I.
CONSTRUCTION OF OUTCOME FUNCTIONS
GUARANTEEING EXISTENCE AND PARETO
OPTIMALITY OF NASH EQUILIBRIA
by
Leonid Hurwicz and David Schmeidler
Discussion Paper No. 76-73, August 1976
..
'I
Center for Economic Research
Department of Economics
University of Minnesota
Minneapolis, Minnesota 55455
CONSTRUCTION OF OUTCOME FUNCTIONS GUARANTEEING EXISTENCE
AND PARETO OPTIMALITY OF NASH EQUILIBRIA
Leonid Hurwicz* and David Schmeidler
Introduction
Recent formulations of social decisions problems focus on rules
by which a group of individuals (the society) arrives at a choice
among available alternatives.
Such a rule, which we shall call the
outcome function (Gibbard's "game form," see [ 3 J), specifies the
permissible strategies of each individual and associates with every
combination of permissible strategies a particular element (outcome) of
the set of alternatives.
The desirability of choices made is evaluated
in terms of preferences each individual is assumed to have among the
alternatives.
Within this framework, Gibbard and Satterthwaite [3, 8) have shown
that where there are more than two alternatives, there does not exist
a non-dictatorial outcome function with dominant strategies for all
preference profiles.
This result encourages the search for non-dictatorial
outcome functions retaining some of the main advantages of
having
dominant strategies even when such strategies do not exist, in particular the Pareto optimality of outcomes and the Nash equilibrium
property of solutions.
Along these lines, . the present paper deals with the question whether
there are non-dictatorial outcome functions with the following two
properties:
for every preference profile there exists a Nash equili-
*The research of Leonid Hurwicz was in part supported by National
Science Foundation grant No. 0620-5371-02.
-2-
brium, and, for every profile, every Nash equilibrium outcome is Pareto
optimal.
(We call such outcome functions acceptable.)
The latter property is not necessarily satisfied by outcome functions possessing dominant strategies, but we regard it as highly desirable
given the possibility of multiple Nash equilibria.
We find that for two person
acceptable outcome functions
societies there are no non-dictatorial
regardless
of the number of alternatives.
(In the Gibbard-Satterthwaite framework,which does not require the
Pareto-optimality of all Nash equilibrium outcomes, there is no
impossibility
result when there are only two alternatives.)
Whenfue society consists of more than two persons, the results
depend on the
are
class
of admissible
profiles.
If indifferences
ruled out, non-dictatorial acceptable outcome functions are shown
to exist.
The
construction
constitute the main
subject
and properties of such functions
of our paper.
On
the other hand, if
all profiles with indifferences are also permitted,
it is
shown that no acceptable outcome functions exist.
However, when
the requirement of Pareto optimality is replaced by that of weak Pareto
optimality(with a corresponding notion of
weak acceptability) it
turns out that every function acceptable for strict preference profiles
is weakly acceptable even where indifferences are·present.
The problem studied in this paper also arises
\
in the context of
recent work in the theory of optimal resource allocation.
Thus, in
models of economies with public goods, Groves and Ledyard
[4
devised
J
have
outcome functions guaranteeing the Pareto optimality of Nash
equilibrium allocations over a certain
range
of preference profiles
-3-
and initial endowments.
Analogous
outcome functions have been
constructed for pure exchange economies (Hurwicz [6]).
However, even
for certain "classical"preference profiles, the Groves-Ledyard outcome
functions either
lack
Nash-equilibria or generate
optimal Nash equilibrium outcomes ( Ledyard
[ 7 ]), and similar
phenomena are likely to appear in pure exchange models.
point
non-
These difficulties
to the importance of requiring the existence of Nash
equilibria
~or
all profiles of a given class and not merely the optimality
of the outcomes whenever the Nash equilibria happen to exist.
The Nash equilibria obtained in our paper, like those in the work
of Groves and
.'
[ 4 ] and Hurwic.z [6 J, are of the
Ledyard
naive
("non-manipulative" in the terminology of [5]) type, and may not be immune
to manipulation
Farquharson
or misrepresentation
C 2 ],
of preferences.
(See
Gibbard [3 J, p. 591, and Hurwicz [5J.)
Notations and Definitions
An outcome function
Cartesian product
set
A.
i,
with
N
= {l, ••• ,n}
S.
as the set
~
denoting the set
The set of transitive total (i.e., complete and reflexive)
binary relations on
profile.
of finite sets into a non-empty finite
A as the set of outcomes and
of strategies of person
element of
maps, by definition, a non-empty
SIX ••• xSn
We refer to
of persons.
f
A
is denoted by
t' , an n-list
(RI, ••• ,Rn )
t' .
Denoting by
of such elements is called a
The set of all profiles is denoted by
antisymmetric relations in
~'
is denoted by
~ will be called strict profiles.
R a generic
t,n.
T,.
The subset of
The elements of
-4-
An outcome function
L.
!n
f
and a profile
R
-
=
(R1, ••• ,R)
n
define an n-person ordinal game is strategic form.
of strategies
x
=
(xl, ••• ,xn )
Slx ••• xSn d~f ~
in
An n-list
is said to be
a Nash equilibrium (NE for short) of this game if, for every
in
N and for every
Xi+l' ••• ,Xn ) .
y
in
S. :
The outcome
f(~)
a
is then called a Nash outcome.
is said to be Pareto optimal (PO
for short) if there is no other outcome
N ,
in
i
bR.a
1
i
1
Given this game, an outcome
all
in
and, far some
j
b
in A such that:
in
A , not
for
aR.b •
J
An outcome function is said to be acceptable for a set of profiles
if, for every profile in this set, there is a
and every
NE
NE
is
PO.
'.
Some writers (e.g., Arrow and Hahn [ 1 ], p. 91) use a concept of
call_~
what we shall
Pareto optimality.
An outcome
to be weakly Pareto optimal if there is no outcome
b P.a
for all
i EN. (bP.a
iff
bR.a
1 1 1
a E A is said
b E A such that
and not
aR.b.)
1
An outcome function is said to pe weakly acceptable for a set of
profiles if, for each profile in the set, there is a
NE
is weakly Pareto optimal.
An
outcome function
f
if there is a person, say,
every
x
in
S
there is a
= a.
in
strategies
is said to be weakly (or
j , such that for every
y
Sj
with
~-)
a
• a
used by others.
a
e-)
dictatorial
in A and
f(xl, ••• ,xj_l,y,
j , such that for every
with
~
in
It is said to be strongly (or
there is a person, say,
y
and every
NE
dictatorial if
in A there is a
regardless of the
-5-
An outcome function will be called non-dictatorial if there is
no weak dictator.
Using this terminology, the problem dealt with in our paper is
whether there exist non-dictatorial outcome function acceptable (or
weakly acceptable) for all profiles, or all strict profiles.
The
question is non-trivial only if there are at least two outcomes in
two persons in
N , and each per"son's strategy set
S.
1.
A ,
contains at
least two elements.
Summary of Results*
Theorem 1 states that for two persons a function that is acceptable
for all strict profiles is dictatorial.
Corollary 1 extends this
result to weak profiles.
In Section 2 we provide a complete analysis of the case involving
three persons, two outcomes, and two-element strategies for strict
profiles.
The "kingmaker" outcome function is shown to be acceptable
for strict profiles, but it lacks symmetry across persons.
It is shown
that there exists only one type of outcome function possessing such
symmetry, viz. one in which split votes yield one outcome and unanimous
votes another.
It should be noted that the simple majority outcome
function, although symmetric, is not acceptable.
In Section 3 we deal with weak profiles (where indifferences are
allowed).
Theorem 2 states that for the class of all such profiles no
acceptable outcome function exists.
It follows, however, from Lemmas
I and 2 of Section 3 that functions acceptable for strict profiles are
weakly acceptable for weak profiles.
*Some of the results presented in this paper were obtained
independently by Eric Maskin.
-6-
Theorem 3a of Section 4 states that there is a non-dictatorial
outcome function ("kingmaker") acceptable for all strict profiles
regardless of the number of persons and alternatives.
