps hold for the following
initial profile R ∈ ℘n (take A = {x, y, z, v, w} and N = {1, 2, 3, 4, 5} as
illustration): xR1 yR1 zR1 vR1 w, yR2 zR2 xR2 vR2 w, zR3 xR3 yR3 vR3 w, and
any other preferences for agents 4 and 5. All the properties GP, N, A and
PO are defined for all subsets S ⊆ N, so if they cannot hold for S = {1, 2, 3}
(which has been proved above simply substituting N by S), they fail to be
true for all S ⊆ N. Notice also that alternatives v and w in profile R ∈
℘n play no role in the proof since they are Pareto-dominated by all the
alternatives in S = {1, 2, 3} , and all the steps in the proof above can be
easily extended to allow for more alternatives in this way. Since for this
particular preference profile R ∈ ℘n and this particular set of alternatives
S = {1, 2, 3} the contradiction can be found in the same way, there cannot
exist a SWF F fulfilling IIF, PO, N and A.
Theorem 1 above is therefore an impossibility result that obliges us to
give up some property to find SWFs such that the most important properties
hold. I think that either N or A can be sacrified in some contexts, although in
the case of general elections of representatives both are important properties,
so the negative imposibility result is strong in this contexts. Next, we provide
examples of SWFs that work when we sacrifice A or N alternatively. The kind
of SWFs that emerge are quite restrictive and unappealing in many contexts
actually, but although I think that a full characterization of them is a rather
difficult task, it is dubious to me that sufficiently interesting SWFs work
in the context of all unrestricted strict preference relations. A possibility
15
to obtain more interesting SWFs could be therefore to restrict the domain
further.
Theorem 2 Let ℘ = =. If #N = 3 and #A = 3, there exist non-dictatorial
SWFs such that IIF, PO, N and SP hold.
following SWF, F : ∀R ∈ ℘n , ∀S ⊆ A,
if {1} ∈ S
if {1} ∈
/ S and {2} ∈ S
if {1} , {2} ∈
/ S and {3} ∈ S

 CA\{CA (F (S,R))} (R2 ) if {2} ∈ S
C
(R3 ) if {2} ∈
/ S and {3} ∈ S
CA\{CA (F (S,R))} (F (S, R)) =
 A\{CA (F (S,R))}
CA\{CA (F (S,R))} (R1 ) if {2} , {3} ∈
/ S and {1} ∈ S
o
n
(F (S, R)) =
CA\ C (F (S,R)),C
A\{CA (F (S,R))} (F (S,R))
©A
ª
= A\ CA (F (S, R)), CA\{CA (F (S,R))} (F (S, R)) .
Note that SWF F is well-defined. Let us first explain the mechanics
underlying SWF F. Given any preference profile and any subset S ⊆ N, it
always takes as the socially most preferred alternative the ideal alternative of
individual 1 when he belongs to the set S, individual 2’s most preferred alternative when 1 is not in set S but individual 2 is and agent 3’s most preferred
alternative in any other case. Then, the socially second-ranked alternative
is chosen among the remaining two social alternatives as follows: it selects
first the most-preferred alternative of individual 2 among the remaining two
alternatives when 2 belongs to the set S, individual 3’s most preferred alternative among the remaining two alternatives when 2 is not in set S but
individual 3 is and agent 3’s most preferred alternative among the remaining
two in any other case. Finally, the socially least-preferred alternative will be
the only alternative left.
Now we proceed to prove that SWF F above fulfills IIF, SP, PO and is not
dictatorial, so the theorem is proved. First we prove that F is such that IIF
holds true. First notice that III holds by construction, since only information
about preferences of individuals in each subset S ⊆ N are used to describe
the ordering F (S, R). Now, take any preference profile R = (R1 , R2 , R3 ) ∈
℘n and let us½rename alternatives x, y, z WLG in the following way: x =
CA (R2 )
if CA (R1 ) 6= CA (R2 )
CA (R1 ), y =
and z = A\ {x, y} .
CA\{CA (R2 )} (R2 ) if CA (R1 ) = CA (R2 )
Now, by construction of F it is clear that xF (N, R)yF (N, R)z (with strict
preference, i.e., q [yF (N, R)x] and q [zF (N, R)y] always by construction, so
we shall omit this in the rest of the proof).
