Fundamentals of social choice theory

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FUNDAMENTALS OF SOCIAL CHOICE THEORY

by Roger B. Myerson, September 1996

http://home.uchicago.edu/~rmyerson/research/schch1.pdf

Abstract.  This paper offers a short introduction to some of the fundamental results of social

choice theory.  Topices include: Nash implementability and the Muller-Satterthwaite

impossibility theorem, anonymous and neutral social choice correspondences, two-party

competition in tournaments, binary agendas and the top cycle, and median voter theorems.  The

paper begins with a simple example to illustrate the importance of multiple equilibria in game-

theoretic models of political institutions.

1.1  An introductory model of political institutions

Mathematical models in social science are like fables or myths that we read to get insights

into the social world in which we live.  Our mathematical models are told in a specialized

technical language that allows very precise descriptions of the motivations and choices of the

various individuals in these stories.  When we prove theorems in mathematical social science, we

are making general statements about whole classes of such stories all at once.  Here we focus on

game-theoretic models of political institutions. 

So let us begin our study of political institutions by a simple game-theoretic model that

tells a story of how political institutions may arise.  Consider first the simple two-person "Battle

of Sexes" game shown in Table 1.1.

   Player 2

f

g

2

2

Player 1

f

0,0

3,6

1

g

6,3

0,0

1

Table 1.1.  The Battle of Sexes game.

The two players in this game, who may be called player 1 and player 2, must

independently choose one of two possible strategies: to defer (f ) or to grab (g ).  If the players

i

     

 

i

both grab or both defer then neither player gets anything; but if exactly one player grabs then he

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gets payoff 6 while the deferential player gets payoff 3.

This game has three equilibria.  There is an equilibrium in which player 1 grabs while

player 2 defers, giving payoffs (6,3).  There is another equilibrium in which player 1 defers while

player 2 grabs, giving payoffs (3,6).  There is also a symmetric randomized equilibrium in which

each player independently  randomizes between grabbing, with probability 2/3, and deferring,

with probability 1/3.  In this randomized equilibrium, the expected payoffs are (2,2), which is

worse for both players than either of the nonsymmetric equilibria.

Now think of an island with a large population of individuals.  Every morning, these

islanders assemble in the center of their island, to talk and watch the sun rise.  Then the islanders

scatter for the day.  During the day, the islanders are randomly matched into pairs who meet at

random locations around the island, and each of these matched pairs plays the simple Battle of

Sexes game once. This process repeats every day.  Each player's objective is to maximize a long-

run discounted average of his (or her) sequence of payoffs from these daily Battle of Sexes

matches. 

One long-run equilibrium of this process is for everyone to play the symmetric

randomized equilibrium in his match each day.  But rising up from this primitive anarchy, the

players could develop cultural expectations which break the symmetry among matched players,

so that they will share an understanding of who should grab and who should defer.

One possibility is that the islanders might develop an understanding that each player has a

special "ownership" relationship with some region of the island, such that a player is expected to

grab whenever he is in the region that he owns.  Notice that this system of ownership rights is a

self-enforcing equilibrium, because the other player does better by deferring (getting 3 rather than

0) when he expects the "owner" to grab, and so the owner should indeed grab confidently.

But such a system of traditional grabbing rights might fail to cover many matching

situations where no one has clear "ownership."  To avoid the costly symmetric equilibrium in

such cases, other ways of breaking the players' symmetry are needed.  A system of leadership can

be used to solve this problem.

That is, the islanders might appoint one of their population to serve as a leader, who will

announce each morning a set of instructions that specify which one of the two players should

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grab in each of the daily matches.  As long as the leader's instructions are clear and

comprehensive, the understanding that every player will obey these instructions is a self-

enforcing equilibrium.  A player who grabbed when he was instructed to defer would only lower

his expected payoff from 3 to 0, given that the other player is expected to follow his instruction

to grab here.

To make this system of government work on our island, the islanders only need a shared

understanding as to who is the leader.  The leader might be the eldest among the islanders, or the

tallest, or the one with the loudest voice.  Or the islanders might determine their leader by some

contest, such as a chess tournament, or by an annual election in which all the islanders vote.  Any

method of selection that the islanders understand can be used, because everyone wants to obey

the selected leader's instructions as long as everyone else is expected to obey him.  Thus, self-

enforcing rules for a political system can be constructed arbitrarily from the equilibrium selection

problem in this game.

The islanders could impose limits to a leader's authority in this political system.  For

example, there might be one leader whose instructions are obeyed on the northern half of the

island, and another leader whose instructions are obeyed on the southern half.  The islanders may

even have ways to remove a leader, such as when he loses some re-election contest or when he

issues instructions that violate some perceived limits.  If a former leader tries to make an

announcement in his former domain of authority, every player would be expected to ignore this

announcement as irrelevant cheap talk.

Of course, the real world is very different from the simple island of this fable.  But as in

this island, coordination games with multiple equilibria are pervasive in any real society.  Thus,

any successful society must develop leadership structures that can coordinate people's

expectations in situations of multiple equilibria.  So the first point of this fable is the basic social

need for leadership and for political institutions that can provide it.

The second point of this fable is that the effectiveness of any political institution may be

derived simply from a shared understanding that it is in effect, as Hardin (1989) has argued.  

Thus, any political system may be one of many possible equilibria of a more fundamental

coordination game of constitutional selection.  That is, the process of selecting a constitution can

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