Theorem 3b
deals with a broader class of non-dictatorial functions acceptable
for all strict profiles; in those functions the person making decisions
is chosen by an elective process.
Examples of alternative (non-
elective)hierachial structures are also given.
Since the outcome functions of Theorems 3a and 3b lack symmetry
across persons, we devote Section 5 to the problem of constructing
functions having this symmetry property.
It is shown (Lemmas I and
2, and Theorem 4) that functions symmetric across persons and acceptable for an arbitrary number of persons over strict profiles can be
constructed when the number of alternatives does not exceed three.
The case of an arbitrary number of alternatives is not treated.
In Section 6 we consider performance functions and games in
characteristic function form as means of evaluating the relative
power of players •
.1
-7-
Section 1:
The Case of Two Persons
A
The simplest non-trivial case is one in which
N
= {1,2} • and #Sl = #S2 = 2.
possible outcome functions.
=
{a,b} ,
Here we can easily exhibit all
Figure 1 contains this list, omitting
outcome functions obtainable from those shown by permutations of
persons, outcomes or strategies.
In Figure 1 an outcome function
is represented by a matrix, with the first person controlling the
rows, the second person controlling the columns, and the entries
representing the corresponding outcomes.
a . a
a a.:
a. b
bl
a
:@'a
a
a ,J>. .
b
~J
f3
f4
"
FIGURE I
It is clear that the outcome function
tatoria1.
Outcome functions
f2
and
f3
f1
is strongli dic-
yield non-optimal Nash
equilibria when both persons strictly prefer
b
to
a.
For
this profile every Nash equilibrium is obviously non-optimal with
f2 ·
For
f3' there is a non-optimal NE for this profile at
the encircled entry (top row, left column).
With
f4
outcome
function, there is no NE when the two persons have opposite
strict preferences.
Thus only a strongly dictatorial outcome
function has the two properties of existence and optimality of
Nash equilibria.
-8-
Remark:
Note that under
f ,
3
which is non-dictatorial, every player
always has a dominant strategy relative to his preferences; hence
f3
is non-manipulable in the Gibbard-Satterthwaite sense on
E,2
This shows that non-manipulability does not imply acceptability.
When there are no more than two persons the following generalization is obtained
Theorem 1
Then
Let n ~ 2, and, let
f
f
be acceptable for
En.
is strongly dictatorial.
Remark
A special case of this result is obtained when
substituted for
E
E
is
in the statement of Theorem 1; this will be
referred to as Corollary 1.
Proof
For
Clearly,
n
=2
f
is strongly dictatorial for
n
=1
by definition.
, we shall think of outcome functions as matrices with
persons I and II respectively controlling rows and columns.
We first
note that, given an arbitrary non-trivial partition of the set of
outcomes
A into
Band
B,
(B #
0,
B
=A \
B ,
B#
0) it is
not the case that every row and every column contains elements of
both
Band
B.
Otherwise, NE cannot exist either for profiles of
the form
I
II
,I
B
B
B
B •
-9-
(We represent preferences of, say, the first person by
I
B
B
when this person prefers every element of
B to every element of
B .)
Therefore, there is either a B-row (i.e., a row consisting
of elements of
B only) or a B-row or a B-column or a B-column.
Consider the first case, i.e., suppose that there is a B-row.
We
then show that there must also be a B-row.
Note first that in this case there can be no B-column.
Other-
wise, the intersection of this column with the B-row is an element
of
B which would constitute a non-PO
I
II
B
B
b*
b*
B \{b*}
NE for profiles of the form
B \ {b*}
Now suppose that there is no B-row.
profiles of the form
I
II
B
B
B
B
Then there is no NE for
b*
~lO-
or for profiles of the form
I
II
B
B
B
B
For person I will always want and be able to prevent a NE with
a B-outcome (since there is no
~-co1umn),
while person II will always
want and be able to prevent a NE with a B-outcome (since, by hypothesis, there is no B-row'.
This contradiction of existence of Nash
~2
equilibria for all profiles in
implies that there must be a
B-row.
Clearly, by symmetry, it follows that, for any non-trivial par-
{B, B}
tition
of
there are either two rows, a B-row and a
A
B-row, or two columns, a B-co1umn and a B-co1umn.
We now apply the last observation to partitions of the form
{ {a}, A \
{a} } where
that, for every
of
a's.
for every
a
in
a
is a single element of
A there is either a row of
It follows
A
ais
or a column
This, in turn, implies that either there is a row of
a's
a
in
in
A, or there is a column of
a's for every
a
In the first case, person I is a strong dictator; in the
second case person II is a strong dictator.
Q.E.D.
Remark
Although we have used the language of matrices, the proof was
not limited to finite
A or
S. .
1
Hence the theorem holds for
infinite outcome and strategy sets, provided there are no more than
two persons.
A.
-11Section 2:
The Case of Three Persons with Strict Preferences
It is helpful to start by considering the simplest non-trivial
subcase, that of two alternatives,
each player.
A
= {a,
b}, with two strategies for
In contrast to the case of two persons,
n = 2,
we find
that there are several types of outcome functions guaranteeing the
Pareto optimality of all Nash equilibria and the existence of Nash
equilibria for all (strict) profiles.
One such type of outcome function will be referred to as
"kingmaker."
An example is given in Figure 2, with person III as
the kingmaker.
III
z2
Y2
It
I
,j
.
--
.'~.-
YI
Y2
.~---
a
a
Xl
a
b
b
b
x
a
b
2
Figure 2
Here person I controls the rows (strategies
columns (strategies
Xl ' x ), II controls
2
YI' Y2) , while III has the choice between
the left and right matrices (strategies
zl' z2)'
It is clear that if III chooses strategy
zl
(the left matrix),
player I acquires dictatorial powers (becomes a "king for a day").
,t
Similarly, if III chooses
z2' player II acquires dictatorial powers.
However, it is clear that none of the three players is a dictator in
the sense of our definitions.
It will be shown below (Theorem 3a)
that a "kingmaker" outcome function guarantees the Pareto-optimality
-12of all Nash equilibria and their existence for any number of persons
and outcomes provided all preferences are strict (no indifference),
~n .
i.e., all profiles are in
The kingmaker outcome function may be criticized for its lack
of symmetry across persons.
least for
It is therefore interesting that at
n = 3, and #A=2=#S.
~
for all i in
functions possessing such symmetry property.
f
symmetric across persons if
N, there do exist outcome
We call an outcome function
[(xi' Yj' zk)' i, j, k
invariant with respect to permutations of
i, j, k.
£
{1,2} is
Thus, for instance
etc. (See also Section 5 below.)
In Figures 3! and
3~
we show two such outcome functions, omitting
the labels for strategies
'-b· a
--·-1
a a
a
a
i
,
a
b
a
b
.....-..
b
b
b
b
b
a
Figure 3~
Figure 3!
We now show that, for the class of cases here considered, there
are no outcome functions symmetric across persons other than those
in Figures
1
2
~ and~.
First, observe than an outcome function symmetric
across persons must be of the form shown in Figure 41 . (Conventions
concerning strategies are the same as in Figure 2, but strategy labels
are omitted.)
Figure 41-
-13-
Now we must require
x I y , since otherwise there would arise a
non-Pareto optimal NE at
x I x,
with
X
€
A.
x
when the profile is
I
II
III
x
x
x
x
x
x
four possible assignments of the values
of these assignments, viz.
y
=
a
=
v
v I w.
Similarly, we must have
x
a, b
to
This leaves
x, y, v, w.
Two
= a = w, y = b = v and x = b = w
yield the functions of Figures
3~ and 3l respectively.
The other two result in the non-existence of NE for certain profiles.
Remark:
Note, in particular, that simple majority voting is not
acceptable because it can yield non-PO Nash Equilibria.
inspection of this outcome function (see Fig.
The
4~) shows that
' ,
ttttf
a
b
b
b I,
?
Fig. 4=there is a NE at the encireled outeome,
corresp~mding
vote for a, regardless of the prevailing profile.