Now, we are going to prove that for the three possible ways society may
form, i.e., ∀S = {1, 2} , {2, 3} , {1, 3} ⊂ N, F (N, F (S, R)S , F (N\S, R)N\S ) =
F (N, R). We therefore distinguish between these three cases:
Proof. Let us consider
the

C
(R
 A 1)
CA (R2 )
CA (F (S, R)) =

CA (R3 )
16
Case 1: S = {1, 2} . Notice that by construction of F, it holds that
xF ({1, 2} , R)yF ({1, 2} , R)z. Therefore, again by construction of F, it is
true that
xF (N, F ({1, 2} , R){1,2} , F ({3} , R){3} )yF (N, F ({1, 2} , R){1,2} , F ({3} , R){3} )z.
So it holds that F (N, F ({1, 2} , R){1,2} , F ({3} , R){3} ) = F (N, R) and IIF
holds for this case of society formation.
Case 2: S = {2, 3} . There are two possibilities now: either CA (R2 ) = x
or CA (R2 ) = y. Note that under both possibilities, by construction of F, it is
true that xF ({2, 3} , R)yF ({2, 3} , R)z. Therefore, again by construction of
F, it is true that
xF (N, F ({2, 3} , R){2,3} , F ({1} , R){1} )yF (N, F ({2, 3} , R){2,3} , F ({1} , R){1} )z.
So it holds that F (N, F ({2, 3} , R){2,3} , F ({1} , R){1} ) = F (N, R) and IIF
holds for this case of society formation.
Case 3: S = {1, 3} . In this case, we must consider again the same
two possibilities considered in Case 2: either CA (R2 ) = x or CA (R2 ) =
y. Again under both possibilities and by construction of F, it holds that
xF ({1, 3} , R)yF ({1, 3} , R)z. Therefore, again by construction of F, it is
true that
xF (N, F ({1, 3} , R){1,3} , F ({2} , R){2} )yF (N, F ({1, 3} , R){1,3} , F ({2} , R){2} )z.
and therefore it holds that F (N, F ({1, 3} , R){1,3} , F ({2} , R){2} ) = F (N, R)
and IIF holds for this last case of society formation.
Notice that there are no more possibilities open, so SWF F must fulfill
property IIF. Now we prove that SP holds for F as well. It is clear that
individual 1 has no incentive to misrepresent his preferences, since for all S ⊂
N, if {1} ∈ S, by construction it holds that CA (R1 )F (S, R)x ∀x ∈ A\CA (R1 )
and there is no way in which he can get his second best-preferred alternative
as the second best-preferred alternative in the social ranking by reporting a
different profile from the true one. Now, we focus in individual 2: he cannot
impose the socially first best alternative by reporting different preferences, so
only the second best is open for manipulation, but he is the one who decides
the second best for society as a whole, so there is no advantage in lying about
his true preferences (he can only either be worse-off or break even under any
possible utility function representing his preferences). Finally, individual 3
can only have an influcence in the social decision for the socially second best
alternative when individual 2 is not present in set S ⊆ N and no lie can make
him be strictly better off compared to reporting his true preferences. When
17
he is the only indivual in set S ⊆ N, SWF F makes him a trivial dictator,
so the SWF is ex-ante strategy-proof.
We must now prove that SWF F is also Pareto-optimal. Notice that
for all S ⊆ N such that individual 1∈ S, since F (S, R) = CA (R1 ), the only
contradiction possible with PO would be for the other two alternatives which
are not the first best in the social ordering, say y, z ∈ A and suppose WLG
that yRi z for all i ∈ {1, 2, 3} . Notice that the social ordering in this case must
also impose yF (S, R)z. For all S ⊆ N such that 1∈
/ S, a similar argument
hold when 2 ∈ S and finally in the case of S = {3} , individual 3 decides the
whole social ranking, so F cannot violate PO.
It is also easy to check that SWF F fulfills neutrality (N), since it does
not make any use of the names of the alternatives, but only the information
about “whatever alternative is the most-preferred one by agent i ∈ N” and
the like.
Finally, we must check that SWF F is not dictatorial. Notice that although individual 1 always decides the socially most-preferred alternative
when 1∈ S, there are cases where the second most preferred alternative is
decided by individual 2 when it happens to be the least-preferred alternative
for individual 1, so 1 cannot be a dictator. The fact that when 1∈ S, there
are cases in which the most preferred alternative for 2 is different from the
ideal for 1 (the socially most preferred alternative in SWF F ) prevents individual 2 from being a dictator. Finally, individual 3 clearly cannot be a
dictator, so F is not dictatorial and the theorem is proved.
Theorem 3 Let ℘ = <. If #N = 3 and #A = 3, there exist non-dictatorial
SWFs such that IIF, PO and N hold.
