Fundamentals of social choice theory

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v (y)  >  v (z)  >  v (x),   

œj 0 N\T,

j

   

j

   

j

and v has everyone preferring x, y, and z over all other alternatives.  By Lemma 2, F(v) must be

in {y,z}, because x and all other alternatives are Pareto dominated (by z).  But F(v) cannot be y,

by Lemma 1 and the fact that T is decisive for (x,y).  So F(v) = z, and T = {i

* v (z) > v (y)}.  So T

i

   

i

is also decisive for (z,y).

Now consider instead a preference profile w such that 

w (x)  >  w (y)  >  w (z),   

œi 0 T,

i

   

i

   

i

w (y)  >  w (z)  >  w (x),   

œj 0 N\T,

j

   

j

   

j

and w has everyone preferring x, y, and z over all other alternatives.  By Lemma 2, F(w) must be

in {x,y}, because z and all other alternatives are Pareto dominated (by y).  But F(w) cannot be y,

by Lemma 1 and the fact that T is decisive for (x,y).  So F(w) = x, and T = {i

* w (x) > w (z)}.  So

i

   

i

T is also decisive for (x,z).

So decisiveness for (x,y) implies decisiveness for (x,z) and decisiveness for (z,y).  The

general statement of Lemma 3 can be derived directly from repeated applications of this fact. 

Q.E.D.

To complete the proof of the Muller-Satterthwaite theorem, let T be a set of minimal size

among all sets that are decisive for distinct pairs of alternatives in X.  Lemma 2 tells us that T

cannot be the empty set, so #T =/  0.

Suppose that #T > 1.  Select an individual h in T, and select alternatives x, y, and z in X,

and let u be a preference profile such that

u (x)  >  u (y)  >  u (z),

h

   

h

   

h

u (z)  >  u (x)  >  u (y),   

œi 0 T\{h},

i

   

i

   

i

u (y)  >  u (z)  >  u (x),   

œj 0 N\T,

j

   

j

   

j

and everyone prefers x, y, and z over all other alternatives.  Decisiveness of T implies that F(u) =/

y.  If F(u) were x then {h} would be decisive for (x,z), which would contradict minimality of T. 

If F(u) were z then T\{h} would be decisive for (z,y), which would also contradict minimality of

T.  But Lemma 2 implies F(u)

0{x,y,z}.  This contradiction implies that #T must equal 1.  

So there is some individual h such that {h} is a decisive set for all pairs of alternatives. 

That is, for any pair (x,y) of distinct alternatives in X, there exists a preference profile u such that

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F(u) = x and {h} = {i

* u (x) > u (y)}.  But then Lemma 1 implies that F(v) =/ y whenever v (x) >

i

   

i

  

 

 

   

 

 

     

 

h

v (y).  Thus, F(v) cannot be any alternative in X other than the one that is most preferred by

h

individual h.  This proves the Muller-Satterthwaite theorem.  Q.E.D.

This theorem tells us that the only way to design a game that always has a unique Nash

equilibrium is to give one individual all the power, or to restrict the possible outcomes to two.  In

fact, many institutions of government actually fit one of these two categories.  Decision-making

in the executive branch is often made by a single decision-maker, who may be the president or

the minister with responsibility for a given domain of social alternatives.  On the other hand,

when a vote is called in a legislative assembly, there are usually only two possible outcomes: to

approve or to reject some specific proposal that is on the floor.  (Of course, the current vote may

be just one stage in a longer agenda, as when the assembly considers a proposal to amend another

proposal that is to scheduled be considered later.  We consider sequential voting in Sections 1.4

and 1.5.  Moore and Repullo, 1988, have shown more generally that more social choice functions

can be implemented by subgame-perfect equilibria in multistage game forms.)

But the Muller-Satterthwaite theorem also leaves us another way out.  The crucial

assumption in the Muller-Satterthwaite theorem is that F is a social choice function, not a

multivalued social choice correspondence.  Dropping this assumption just means admitting that

political processes might be games that sometimes have multiple equilibria.  As Schelling (1960)

has emphasized, when a game has multiple equilibria, the decisions made by rational players may

depend on culture and history (via the focal-point effect) as much as on their individual

preferences.  So we can use social choice procedures which consider more than two possible

outcomes at a time and which are not dictatorial, but only if we allow that these procedures might

sometimes have multiple equilibria that leave some decisive role for cultural traditions and other

factors that might influence voters' collective expectations.

1.3  Anonymity and neutrality

Having a dictatorship as a social choice function is disturbing to us because it is

manifestly unfair to the other individuals.  But nondictatorship is only the weakest equity

requirement.  In the theory of democracy, we should aspire to much higher forms of equity than

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nondictatorship.  A natural equity condition is that a social choice function or correspondence

should treat all the voters in the same way.  In social choice theory, symmetric treatment of

voters is called anonymity. 

A permutation of any set is a one-to-one function of that set onto itself.  For any

preference profile u in L(Y)  and any permutation 

B:N

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N of the set of voters, let u

CB be the

N

preference profile derived from u by assigning to individual i the preferences of individual 

B(i)

under the profile u; that is

(u

CB) (x) = u (x).

i

   

B(i)

A social choice function (or correspondence) F is said to be anonymous iff, for every permutation

B:N 

6

 N and for every preference profile u in L(Y) , 

N

F(u

CB) = F(u).

That is, anonymity means that the social choice correspondence does not ask which specific

individuals have each preference ordering, so that changing the names of the individuals with

each preference ordering would not change the chosen outcome.  Anonymity obviously implies

that there cannot be any dictator if #N > 1.

There is another kind of symmetry that we might ask of a social choice function or

correspondence: that it should treat the various alternatives in a neutral or unbiased way.  In

social choice theory, symmetric treatment of the various alternatives is called neutrality.  (A bias

in favor of the status quo is the most common form of nonneutrality.)

Given any permutation 

D:Y 

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 Y of the set of alternatives, for any preference profile u, let

u

BD be the preference profile such that each individual's ranking of alternatives x and y is the

same as his ranking of alternative 

D(x) and D(y) under u.  That is,

(u

BD) (x)  =  u (D(x)).

i

   

i

Then we say that a social choice function or correspondence F is neutral iff, for every preference

profile u and every permutation 

D:Y 

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 Y on the set of alternatives, 

D(F(uBD)) = F(u).

(When F is a correspondence, 

D(F(uBD)) is {D(x)* x 0 F(uBD)}.)  Notice that neutrality of a social

choice function F implies that its range must include all possible alternatives, that is,

F(L(Y) ) = Y.

N

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