PO when everyone prefers
b
to
a.
to a unanimous
Such NE is non-
There also is a NE corres-
ponding to a unanimous vote for b (lower right-hand corner of the
second matrix), which is non-PO when everyone prefers
a
to
b.
-14-
We notice immediately that the outcome function in Figure
is obtained from that in Figure 3! by replacing
vice versa.
a
with
3~
band
However, neither function by itself is symmetric with
respect to outcomes, i.e., as between
a
and
b
On the other
hand it is clear that the kingmaker function is symmetric with respect to outcomes, though the precise meaning of this symmetry must
still be supplied.
It, therefor.e, becomes natural to ask whether
there are outcome functions that are symmetric both across persons
and across outcomes.
An outcome function is said to be symmetric across outcomes
~
if, for every permutation
n
permutations
for all
cr
i
, cr
(xl"" ,x )
n
8.
i
in
A
1
~
A
+
8 , i
of the outcome set, there are
€
N , such that
8
Clearly, the kingmaker function is symmetric across outcomes
by this definition, while the functions in Figures 31 and
not.
3~
are
But we demonstrated above that the latter were the only
functions (for
n -3
-
,
#A-2
- , #8.
1
=2
the property of symmetry across persons.
for
i
€
N )
possessing
It follows that for this
case there are no outcome functions symmetric both across outcomes
and persons.
Are there other outcome functions that cannot be derived from
either kingmaker or Figure 3 functions by suitable permutations ?
-15It turns out that there are.
One is shown in Figure ~.
IillI IillI
IillI lilll
Figure
s!.
We note that interchanging
Figure
a
and
b
#
together with the
permutation of the two strategies of player III leaves this outcome function unchanged.
Hence we have here, as with the king-
maker function, symmetry across outcomes.
By interchanging persons II and III we obtain an outcome function
(see Figure
of Figure
2
~)
5!,
that is somewhat different in appearance from that
It is seen that for the function of Figure
5~
player
III, by selecting his first strategy, can give dictatorial powers
to player II; however, unlike with the kingmaker function, he cannot
do this for player I.
Must an outcome function by symmetric across either persons
or outcomes?
That the answer is in the negative is shown by yet
another pair of outcome functions (equivalent by interchange of
players) presented in Figures 6! and
6~.
fa 'bi
i at:~~
I
Figure 61.
P
:
Figure
6~
Clearly, these functions are symmetric neither across persons nor
across outcomes.
A systematic inspection reveals that for
#8.~ = 2 , i
€
n
= 3 , #A = 2 ,
N , every outcome function acceptable for
En
-16is reducible to one of the four types (kingmaker, Figure 3, Figure 5,
Figure 6) by suitable permutations of persons, strategies, or outcomes.
-17-
Section 3:
The Case of Unrestricted Domain
E,n
An Impossibility Result
Theorem 2
n ~ 2 ,
Let
/FA
~
f
2
is a pro'file
S
~
A be an outcome function with
If f or every
R
where
Let
C
Suppose
f
R*
is non-PO.
acceptable for
E,n.
R be a profile of the form
=A\
a
there is a NE, then there
such that a NE outcome for
R*
I.e., for n ~ 2, #A ~ 2, there is no
Proof
"'~In
-'
1'n
I
II
III
n
a
b
b
b
b
a
a
a
C
C
C
C
{a, b}.
Then only
is a NE outcome.
a
and
b
are Pareto optimal.
Then consider the profile
R*
of the
form
where I
"
I
II
III
n
ab
b
b
b
C
a
a
a
C
C
C
is indifferent between
a
and
a NE outcome, but i t is no longer PO.
b
.
Clearly
a
is still
-18-
On the other hand if
profile
R**
b
is a NE outcome then the following
yields a non-PO NE
I
II
III
n
a
ab
ab
ab
b
c
c
c
C
If neither
outcome for
a
nor
b
is' a NE outcome for
R, then any NE
R is non-PO.
Q.E.D.
Remark
Note that even a strong dictator does not guarantee the optimality of all NEls when indifferences is allowed.
Weak Acceptability
We now establish two lemmas which imply that outcome functions
acceptable for En (strict preferences) are weakly acceptable for
'n
E
(allowing indifferences).
-19-
Lemma 1
every
~,n
Let
~n
R in
S
~
A be an outcome function such that for
each NE outcome is PO.
Then for every
R'
in
each NE outcome is weakly PO.
Lemma 2
every
R'
f:
Let
R
in
f:
~n
in
S
A be an outcome function such that for
~
there is an NE.
Then there is a NE for every
~,n .
Remark
In virtue of Theorems 3a and 3b below, these lemmas
imply that there exist outcome functions weakly acceptable
~,n
for
when
n > 2 .
Proof of Lemma 1
f(~)
R'
= a.
be a NE for a profile
For each
that, for all
(b
1.
x
Then define the profile
as follows.
a R~b
Let
b
p~a
~
Clearly
in
i
A,
R
N ,R.
€
1.
b R'a
is still a NE for
is also preferable under the ordering
Furthermore, by the assumption on
such that
b R,a
l.
for all
follows that there is no
Hence
a
i
b
€
N
a
such
and
a R.b
1.
R~b.)
1.
This is so because for
R
every person any outcome preferable to
,with
~
b pta
i
but not
i
n
~'
corresponding to
i f and only i f
means, as usual,
in
is any ordering in
1.
b R.a
~n
in
R'
a
under the ordering
R.1.
R'
i
,
there is no
b
in
But by definition of
R
it
f
in A such that
b
P~a
1.
for all
i
A
€
N.
is weakly Pareto optimal.
Q.E.D.
if
-20Proof of Lemma 2
in
2:
R'
in
(I.e.,
Q be a fixed (arbitrarily chosen) ordering
Q is reflexive antisymmetric.)
Given a profile
2:,n , we define the corresponding profile
follows.
or
Let
For each
a I!b
1
difference.
and
aQb
1. e. ,
1
(Here, as usua 1,
means
a
R~b
I.
1
a P'b
and
i
is the symbol for inb
R~a
1 1 1
f , there is a NE, say
By our assumption on
as
if and only if either
a R.b
a I!b
R
.)
for
~ ~
R.
But clearly
x
is also a NE for
some person
i
who by changing strategy could obtain an outcome
strictly preferred under
R'
i
R' .
Otherwise there would be
But the same strategy
to
change would have been feasible and preferable under
tradicting the NE property of
x
under
R. , con1
R
Q.E.D.
-21Section 4:
n
Existence of Outcome Functions Acceptable for L When n > 2
In this section we shall first show that the kingmaker outcome
function introduced earlier for three persons (see Figure 2) can be
defined for arbitrary
n > 2
and turns out to be acceptable.
We
also construct other types of outcome functions acceptable for
arbitrary large
n
One category involves what we shall call in-
direct kingmakers; another category provides for the election of a
"king for a day," i.e., someone with dictatorial powers.
Theorem 3a
The following ("kingmaker") outcome function
~n
is acceptable for
and nondictatorial for
f
n > 2:
for all i in {l, ... ,n - I}
A
Sn = {l, ... ,n - I},
and
x
x
for every
x
in
S
n
(Note that the last equation is equivalent to
f(x1,···,x n _ l , j)
where
j
= xn
= x,
J
j
is the "king for a day.")
Remark
Theorem 3a makes the n-th person a kingmaker.
person can be selected for this role.
for
n > 2 •
Clearly
f
Obviously, any
is non-dictatorial
-22-
Proof
Let
x
be a NE for the profile
in
R
i.e., with j-th person as "king for a day,"
Hence, by the definition of NE,
R
x. R.a
J
J
~n , with
for all
a
~n , this implies Pareto optimality of
is in
f(~)
Then
€
A
x.
J
x
n
j
,
x.
J
Since
Hence every
NE is PO.
Given a profile
R,
define
outcome under the ranking
for
XjRnXi
j
Ri
i=I,2, ••• ,n-1.
,
i
to be the first (most preferred)
= {l, ••• ,n-l}.
The n-list
x
Then set x =j
n
where
so defined is a NE.