Proof. Let ℘ = ℵ be the domain of all logically possible preference
profiles such that all agents have a single best preferred alternative and let
Fb be the CSWF defined in Theorem 2 for the domain ℘ = ℵ (it is easy to
check that PO, N and IIF still hold for this slightly bigger domain). Let us
consider the following SWF, F : ∀R ∈ <n , ∀S ⊆ A,
½
Fb(S, R) iff R ∈ ℵ
F (S, R) =
F (S, R) iff R ∈
/ℵ
Theorem 4 Let ℘ be the set of all logically possible strict orderings among
alternatives. If #N ≥ 3 and #A = 3, there exist non-dictatorial SWFs such
that IIF, PO and A hold.
18
Proof. Let {x, y, z} = A. Given any set S ⊆ N, alternative x ∈ N
and any preference profile R ∈ ℘n , let B(x, S, R) ⊆ N be the set of individuals in S whose best preferred alternative is x ∈ A in profile R ∈
℘, SB(x, S, R) ⊆ N the set of individuals in S whose second best preferred alternative is x ∈ A in profile R ∈ ℘ and L(x, S, R) ⊆ N the
set of individuals in S whose least preferred alternative is x ∈ A in profile R ∈ ℘. Now, given any set S ⊆ N and preference profile R ∈ ℘n ,
let us consider the following set: K(S, R) ⊆ N defined as K(S, R) =

 B(x, S, R) ∩ SB(y, S, R) iff B(x, S, R) ∩ SB(y, S, R) 6= ∅
B(x, S, R) ∩ SB(z, S, R) iff B(x, S, R) ∩ SB(y, S, R) = ∅ and B(x, S, R) ∩ SB(y, S, R) 6= ∅ a

consider the following SWF, denoted as F and parameterized by the ordering
x, y, z ∈ A: ∀R = (R1 , ..., Rn ) ∈ ℘n , ∀S ⊆ A,
F (S, R) = Ri iff ∃i ∈ S such that i ∈ B(x, S, R) ∩ SB(y, S, R)
F (S, R) = Rj iff B(x, S, R) ∩ SB(y, S, R) = ∅ and ∃j ∈ S such that i ∈ B(x, S, R) ∩ SB(z, S,
yF (S, R)z
First, we must prove that SWF F is a well-defined SWF such that ∀R ∈
℘ , ∀S ⊆ A, F (S, R) ∈ ℘. The only possibilities to check for F (S, R) to be
intransitive for some R ∈ ℘n and S ⊆ A are the following: Case 1: ∃i ∈ S
such that xRi y and ∀j ∈ S such that zRj x, and therefore it holds that
xF (S, R)y and zF (S, R)x. Then, transitivity alone implies that zF (S, R)y,
and notice that we are also under the conditions for this to occur in the
definition of the function. Case 2: ∃j ∈ S such that xRj z and ∀i ∈ S such
that yRi x, and therefore it holds that xF (S, R)z and yF (S, R)x. Hence,
transitivity alone implies that yF (S, R)z, but notice that we are again under
the conditions for this to be true in the definition of the function for the
ranking of the pair yz ∈ A: actually, individual j 0 s preferences must be
transitive since R ∈ ℘n , so it must hold that yRj xRj z and yRj z is true, so
although it is not true that {∃i ∈ S such that xRi y and ∀j ∈ S, zRj x} , the
definition of SWF F establish that yF (S, R)z.
Now, we proceed to prove that SWF F fulfills IIF. Notice that III holds
trivially, since only preferences of agents in subset S ⊆ N are considered
when constructing F (S, R). Now, to prove that GP holds, we must take any
preference profile R = (R1 , ..., Rn ) ∈ ℘n , any set S ⊆ N and any subset
S 0 ⊂ S and it is sufficient to prove that for any pair of alternatives xy ∈ A,
xF (S, R)y if and only if xF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )y. We shall distinguish between the three possible (unordered) pairs allowed: xy, xz, yz ∈ A
and prove it for each one:
Pair xy ∈ A: Given the definition of SWF F, xF (S, R)y if and only if
∃i ∈ S such that xRi y. Now, if i ∈ S 0 ⊂ S, it holds that xF (S 0 , R)y and
n
19
therefore, again by construction of F , xF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )y.
