Fundamentals of social choice theory

6

L(Y) , and for every x in Y, if x 

0 G(u) and 

N

(1)

{y

*  v (y) 

>

 v (x)} 

f

 {y

*  u (y) 

>

 u (x)},  

œi 0 N,

i

   

i

   

 

i

   

i

then x 

0 G(v).

Maskin (1985) showed that any social choice correspondence that is constructed as the set

of Nash equilibrium outcomes of a fixed game form must be monotonic in this sense.  A game

form is a function of the form H:×

 S

6Y where each S  is a nonempty strategy set for i.  The

i0N

 

i

 

 

 

i

pure Nash equilibrium outcomes of the game form H with preferences u is the set

E(H,u) = {H(s)

* s 0 ×  S ,  and, œi0N, œr 0S , u (H(s)) $ u (H(s ,r ))}.

i0N

 

i

  

 

 

i

i

 

i

   

i

-i i

That is, x is a Nash equilibrium outcome in E(H,u) iff there exists a profile of strategies s such

that x=H(s) and no individual i could get an outcome of H that he would strictly prefer under the

preferences u  by unilaterally deviating from s  to another strategy r .  Condition (1) above says

i

 

 

 

 

 

i

   

 

 

i

that the set of outcomes that are strictly better than x for any player is the same or smaller when

the preferences change from u to v, and so x=H(s) must still be an equilibrium outcome under v. 

So if  x

0E(H,u)  and the preference profiles u and v satisfy condition (1) for x, then we must have

x

0E(H,v).  Thus, for any game form H, the social choice correspondence E(H,C) is monotonic.

Given any social-choice function F:L(Y)  

6

 Y, let F(L(Y) ) denote the range of the

N

   

 

 

N

function F.  That is,

F(L(Y) ) = {F(u)

* u 0 L(Y) }.  

N

   

     

N

So #F(L(Y) ) denotes the number of elements of alternatives that would be chosen by F under at

N

least one preference profile.  The Muller-Satterthwaite theorem asserts that any monotone social

choice function that has three or more outcomes in its range must be dictatorial.

Theorem 1.1. (Muller and Satterthwaite, 1977.)  If F:L(Y)  

6

 Y is a monotone social

N

choice function and #F(L(Y) ) > 2, then there must exist some dictator h in N such that

N

F(u) = argmax

 u (x),  

œu 0 L(Y) .  

x0F(L(Y) )

N

h

N

Proof.  Suppose that F is a monotone social choice function.  Let X denote the range of F,

X = F(L(Y) ).  

N

We now state and prove four basic facts about F, as lemmas.

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7

Lemma 1.  If F(u) = x, x =/ y, and 

{i

*  u (x)  >  u (y)} 

f

 {i

*  v (x)  >  v (y)}

i

   

i

   

 

i

   

i

then y =/ F(v).

Proof of Lemma 1.  Suppose that F, x, y, u, and v satisfy the assumptions of the lemma

but y = F(v), contrary to the lemma.  Let u be derived from u by moving x and y up to the top of

^

every individual's preferences, keeping the individual's preference among x and y unchanged. 

Derive v from v in the same way.  By monotonicity, we must have x = F(u) and y = F(v).  But the

^

 

     

 

 

  

 

 

 

 

     

^

 

     

^

inclusion assumed in the lemma implies that monotonicity can also be applied to u and v, with

^

 

 

^

the conclusion that F(v) = F(u) = x.  But x =/ y, and this contradiction proves Lemma 1.  Q.E.D.

^

   

^

Lemma 2. F(v) cannot be any alternative y that is Pareto-dominated, under the preference

profile v, by any other alternative x that is in X.

Proof of Lemma 2.  Lemma 2 follows directly from Lemma 1, when we let u be any

preference profile such that x = F(u).  Pareto dominance gives the inclusion needed in Lemma 2,

because {i

* v (x) > v (y)} is the set of all voters N.  Q.E.D.

i

   

i

Following Arrow (1951), let us say that a set of voters T is decisive for an ordered pair of

distinct alternatives (x,y) in X

×

X under the social choice function F iff there exists some

preference profile u such that 

F(u) = x  and  T = {i

*  u (x)  >  u (y)}.

i

   

i

That is, T is decisive for (x,y) iff x can be chosen by F when the individuals in T all prefer x over

y but everyone else prefers y over x.  Lemma 1 asserts that if T is decisive for (x,y) then y is

never chosen by F when everyone in T prefers x over y.  

Lemma 3.  Suppose that #X > 2.  If the set T is decisive for some pair of distinct

alternatives in X

×

X, then T is decisive for every such pair.

Proof of Lemma 3.  Suppose that T is decisive for (x,y), where x 

0 X, y 0 X, and x =/  y. 

Choose any other alternative z such that z 

0 X and x =/  z  =/  y.   

Consider a preference profile v such that 

v (z)  >  v (x)  >  v (y),   

œi 0 T,

i

   

i

   

i

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