Since
N is the "king for a day", the outcome
in
under the ranking
R .
someone other than
j
the set
i
choice under
j
If player
(the kingmaker) were to choose
as "king for a day," the outcome would still be in
= 1, .•. ,n-l}
R.
n
f(x) is j's first choice
in which
x.
is the kingmaker's first
J
As for players other than j or n, their strategy change
n
would not affect the outcome.
Thus no person has an incentive to
depart from his component of l£, and so
~
is a NE.
Q.E.D.
We now proceed to give an example of an indirect kingmaker
outcome function.
Le t
n = 7 , S1 = {2, 3} , S2
and
Si
=A
for
i
€
{4,
= {4, 5, 7} , S3 = {5, 6, 7} ,
5, 6,
7}.
Here person 1 is the indirect
kingmaker, while 2 and 3 are direct kingmakers subordinated to him.
A "king for a day" will be selected from {4, 5,6, 7}.
If 1
chooses 2 as the direct kingmaker, the latter can pick a king for
a day from among {4, 5, 7,}; if 1 chooses 3 as the direct king-
--23-
maker, the latter picks a king for a day from among {5, 6, 7} .
Finally, the chosen king for a day selects an outcome from A
This type of outcome function is represented in Figure 71 as directed
graph.
The proof of acceptability given for Theorem 3a carries over
to this type of outcome function, which is also clearly non-dictatorial.
It is not necessary that each player appear only at one point
of the graph.
This is shown by a modified form of the previous
function in which 1 is substituted for 7 as a potential king for
a day, while retaining his indirect kingmaker powers (see Figure
1
1
7
5
4
.
F~gure
4
6
3
=
{5, 6, l}, and
5
1
6
2
7-1
Figure 7-
Formally, we have here
8
7~).
n
= 6,8 1 = A x {2, 3} ,8 2
8. = A for i
~
€
{4, 5, l},
{4, 5, 6} .
We shall not try to provide a characterization of the class
graphs of this type yielding acceptable non-dictatorial outcome
functions.
-24Another class of variants of the kingmaker function yielding acceptable nondictatorial outcome functions, is given by systems in which the
king for a day is chosen by an elective process.
where
may overlap_
#N
Each member of
from among members of
N .
2
~
l
1
Nl
and
#N
2
~
Thus, for
2
n > 3 ,
and the two sets
votes for his choice of king for a day
Different voting systems may be used.
In
one example of such a system the median of the frequency distribution 01
the voters' choices (under any fixed order of
turn, chooses an element of
#N
l
A.
N ) is elected and, in
2
(This system will be applied only when
is odd.)
As another example of an admissible type of election procedure
we consider the plurality principle together with the following
tie-breaking rule:
if the maximal number of votes is obtained by
more than one member of
N2 ' the one designated by the smallest
integer is elected king for a day.
It is clear that under either voting system any Nash equilibrium
outcome will be Pareto optimal because it is the first choice of the
king for a day.
On the other hand an N-list of strategies in which
all the members of
of
N2
the sets
Nl
vote for the same member of
votes for his first choice in
Nl
and
N2
A is a NE for
N2
and every member
#Nl~
3.
overlap, those belonging to both sets simultane-
ously vote for a king for a day and also choose an element of
Nl
Another NE, valid for
#N
vote for the member of
N2
integer.
(When
l
~
2
A.)
is obtained when all members of
designated by the smallest
(in
N )
2
-25-
UN I = 1, these voting systems are equivalent to the king-
When
maker
ou~come
belong to
function af Theorem 3a.
be~omes
N ; otherwise he
2
(But then the kingmaker cannot
a dictator.)
To give a more general formulation, we introduce the following
class of outcome functions to be used in the election process.
An outcome function
g
with
as the strategy set of each
i
M as the set of persons,
in
in
z
such that for every
T
in
i
if there is an M-list
b
M and every
g(~
g
in
v.
~
b
An outcome function
~
B as the set of out-
M and
comes, is said to be fixed at an outcome
T.
1'.
~
°
is said to be fixed if it is fixed at some
outcome!
Now consider as such an outcome function, with
for all
i
in
Nl ,
rule described above.
N = M
1
the plurality voting system with the tie-breaking
Let
j*
be the smallest integer in
this system constitutes an outcome function fixed at
Furthermore, for
and
UN I
~
j*
N •
2
for
Then
UN
l
> 2.
3, both the plurality and the median
voting systems are fixed at every member of
N
2
A "fixing" strategy
0
is a unanimous vote for that member.
To state the condition needed to ensure non-dictatorship we
introduce the following definitions.
Given an outcome function
a person
in
B
k
in
M is said to
if there is a strategy
for any list of strategies
g
with domain and range as above,
a-dictate via
Y
k
in
Tk
(Yj)j€M\{k}.
g
an outcome
such that
g(y)
=b
b
-26-
Person
k
is said to a-dictate via
if for any list of strategies
Yk
€
Tk
yielding
(Yj)j€M,{k}
g
an outcome
b
in
B
there is a strategy
g(y) = b •
If a person
k
a-dictates [resp.
then such a person is an a-[resp.
~-dictates]
~-dictator]
every outcome,
as defined in the
Introduction.
Theorem 3b
Let
N
= NlU
N
2
,
N ;'
l
~;.
N
2
0
Let
g (lithe voting
system") be an outcome function with the set of persons (voters)
N ,
l
the set of outcomes (potential kings for a day)
for each
i
€
N ,
l
the set of strategies denoted by
an outcome function
outcomes
A,
Ti x A
f
with the set of persons
N ,
2
Tio
N,
i
€
Nl
n
N2
0
For an arbitrary list of strategies
we define the corresponding value of the function
f~)
= Zj in A,
f
Define
the set of
and the following strategy sets:
for
and
by
-27-
is the elected king for a day.
where j = g «y.). N )
~
~€
1
Then:
The function
I.
II.
The function
f
is acceptable for ,ttl
f
if
has no a-dictator (resp.
g is fixed.
~-dictator)
1f and only i f
#{range of
(i)
g}
~
2
and
#A
Nl
~
2,
and
p, 1= k
(ii)
whenever
k
in
via
g
the outcome
~-dictates)
a-dictates (resp.
in
t
N2
.
Proof
I.
Z.
J
,
hence
for
x
Let
j
in
R
= g (y)
~
n
be given.
Clearly
Zj
Let
be a NE, say
x
is the first choice of
f(!)
j
,
is PO.
To prove the existence of NE for
the voting system
a strategy X.
g
Set
any strategy list
~
~,
we use the assumption that
is fixed at some potential king
zk
to be the first choice of
compatible with Z and
k
k. in
in
is a NE.
N2
A.
via
Then
Indeed
is the only person who can affect the outcome and he has no incentive
to do so.
Thus
f
is acceptable.
k
-28-
We first show that if conditions (i) and (ii) with a
II.
~]
[resp. (ii) with
~-dictator]
are satisfied then there is no a-dictator [resp.
•
Given an arbitrary
that
i
i
in
N and an arbitrary
cannot a-dictate Crespo a-dictate]
a.
a
in
A, we show
We consider in tum
three categories of persons.
1.
Let
i
€
Nl \ N2 • Consider any strategy list
x
such that
for all
z.=bl=a
J
Then
does not
f~)
=b
~-dictate
regardless of
a.
j
N2 •
€
y,
~
== x, •
~
Hence he is neither a
Hence
i
~-dictator
nor an a-dictator.
2.
g~)
Let
1= i .
Z
j
i
€
By (i), there is
N2 \ Nl
Now consider any strategy list
for all
=bl=a
j
€
x
y
such that
such that
N2 \ {i}
and
:t.
such that
Then
f~)
is neither a
3.
sider
x
g(y) 1= i •
= b regardless of x. • Hence again
~
~-dictator
i
nor an a-dictator.
and let
Let
y.
~
such that
for all
j
€
N2 \ {i}
€
T.
~
be given.
Con-
-29-
and
such that
g (I) 1= i •
follows from (ii) with
(The existence of such
Then
~.)
f(~)
band
c
other hand, if (ii)
i
~ith
is not an a-dictator.
