Fundamentals of social choice theory

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alternatives have been eliminated.  For any given tournament (Y,>>), the set of alternatives that

can be sophisticated outcomes of such successive-elimination agendas is called the Banks set.  

Given a tournament (Y,>>), an alternative x is covered iff there exists some other

alternative y such that y >> x and

{z

* x >> z} 

f

 {z

* y >> z}. 

If x is Pareto-dominated by some other alternative then x must be covered.  The uncovered set is

the set of alternatives that are not covered.  (See Miller, 1980.)  The uncovered set is always a

subset of the top cycle, because any alternative not in Y*(3) is covered by any alternative in

Y*(3).  On the other hand, the Banks set is always a subset of the uncovered set.  (See Shepsle

and Weingast, 1984; Banks, 1985; McKelvey, 1986; or Moulin, 1986.)  Thus, Pareto-dominated

alternatives cannot be sophisticated outcomes of successive-elimination agendas.

1.6  Two-party competition

We have studied binary agendas because they allow us to reduce the problem of choosing

among many alternatives to a sequence of votes each of which is binary, in the sense that it has

only two possible outcomes.  However, the number of binary votes that are needed to work

through a large number of alternatives goes up at least as the log (base 2) of the number of

alternatives.  When we move from voting in small committees to voting in large democratic

nations, the increased cost of each round of voting makes it impractical to work through a long

sequence of votes.  So democracies generally rely on political leaders to select a small subset of

the potential social alternatives, and then only this small selected set of social alternatives will be

considered by the voters in the general election.  The hope for a successful democracy is that

competition among political leaders should ensure that they will try to select alternatives that are

highly preferred by a large fraction of the voting population.

So let us consider a simple model of how political leaders might select alternatives to put

before the voters in a general election. We assume that the set of all possible social alternatives Y

is a nonempty finite set.  To keep voting binary, we assume here that there are only two political

leaders, each of whom must select an alternative in Y, which we may call the leader's policy

position.  Making the simplest assumption about timing, let us suppose that the two political

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leaders must choose their policy positions simultaneously and independently.  

Let >> denote the majority preference relation, satisfying the completeness and

antisymmetry properties of a tournament.  We assume that the leader whose policy position is

preferred by a majority of the voters will win the election if they choose different positions, and

each leader has a probability 1/2 of winning if both leaders choose the same policy position. 

Assuming that each leader is motivated only by the desire to win, we get a simple two-person

zero-sum game.  In this game, when leader 1 chooses position x  and leader 2 chooses position

1

x , the payoffs are

2

+1 for leader 1 and -1 for leader 2 if x  >> x  

1

 

 

2,

-1 for leader 1 and +1 for leader 2 if x  >> x ,

2

 

 

1

and 0 for both leaders if x  = x

1

   

2.

If Y contains a Condorcet winner that beats every other alternative in Y, then the unique

equilibrium of this game is for both political leaders to choose this Condorcet winner as their

policy position.  But if no alternative is a Condorcet-winner then this game cannot have any

equilibria in pure strategies, because any position could be beaten by at least one other

alternative, and so each leader could make himself the sure winner if he knew which position

would be chosen by his opponent.

The general existence theorems of von Neumann (1928) and Nash (1951) assure us that

this game must have at least one equilibrium in mixed strategies, even if a Condorcet winner

does not exist.  We know that all equilibria must give the same expected payoff allocation,

because this game is two-person zero-sum.  The ex-ante symmetry of the leaders who are playing

this game makes it obvious that each player must have the same set of equilibrium strategies, and

the expected payoff allocation in equilibrium must be (0,0).  That is, each leader's probability of

winning the election must be 1/2 at the beginning of the game (before the randomized strategies

are implemented).  

For the example of the ABC paradox from Section 1.3, the unique equilibrium strategy

for each leader is to randomize uniformly over the three alternatives, choosing each with

probability 1/3.  Then there is a 1/3 probability of leader 1 choosing a position that beats leader

2's position (in the sense that x  >> x ); there is a 1/3 probability of leader 1 choosing a position

1

 

 

2

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that is beaten by leader 2's position; and there is a 1/3 probability of leader 1 choosing the same

position as leader 2, in which case each has an equal probability of winning the election.

If we add a fourth alternative d such that every voter ranks d immediately below c, then

the unique equilibrium strategy remains the same.  That is, the alternative d would not chosen by

either leader, even though d is in the top cycle.  Notice that d is covered by c in this example.  In

fact, the covered alternatives in any tournament are precisely the dominated pure strategies for

the leaders in this policy-positioning game.

Remarkably, Fisher and Ryan (1992) showed that there is always a unique equilibrium

strategy in this game.  Our formulation and proof of this uniqueness theorem here is based on

Laffond, Laslier, and Le Breton (1993).

Theorem 1.4.  The two-person game of choosing positions in a finite tournament (Y,>>)

has a unique Nash equilibrium.  In this equilibrium, every alternative that is a best response is

assigned positive probability.

Proof.  Let p and q be randomized strategies in 

)(Y) in two Nash equilibria of this game. 

The two players in this game are symmetric, and Nash equilibria of two-person zero-sum games

are always interchangeable, and so (p,p), (q,q), and (p,q) must all be Nash equilibria of this game. 

Let 

B = {y

0Y* p(y) > 0 or q(y) > 0}.

Because randomizing according to p or q is optimal for a player against p or q, all of the

alternatives in B must offer the equilibrium expected payoff against both p and q.  But we know

that the equilibrium expected payoff is 0 in this game.  So choosing any alternative in B must

give a player a probability of winning that equals his probability of losing, when the other player

randomizes according to p or q.  That is,

3

 p(x) = 

3

 p(x),  

œy 0 B

x>>y

 

   

x<3

 q(x) = 

3

 q(x),  

œy 0 B.

x>>y

 

   

x<Now let d(x) = p(x) - q(x) for every x in B.  So we have

3

 d(x) = 

3

 d(x),  

œy 0 B,

x>>y

 

   

x<3

 d(x) = 0.

x0B

(The latter equation holds because the p(x) and q(x) both sum to 1.)  To show uniqueness, we

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