Fundamentals of social choice theory

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be viewed as an equilibrium-selection problem, but it is the equilibrium-selection problem to

solve all other equilibrium-selection problems.

The arbitrariness of political structures from this game-theoretic perspective validates our

treating them as exogenous explanatory parameters in political economics.  A question might be

posed, for example, as to whether one form of democracy might generate higher economic

welfare than some other forms of government.  Such a question would be untestable or even

meaningless if the form of government were itself determined by the level of economic welfare. 

But our fable suggests that the crucial necessary condition for democracy is not wealth or

literacy, but is simply a shared understanding that democracy will function in this society (so that

an officer who waves his pistol in the legislative chamber should be perceived as a madman in

need of psychiatric treatment, not as the new leader of the country).

1.2  A general impossibility theorem

There is an enormous diversity of democratic political institutions that could exist.  Social

choice theory is a branch of mathematical social science that tries to make general statements

about all such institutions.  Given the diversity of potential institutions, the power of social

choice theory may be quite limited, and indeed its most famous results are negative impossibility

theorems.  But it is good to start with the general perspective of social choice theory and see what

can be said at this level.  Later we can turn to formal political theory, where we will focus on

narrower models that enable us to say more about the specific kinds of political institutions that

exist in the real world.

Modern social choice theory begins with the great theorem of Arrow (1951).  This

theorem has led to many other impossibility theorems, notably the theorem of Gibbard (1973)

and Satterthwaite (1975). (See also Sen, 1970.)  In this section, we focus on the theorem of

Muller and Satterthwaite (1977), because this is the impossibility theorem that applies directly to

Nash-equilibrium implementation.  The Muller-Satterthwaite theorem was first proven as a

consequence of the Gibbard-Satterthwaite and Arrow theorems, but we prove it here directly,

following Moulin (1988). 

Let N denote a given set of individual voters, and let Y denote a given set of alternatives

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or social-choice options among which the voters must select one.  We assume that N and Y are

both nonempty finite sets.  Let L(Y) denote the set of strict transitive orderings of the alternatives

in Y.  Given that there are only finitely many alternatives, we may represent any individual's

preference ordering in L(Y) by a utility function u  such that u (x) is the number of alternatives

i

 

 

 

i

that the individual i considers to be strictly worse than x.  So with strict preferences, L(Y) can be

identified with the set of one-to-one functions from Y to the set {0,1,...,#Y-1}.  (Here #Y denotes

the number of alternatives in the set Y.)

We let L(Y)  denote the set of profiles of such preference orderings, one for each

N

individual voter.  We denote such a preference profile by a profile of utility functions u = (u )

,

i i0N

where each u  is in L(Y).  So if the voters' preference profile is u, then the inequality  u (x) > u (y)

i

    

 

      

 

 

   

   

 

 

i

   

i

means that voter i prefers alternative x over alternative y.  The assumption of strict preferences

implies that either u (x) > u (y) or u (y) > u (x) must hold if x =/  y.

i

   

i

   

i

   

i

A political system creates a game that is played by the voters, with outcomes in the set of

alternatives Y.  The voters will play this game in a way that depends on their individual

preferences over Y, and so the realized outcome may be a function of the preference profile in

L(Y) .  From the abstract perspective of social choice theory, an institution could be represented

N

by the mapping that specifies these predicted outcomes as a function of the voters' preferences. 

So a social choice function is any function F:L(Y)  

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 Y, where F(u) may be interpreted as the

N

alternative in Y that would be chosen (under some given institutional arrangement) if the voters'

preferences were as in u.

If there are multiple equilibria in our political game, then we may have to talk instead

about a set of possible equilibrium outcomes.  So a social choice correspondence is any point-to-

set mapping G:L(Y)  

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 Y.  Here, for any preference profile u, G(u) is a subset of Y that may

N

be interpreted as the set of alternatives in Y that might be chosen by society (under some

institutional arrangement) if the voters' preferences were as in u.  

A social choice function F is monotonic iff, for every pair of preference profiles u and v

in L(Y) , and for every alternative x in Y, if x = F(u) and 

N

{y

*  v (y) 

>

 v (x)} 

f

 {y

*  u (y) 

>

 u (x)},  

œi 0 N,

i

   

i

   

 

i

   

i

then x = F(v).  Similarly, a social choice correspondence G is monotonic iff, for every u and v in

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