On the
is assumed, there exists a
~
strategy list
such that
g(y) 1= i
regardless of
be such that
Let
x
and
(Yk)k€Nl\{i}
as above.
and so
is not a
of
x.
~
i
Z
j
=
Yi'
b 1= a
for all
Then
f~) =
€ NZ \ {i}
j
b
regardless of
~-dictator.
To complete the proof we show that the violation of either
of condition (i) or (ii) yields the corresponding type dictator.
If the range of
then clearly
is an
p
Now let
k
tate any outcome
Z
k
in
a
g
is a one element set, say
~-dictator
Nl
n NZ
by choosing a strategy
in
Then he can dic-
(Y , zk)
k
such that
= a
Then given any
k
in
Nl
n
NZ '
~-dictator).
dictate himself.
On the other hand, suppose
himself.
(and hence a
1.
NZ
and any
~-dictates
-30person
Yk
k
has a strategy
with
zk
=a
,person
Yk
k
such that
can
g(y)
~-dictate
= k.
By combining
any outcome
a.
Hence
he is a a-dictator •
Q.E.D.
To illustrate the conditions required to ensure non-dictatorship, we consider two examples.
Example 1.
i # j •
Let
N2
=
{1, 2}
and
#N1
=2
, say
N1 = ii, j} ,
Let the plurality voting system prevail, with candidate desig-
nated by the lower number winning in case of a tie.
For such
g, person
1 is a fixed outcome of the voting system that can be dictated by either
i
or
j, while 2 cannot be dictated by anybody.
1
dition (ii) that
t
N1
and any membership of
would ensure non-dictatorship.
or
N1
=
It then follows from con-
N1
excluding 1
N1
E.g., we could have
= {2,
3}
{3, 4} •
Example 2.
Let
N = N1
= N2 =
{1, 2, 3}
and let
voting system with the above tie-breaking rule.
g
be the plurality
Here every parti-
cipant is both a voter and a potential king for a day, but the tiebreaking rule introduces an asymmetry across persons.
It can be
verified that both conditions (i) and (ii) above are satisfied and
hence
f
is acceptable and non-dictatorial.
Remark
It is possible to combine the elective elements of Theorem 3b
with the indirect appointive elements of the type represented by
the graphs in Figures 7 above to yield non-dictatorial acceptable
outcome functions.
-31-
Section 5:
Outcome Functions Symmetric Across Persons
In Section 2 above, we discussed the problem of existence
of symmetric outcome functions.
It was seen that, in general, acceptable
non-dictatorial functions that are symmetric both across outcomes and
across persons do not exist.
lSi = 2
for all
persons do exist.
i
€
It was shown that when
n=3,
#A
= 2,
N , outcome functions symmetric across
We shall now show that this result can be
generalized.
An outcome function
f:
Slx ••. x Sn
A is said to be
-+
symmetric across persons if and only if:
= Sn
(2)
for any permutation
It follows that, for
n
of the set
f
Sl = S2 = ••• Sn
is symmetric
across persons if and only if: for every
I{i e:N: x.~
implies
f~)
= f<1:)
= s}
=
#{i
€
N:
N, we have
=
Yi
=
s}
t
-32-
We shall now restate this in terms of what may be called the distribution of votes.
Let
Nand
S
o
also fixed), and write
S
= Sox •••
duct).
~
€
S
= #- {i
€
N
Then, for each
S
t
be fixed (so that
x So
n
is
(n-fold Cartersian pro-
there exists a unique distribution
given by
o
P
x
Denoting by
J
(8)
x.1.
=
s}
the set of nonnegative integers, we shall write
S
D (S )
n
0
= {p
d
with an element
of
x
We see that
, :t.
€
J
0
~
s€S
We shall denote by
v~
€
the mapping,
= n} •
o
d :
S -+ D (S )
-
n
associating
is symmetric across persons if and only if:
§. ,
d(~)
= d(X)
Therefore,
implies
f~)
= f<l)
an outcome function
f
persons if and only if there exists a function
that
0
the corresponding distribution px •
S
f
p(s)
•
is symmetric across
g:
D (S ) -+ A such
n
0
f = god •
Note that for outcome functions symmetric across persons
#S2 = ••• = #Sn.
#Sl
It is clear that one cannot hope to find acceptable
outcome functions with
Is.1.
< #A.
On the other hand if acceptable
=
-33-
functions are round with
= #A
#Si
, one can trivially enlarge the
strategy spaces to any desired extent.
seek outcome functions with
It is therefore natural to
= #A.
#Si
Since in cases we have in-
vestigated such functions do exists, there is no loss in assuming
#S.~
= #A
for all
i
N.
€
In fact, we may take
S
o
= A.
We do not at this time have a result for arbitrary
#A •
In what follows, we shall confine ourselves to separate treatments of two special cases:
The Case
#A
= 2,
n
~
=2
#A
,and
#A
=3
•
3
As indicated above, we shall in this subsection require
Si
= A for
all i, so that #Si = #A
=2
for all
a symmetric across persons outcome function
if we find a function
Write
A
= {a,
D (A)
8:
b}
n
= So
-+
is given, by
g
D (A)
n
~
p(a)
alone.
€
N.
Therefore,
will be obtained
A.
An element of
characterized by the pair of integers
n
f
i
so =
D (A)
(p(a) , p(b»
Hence, for a given
n
is uniquely
, or, since
n, a function
A is represented by the sequence
< g(O) , g(l), ••. ,8(n) >
where
g(i)
is the outcome for
p(a)
=i
.
In this notation, the symmetric outcome function of Figure
~ is represented by the sequence < b, a, a, b > •
-34We shall now characterize the functions
yielding acceptable outcome functions
g, for
= god
f
n
so that
~
3
f
is
symmetric across persons.
Lemma 1
n ~ 3.
Let
if the function
g(O)
g
~
Every NE is PO for all profiles in
En if and only
has the following property:
g(l) ; g(n - 1)
~
g(n) ; and, for all
g(i + 1)
~
g(i)
(*)
o<
Proof
(i)
i < n ,either
Sufficiency.
the distribution
p (a)
x
that every person in
Let
=i
x
and
or
g(i - 1)
~
g(i)
be a NE strategy list inducing
p (b)
x
=n
- i , and suppose
N prefers the other outcome to
g(i) •
Then
by condition (*) of the Lemma there is always some person who can
swing the outcome to either
g(i - 1)
or
g(i + 1)
which he would
prefer.
(ii)
Necessity.
Suppose that for some
violates condition (*) of the Lemma.
everyone prefers the other outcome to
list where
i
persons vote
non-optimal outcome
g(i) •
a
and
0
~
i
~
n , g(i)
Consider the profile where
g(i) •
n - i
Then a strategy
vote
b
yields the
However, this strategy list is a NE,
since a change in one person's vote cannot swing the outcome to
anything different from
g(i) •
-'35-
Lemma 2
Let
n
~
Under condition (*) of Lemma 1, there exists
3.
a NE for every profile in
t
n
if and only if
either
g(2)
g (1)
or
= g(n
g(n - 2)
-1)
or
gj
such that
g(j)
= g(j + 1) = a
such that
g(k)
= g(k +
and
~k
Proof
(i)
Sufficiency.
1)
=b
•
For the unanimous profile case, a NE always
exists, since by (*) the range of
g
is the whole set
A.
When the profile is not unanimous, we consider in turn the
cases where (**) and (***) hold.
Suppose (**) holds with
= g(2)
gel)
•
Since the profile is
not unanimous there is at least one person preferring
g(O) .
The strategy list where such a person votes
one else votes
b
is a NE with the outcome
gel) to
a
g(l).
and every-
Indeed this
person has no reason to change his vote, and a vote change by anyone else would yield
By
('b'~),
if
=2
pea)
and so not affect the outcome.
gel) 1= g(2) , we have
g(n - 1)
= g(n) , and
the proof is analogous.
Suppose now the profile is not unanimous and (***) holds.