Otherwise, if i ∈
/ S 0 ⊂ S,then i ∈ S\S 0 and by definition of F, xF (S\S 0 , R)y,
which also implies that xF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )y. Now, given the
definition of F, yF (S, R)x if and only if ∀i ∈ S, yRi x. Therefore, since it
is also true that ∀i ∈ S 0 , yRi x, and ∀i ∈ S\S 0 , yRi x, both yF (S 0 , R)x and
yF (S\S 0 , R)x hold so, by construction of F , there does not exist any i ∈ S
such that xRi y in preference profile (F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 ) ∈ ℘n , so
by definition of F, yF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )x and the proof for this
unordered pair is complete.
Pair xz ∈ A: This case is virtually symmetric to the previous one. By definition of SWF F, xF (S, R)z if and only if ∃j ∈ S such that xRi z. If j ∈ S 0 ⊂
S, it holds that xF (S 0 , R)z and by construction of F , xF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )z.
The other case occur when j ∈
/ S 0 ⊂ S,so j must belong to the set S\S 0 and by
definition of F, xF (S\S 0 , R)z, implying that xF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )z.
The cases in which zF (S, R)x is true entail by construction that both ∀j ∈ S 0
and ∀j ∈ S\S 0 , zRj x, so zF (S 0 , R)x and zF (S\S 0 , R)x are true by construction of SWF F , so there does not exist any j ∈ S such that xRj z in
preference profile (F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 ) ∈ ℘n , and by definition of F,
zF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )x, completing the proof for this pair.
Pair yz ∈ A: By definition of SWF F , yF (S, R)z is true if and only if
∃h ∈ S such that yRh z and it is not true that {∃i ∈ S such that xRi y and ∀j ∈ S, zRj x} .
Take any S 0 ⊂ S ⊆ N and consider the following cases:
Case 1 : ∃h ∈ S such that yRh z and ∀i ∈ S, yRi x. There are two possibilities now: either h ∈ S 0 or h ∈ S\S 0 . Take the first one WLG: h ∈ S. Then,
by definition of F, yF (S 0 , R)z since it is true that ∀j ∈ S 0 , yRj x. Moreover,
since ∀j ∈ S\S 0 , yRj x, it is true that by definition of F, yF (S\S 0 , R)x, so
yF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )x holds true and ∃h ∈ S whose preferences
in profile (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 ) are such that yRh z, so by construction of SWF F, it holds that yF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )z.
Case 2 : ∃h ∈ S such that yRh z and ∃j ∈ S such that xRj z. Again,
there are two possibilities open: either h ∈ S 0 or h ∈ S\S 0 . Take the first one
WLG: h ∈ S. Again, there are two possibilities now:
Subcase 2.1.: j ∈ S 0 . Then, by construction of F , yF (S 0 , R)z and xF (S 0 , R)z,
so ∃h, j ∈ S whose preferences in profile (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 ) are
such that yRh z and xRj z, so by construction of SWF F, it holds that
yF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )z.
Subcase 2.2.: j ∈
/ S 0 . Then, j ∈ S\S 0 and
Using the definition of F, yF (S 0 , R)z since it is true that ∀j ∈ S 0 , yRj x.
Moreover, since ∀j ∈ S\S 0 , yRj x, it is true that by definition of F, yF (S\S 0 , R)x,
so yF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )x holds true and ∃h ∈ S whose preferences in profile (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 ) are such that yRh z, so by
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construction of SWF F, it holds that yF (S, F (S 0 , R)S 0 , F (S\S 0 , R)S\S 0 )z.
References
[1] Aghion, P., Alesina, A. and Trebbi, F.: “Endogenous Political Institutions”, mimeo: Harvard, 2002.
[2] Arrow, K.: “Social Choice and Individual Values”, 1963.
[3] Barberà, S. and Jackson, M.O.: “Choosing How to Choose: self stable majority rules and constitutions”, Quarterly Journal of Economics,
2004a, forthcoming.
[4] Barberà, S. and Jackson, M.O.: “On the Weights of Nations: Assigning Voting Weights in a Heterogeneous Union”, 2004b, mimeo: Caltech,
“http://www.hss.caltech.edu/Jacksonm/Jackson.htm”.
[5] Buchanan, J. and Tullock, G.: “The Calculus of Consent: Logical Foundations of Constitutional Democracy”, 1962, University of Michigan
Press.
[6] Koray, S.: “Self-selective Social Choice Functions Verify Arrow and
gibbard-Satterthwaite Theorems”, Econometrica, LXVIII (2000), 981996.
[7] Persson, T.: “Do Political Institutions shape Economic Policy?”, Econometrica, LXX, (2002), 883-906.
[8] Persson, T., Tabellini, G.: “Political Economics: explaining economic
policy” (Cambridge: MIT Press, 2000).
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