Denote by
m the number of persons preferring
Case 1
There exist
Subcase 1.1:
a NE here:
j
j
m~ j .
and
k
b.
to
b •
satisfying (***) such that
The fo~lowing strategy list yields
persons preferring
everyone else votes
a
a
to
b
vote
a, while
This is a NE because no a-voter has an
incentive to switch his vote to
g(j - 1) , which by (*) equals
b
b,
and so to swing the outcome to
j < k •
-36-
and nobody else can affect the outcome
gO + 1)
since
=
g(j) •
Subcase 1.2:
m< j
The following strategy list yields a NE:
b
to
a
vote
since
Here n - m > n - k
n - k
b, while everyone else votes
k > j .
persons preferring
a.
The argument
is like that in the preceeding subcase.
Case 2
the names
j > k.
a
and
This can be reduced to Case 1 by exchanging
b.
Suppose that (*) is satisfied but neither
(ii) Necessity.
(*-k)
nor ("in'(*) hold.
This means that in the sequence
< g(O) , g(l), .•. ,g(n) > either
g(i + 1)
then
g(i)
= g(i +
1)
=a
is absent or
g(i)
= b is absent, and if, for some i , g(i) = g(i + 1) ,
i '- {O, 1, n - 2, n - 1} .
g(i) = g(i + 1) = b
Suppose first that
for the profile where person 1 prefers
prefers
b
to
a.
those preferring
b
If the outcome is
a
to
a
b
then person 1 can swing the outcome to
a
b
and everyone else
Suppose first that
a
and
b.
adjacent in the g-sequence
because on either side
in the g-sequence there is an
be at an endpoint of the
ferring
Then there is no NE
a, there is someone in this group who by changing his
b
b
b
split their votes between
If the outcome is
of an interior
to
This is seen as follows.
vote can swing the outcome to an outcome
< g(O), ••• ,g(n) >.
is absent.
a , and we cannot
g - sequence when the votes of those pre-
are split.
Now suppose that all those preferring
b
to
a
(i.e.,
-37persons
2, ... ,n) vote the same way.
one of the following:
Here the outcome must be
g(O), g(l), g(n - 1), or g(n) .
When (*)
holds but neither (**) nor (***) do, the g-sequence begins either
with
< g(O), g(l), g(2) >
< b, a, b >
= < a, b, a > or with
and ends either with
< a, b, a >
or with
< g(O), g(l), g(2) >
< g(n - 2), g(n - 1), g(n) >
< g(n -2), g(n -
=
1), g(n) > = < b, a, b > •
A direct check shows that in none of these cases is there a
NE yielding any of the outcomes. g(O), g(l), g(n - 1), or g(n)
for the above profile.
The proof for the case where
is obtained by exchanging the names
= g(i +
g(i)
a
and
1)
=a
is absent
Q.E.D.
b
Lemmas 1 and 2 prov,ide a complete characterization of outcome
functions that are acceptable and symmetric across persons for the
case
#A
= #S.1. = 2
For
n
=3
for all
i
in
N, n
~
3 .
, the two Lemmas imply that the outcome functions
1
2
in Figures 3- and 3- are the only ones possible.
that for every
n
~
3
It is obvious
there exist functions satisfying the con-
ditions of the two Lemmas.
The following is one example (with (,,,"*)
satisfied) :
g(O)
= g(2r +
1)
=a
for all
n-1
r = 1,2""'[-2-]
and
g (i)
where
[x]
b
otherwise,
is the largest integer not exceeding
x •
The smallest number of persons where (*) and (***) can be
=
-38satisfied with (**) violated is
n
= 7.
An example is given by
the following function:
= g(2) = g(3)
g(O)
a ,
g (6)
gel) = g(4) = g(S)
g (7) = b •
Some General Considerations When
~
#A
2 ,and
The method of analysis used above for
for bigger sets of outcomes, with
S
o
n
~
#A
3
2
can be generalized
= A •
We start by defining the set of distributions accessible from
a given distribution by (at most) a single person's vote switch.
Definition
For every
~(g)= {q
#{a
Clearly,
A
€
q
€
I:
D (A)
n
,
q
Va
D (A)
n
I:
q (a)
Vq , q
reached from
q
€
in
q
€
q
c'a)}
D (A) , i f
n
D (A) , we shall write
n
A , Iq(a) - q(a)1 ~ I , and
€
~
2} .
q I: q
and
q
€
'n (g)
by one person switching his vote.
q(a)
~
= n , it follows that
,
then
q can be
Since, for any
=2
for
n ~ 3 , #A ~ 2 , and for every profile in
~n,
Iq(a) - q(a)1
a€A
q •
We now state a generalization of Lemma I above.
Lemma 3
For every
each NE is PO if and only if
(#)
vq
€
D (A)
n
, q
(JI. (g»
=A
•
-39-
Remark
The "accessibility condition"
("k)
(#)
is an extension of Condition
q
is a distribution induced by
in Lemma 1.
Proof
(i) Sufficiency.
Suppose that
a NE strategy list for some profile in
~.
If
g(q)
were .not
PO for
this profile, there would be another outcome preferred by everyone,
and this outcome is accessible from
virtue of Condition (#).
g(q)
by a single vote switch in
This contradicts the supposition that
q
is
NE.
Suppose that the Condition (#) is violated so
(ii) Necessity.
that there is a
q
in D (A)
n
and
a
€
A such that
Consider a profile in which everyone prefers
and
g(q)
as his second choice.
a
a
t
g('J?(q)) •
as his first choice
Then the strategy list inducing
q
in a non-PO Nash Equilibrium.
Q.E.D.
The Case
#A
=3
, n
~
3
Here, again, we take
S
o
= Si = A
for all i, so that
lis.~ = IIA = 3.
In this case, it is possible to provide a simple representation
of the function
g
by means of the following two-way table where
the entry in the i-th row, j-th column represents the value of
when
n-i-j
a
receives
votes,
i,j
i
votes,
~ ~
, i + j
b
receives
~
j
votes and
n , and we write
A
c
g
receives
= {a,b,c} .
-40-
no
of votes 'or b
"-
no. of votes for a
0
I--
1
t---
0
n-l
t----
n
,
i~
•
!
1
x
!
l
(X
indicates a voting pattern that is impossible because
The accessible set
~(q)
can here be written as
i + j > n.)
~(i,j)
and
is given by
'J? (i , j
)
=
{(k,.e)
k ~ 0 , ~ ~ 0 , k
Ik + t
~
il ~ 1 ,
I~
- j!
~ 1 ,
Ik
+~ -
n}
and can be represented graphically as follows
(i-l,j)
(i-l,j+l)
/
(i,j-l)~-----------*---___________~
(i+l,j-l)
/
(i+1 ,j)
(i ,j+l)
i
-
j!
~ 1 ,
-41-
(It is understood that all points indicated by arrows are accessible
provided their indices are non-negative and do not add up to more
than
n
.)
n
The construction for
obtaining
acceptable g's
~
3
for
of extension for arbitrary
,
#A = 3
n
= 3.
is accomplished by first
4
and then providing an algorithm
n >4 .
The following table gives a function
#A
= 3.
(It may be noted that other
g
g
acceptable for
n = 3
t
functions, not obtainable
from considerations of symmetry, have also been found.)
I
i
~~llI
votes
for a
!
0
II
Ii
.-
1
1
2
3
!,
I
i
_.'r--._-
,
--=9
a
b
b
1
c
a
a
X
,
2
c
a
X
X
!
3
b
X
X
X
0
i
c
f
t
1
I
I
It may be verified that for every entry
the accessible set
profiles in
~3.
every profile in
=
~(i,j)
{a,b,c}.
(i,j)
Hence, every
It remains to show that this
g
in this table
NE is PO for all
possesses a NE for
~ 3.
Since the range of
g
is
A, NE's exist for all profiles with
unanimity as to the first choice.
If at least one person has
voting pattern yields a NE:
for
one
a
as first choice, the following
perso~whose
a, while the other two vote for
b
first choice is
a, votes
The a-voter has no in-
centive to switch his vote since the outcome at
(1,2)
ii
a» which is his
-42first choice; neither b-voter can affect the outcome by switching his vote
either to
a
a
or to
c.
Hence there is a NE at
(i,j)
=
(1,2) , with
as outcome.
Suppose
a
is nobody's first choice and there is no unanimity
as to first choice.
Consider first the case where
is undominated (hence PO).
a
there exist two (distinct) persons, say
a
to
c
and
j
prefers
yields a NE at the
b , j
for
a
(1,1)
to
b.
i
and
j , such that
a
prefers
Here the following voting patterns
entry in the above table:
a, while the third person votes for
Second, let
i
Then
be dominated, say by
let
i
vote for
c .
b.
Then, since we are at
present assuming the absence of unanimity as to first choice, there must
be a person whose first choice is
obtained when this person votes
the other hand, a
b.
b
(0,1)
is therefore
while the others vote
is dominated by
c.
If, on
c, there is an analogous NE at
with a person whose first choice is
for
A NE at
c voting for
(1,0)
a while others vote
c.
Thus, we have shown that the function
When there are four persons
g
0
1
b
c
aI
a
o!
I
= 4)
(n
2 I, a
c
2
b
II
,
-3
-~-
4
b
c
b
a
X
c
X
X
c
X
X
X
X
X
X
III
i
Ib
4 I a
I
3
, with
#A
remaining at 3,
function will be shown to be acceptable.
i
I
in the above table is
~3 •
acceptable for all profiles in
the following
g
.
X
-43i~
The Pareto optimality of all NE's is proved here as
As for existence, we distinguish 3 types of profiles.
n = 3
let
was for
a
be at least one person's first choice.
First,
Then the entry
(i,j)
=
(1,0) , marked I, yields a NE when such a person votes
a
vote
is the first choice
c
If
a
is nobody's first choice, suppose
of at least two persons.
marked II (i.e., at
others vote
b
For such profiles, a NE exists at the entry
(0,2)) when such two persons vote
b
and the
c .
For the remaining profiles at least three have
choice.
and others
c
as their first
Then NE are attained at the entry marked III (i.e., (2,1))
when two of such three persons vote
fourth person votes
c
a, one votes
Hence the function
is acceptable for all profiles in
~4
g
and the
b
of the above table
•
Note that all of the locations of the NE's mentioned above
(I, II, III)
satisfy
the condition
n , we shall say that a function
if, for every profile in
i + j < n.
n = 4
a
In general, for any
has the "interiority property"
~n , there is a NE at a location
(Recall that
of votes cast for
g
i + j < 4.
i
and
and
b.)
has the "interiority
j
(i,j)
with
denote respectively the numbers
So, for instance, the above
property," while that for
n = 3
g
for
does
not.
Although the preceding argument only shows the existence of acceptable functions g
for
n = 3; 4, #A = 3 , it turns out that these
results can be used to construct acceptable
n
~
3
as long as
#A
Theorem 4:
If
a function
g
= 3.
g
functions for all
Formally, we then obtain the following
IIA - 3, then for every
n
~
acceptable for all profiles in
3, there exists
~n.
-44Proof
For
For
n > 4
n
= 3,4 the Theorem has already been proved by construction.
the proof is carried out by induction whose validity is
established in the Induction Step Lemma below.
Before stating the Lemma, we show for illustrative purposes how
the induction step is carried out for
n
= 4.
The problem is to fill the blank spaces (forming what we shall
call the "new diagonal"(*)) in the table below in such a way that both
the existence and Pareto-optimality properties present for
n
= 4 are
preserved.
I
o
5
o
b
c
b"'~'
I
a*
a
b
2
a
c*
3
b
!
4
5
x
I
b
alJ.
I clJ.
I
I
x
x
x
x
x
x
x
x
x
x
x
x
x
x
In this table, the letter entries define the function
n
in
g
for
= 4 which is acceptable and with "interior" NEls for all profiles
y:,4
marked by asterisks.
It can be verified that any extension
g
of this
g
(i.e.,
any way of filling of the new diagonal) will possess "interior" NEls
for all profiles in
5
L , since the NEls for
"interior" NE status for
n
=5
n
=4
retain their
, and there are enough of them for
(*) To be dis tinguished from the "old (boundary) diagonal,"
by triangles in the above table.
marked
-45-
all the new profiles. So the problem reduces to that of assuring the
Pareto-optjmality of all the NE's that appear in the extension
~
g •
It may be verified that the extension presented in the following
table satisfies this requirement, and is therefore acceptable for
n
also it has "interior" NE's (marked by asterisks)
=5
~_O~~1
r-_4~
__5__
b
c
a
__+-_2__r-3__
o
c
b
1
a
b
a
b
x
2
c*
c
a
x
x
3
c
b
x
x
x
4
a
b
x
x
x
x
5
c
x
x
x
x
x
for all profiles in
2::
Induction Step Lemma
5
.
For
#A
=
3
be acceptable for all profiles in
in
2::
-g :
n
there is an "interior" NE.
D + (A)
n 1
profiles in
Proof
~
2::
A of
g
and
2::
n
n
~
4
,
let
g :
D (A)
n
~
A
and such that for every profile
Then there exists an extension
having the corresponding properties for all
n 1
+
First we show that any extension
a NE for each profile in
Let a profile in
2::
n 1
+
2::
n 1
+
g
of
g
to
D + (A)
n 1
possesses
and has the interiority property.
be given.
Consider the profile in
2::
n
obtained by
-46deleting (the preferences of) one of the
n + 1
persons.
thesis the function g has a NE at some(i,j) ,with
g
then obtain a NE of
location
(i,j)
i + j < n.
We
for the original profile at the same
when the "deleted" person votes
c
retain the strategies used in the n-person game.
choices yield a NE at
By hypo-
(i,j)
affect the outcome (since
and the others
These strategy
because the "deleted" person cannot
g(i,j + 1)
= g(i
+ 1, j)
=
g(i,j»
, and th-e
others either cannot or have no. incentive to destroy the NE status of
(i,j)
in the n-person game.
property is preserved since
Also, for the n+1-game the interiority
i + j < n < n +. 1 .
It remains to show that there exists an extension
which all NE are Pareto-optimal.
for any
q
in
,g m(q»)
Dn+1 (A)
"neighborhood" of
q
the domain of
g
we shall write
of
n
q
in the
x
in the
n
we have
g(~(i,j))
By Lemma 3, we must require that,
=
(n + 1)
q
by
'lI(q)
(n + 1)
(i,j)
gcn(i,j»
i + j = n + 1
i
g.)
such that
= A.
g
such
i.
= 0,1 , the extension g is defined by
g(O,n + 1)
g (1, n)
i + j
= g(l,n
g(O,n - 1)
- 1)
=
=
g(l,n - 1)
g(O,n - 1) •
~
n ,
Since
Therefore, we need
that~
(Le., on the new diagonal),
We shall establish this by induction on
For
table, Le., in
= A by the assumed acceptability of g.
only verify that there exists an extension
with
denotes the
to denote the "neighborhood"
represented by
~(i,j) ~ n(i,j), it follows that
i,j
n(q)
(Here
A
table, i.e., in the domain of
Note that for
g for
for every
g(77(i,j»
= A
-47~(O,n)
First note that
hence, by Lemma 3,
also
g(l,n)
{g(O,n), g(O,n - 1), g(l,n - I)}
= A.
g(0(0,n + 1»
~
= {(O,n), (O,n - 1), (l,n - I)} and
Therefore,
Furthermore, g(~(l,n) \ {(2,n - I)}) = A , and
g(l,n -
defined the values for
in such a manner that
= A.
1)
•
Proceeding by induction, suppose that we have
g(i,n+!-i)
up to and including
i
g(~(i,n + 1 - i ) \ {(i + 1, n -i)})
Then we define
g(k,n + 1 - k) # g(k,n - k)
=k
n ,
~
= A and
g(k + 1, n - k)
as
follows.
If
g(k + 1, n - k - 1)
g (k, n - k)
or
g(k + 1, n - k - 1)
then set
both
S(k + 1, n - k)
g (k , n - k) and
= g(k,
n + 1 - k)
to be the unique element in A different from
g (k, n + 1 - k) .
On the other hand, if
g(k + 1, n - k - 1) # g(k, n - k)
and
g(k + 1, n - k - 1)
~
g(k, n + 1 - k)
then set
g(k + 1, n - k)
Finally, for
--
k
= g(k,
n - k) .
= n , apply the first alternative (i.e., set g(n+L,O) equal
to -the unique clement in A different from both
g(n,O) and
g(n,!»
0
-48-
Clearly, the two properties,
and
g(~(k
g(k + 1, n - k).
+ 1, n - k)) \ {(k + 2, n - k - I)} = A
k < n ; and, when
k = n,
g(~(k
F g(k +
1, n - k - 1)
are satisfied for
+ 1, n - k)) = A
Q.E.D.
-49-
Section 6:
Performance Correspondences and Characteristic Functions.
In general, a choice correspondence associates a subset of the
set of outcomes with a given profile.
Given an outcome function
~
f , its performance correspondence
is defined as the choice correspondence which associates with a
profile
R
the set of
NE
of NE of the ordinal game
Clearly
f
if and only if
in
W is
f
for
R.
~(~)
Thus
is the set
(f,~).
is acceptable on a class of profiles containing
~(~)
optimal subset of
pondence
of
R
is nonempty and is contained in the R-Pareto-
A for every
R
in the class.
A choice corres-
said to be dictatorial if there is exactly one person
N for whom, given any profile
contains an element
a
in
R
in the domain of
A such that
aR.b
W'
the set
for all
1
b
in
Clearly, if an outcome function is dictatorial, so is its performance correspondence.
It is also obvious that the performance
correspondence associated with an outcome function that is symmetric
across persons is not dictatorial.
We have not determined whether,
in general, a non-dictatorial outcome function acceptable on
generates a non-dictatorial performance correspondence.
easily be seen, however, that the outcome function
n
~
It can
in Theorem 3a
generabesnon-dictatorial performance correspondences.
In the case of
the kingmaker outcome function (Theorem 3a), for each person there is
a profile such that the only NE outcome is his least preferred alternative:
(An example; the given person's last choice is the first
-50..,.
choice of all others).
In the case of Theorem 3b, we restrict our
attention to voting systems
set
1\r(~)
N2
g
that are fixed at all points of the
~
of potential kings. Then for every profile
,
the value
of the performance correspondence contains the first choices of
all potential kings for a day.
Hence the uniqueness requirement in
the definition of a dictatorial choice correspondence is violated.
We may note that had the uniqueness requirement
been deleted
from the definition of a dictatorial choice correspondence (thus
making the class of non-dictatorial correspondences smaller), the
performance correspondence associated with the kingmaker outcOme
function would have remained non-dictatorial.
Furthermore, no person is a "dununy" in the kingmaker outcome function. (The i-th person is called a performance correspondence dummy
if on the domain of
is dependent of
cp , the set
R. .)
~
Note however, that for outcome function of Theorem 3b, with
and
N2
disjoint, every person in
N1
N1
is a dununy of the associated
performance correspondence - without necessarily being a dununy of the
outcome function.
For example, if the fixed function
g
is that of
plurality voting with some tie-breaking rule (see Section 4 above),
then no person is a dununy of the outcome function.
Yet
cp(E)
con-
sists of first choices of potential kings for a day (members of
and hence is independent of the preferences of the members of
N ),
2
N1 •
Dictatorship and dununy status constitute the extremes of power
positions of game participants.
An alternative approach to the analysis
of power relationships is to investigate games in von Neumann and Morgenstern
characteristic function form corresponding to a given outcome function.
-51-
First, for an outcome function
istic function
MeN,
v (.)
f.'!I
v (M) = 1
outcome
(X
a
in
A
~-dictate
the outcome v
v
for a set of persons
otherwise we set
v (.)
the outcome
as follows:
if and only if the coalition
function
v (M)
q
is defined analogously.
a
we define the character-
f:S~,
=
M
0
~-dictates
The characteristic
(A coalition
if there is an M-1ist of strategies yielding
A coalition
M is said to
S-dictate the outcome
if for any choices of strategies by persons outside
st.rategies for members of
M yielding
Characteristic function
v(.)
if and only if
belongs to
M.
M there are
is said to have a dictator if
i such that, for every coalition
i
a
a.)
there is a player
M, v(M)
=1
It follows from our definitions
in the Introduction that an outcome function
v
M is said to
regardless of the strategies chosen by those outside
the coalition M.
and only if
every
f
has an
~-dictator
if
has a dictator.
(Y
It is also evident that an outcome function dummy is necessarily
a characteristic function dummy (in the sense that he does not affect
the value of a coalition by joining it).
It is not known, however,
whether the converse is true.
An advantage of the characteristic functions approach is that it
reveals certain symmetries that are not otherwise apparent.
Thus,
for instance, in the three-person kingmaker outcome function of
Section 4, the characteristic function has the values
for
#M~
2
and zero otherwise.
v (M)
a
=v
8
(M)
=1
Hence there is symmetry of the char-
acteristic function with respect to players even though the outcome
function is not symmetric across persons.
-52-
Another example of a symmetric characteristic function associated
with an asymmetric outcome function is a special case covered by
Theorem 3b where
Nl
= N2
,#N l
is odd, and voting is by plurality with
some tie-breaking rule.
Clearly, the tie-breaking rule makes
both
v
and
ex
f
asymmetric.
However,
are those of a simple majority game and hence are
symmetric.
A more general notion of a characteristic function can be introduced as follows.
We define
V: M ~ B where
M ~ Nand
B
~
A ,
QI
where
b
E
V(M)
analogously.
vQl(v
a)
if and only if
It is seen that
M ~-dictates
V (V )
~
S
b •
V~
is defined
is more informative than
and thus its symmetry properties may be regarded as more
plausible indicators of equal power of the players.
So far we have been discussing the construction of characteristic functions given the outcome functions.
It is of considerable
interest to consider the reverse process of finding outcome functions
that will "implement" given characteristic functions.
For instance,
sons (n odd)
a simple majority characteristic function game for n per-
can be "implemented" by an outcome function of the type
described in Theorem 3b, with
Nl = N2 = N, lIN = n ,
voting system and some tie-breaking rule.
and with plurality
The use of such an outcome
function provides a well-defined procedure for realizing the given
characteristic function, with a guarantee that the outcome will necessarily be Pareto-optimal.
It would be of interest to develop a method
of specifying outcome functions for broader classes of characteristic
function.
References
[lJ
Arrow, K. J., and F. H. Hahn, General Competitive Analysis,
Holden-Day, San Francisco 1971.
[2J
Farquharson, R., Theory of Voting, Yale University Press, 1969.
[3J
Gibbard A., ''Manipulation of Voting Schemes: A General Result,"
Econometrica, July, 1973, v.41, pp. 587-601.
[4J
Groves, T., and J. Ledyard, "Optimal Allocations of Public
Goods: A Solution to the 'Free Rider' Problem," mimeo. 1974
Center for Mathematical Studies in Economics and
Management Science, Northwestern University, Discussion
Paper 119, (Revised DP 144, 1975), (to appear in
Econometrica) •
,<
[5]
Hurwicz,L., "On the Existence of Allocation Systems Whose
Manipulative Nash Equilibria are Pareto-Optimal: The Case
of One ,Public Good and One Private Good."
Presented at
World Econometric Congress, Toronto, August 1975.
(6)
"On Infonnational Requirements for Non-Wasteful
Resource Allocation Systems,"
Presented at the Conference
on Mathematical-Economic Models, Moscow, June,1976.
[7J
Ledyard, J., A Private Communication, December 23, 1975.
[8J
Satterthwaite, M., "Strategy-Proofness and Arrow's Conditions:
Existence and Correspondence Theorems for Voting Procedures
and Social Welfare Functions,"
October, 1975, pp. 187-217.
Journal of Economic Theory,
[9J
Schmeid1er, D., and H. Sonnenschein, The Possibility of Nonmanipulable Social Choice Functions: A Theorem of A. Gibbard
and M. Satterthwaite, mime 0 , 1974. Center for Mathematical
Studies in Economics and Management Science, Northwestern
University, Discussion Paper'89.