Fundamentals of social choice theory

22

need only to show that this system of equations for d has no nonzero solutions.

So suppose to the contrary that this system of equations has some nonzero solution for d. 

Then it must have at least one nonzero solution such that all d(x) numbers are rational, because

all coefficients are rational in these linear equations.  Furthermore, multiplying through by the

lowest common denominator, this system of equations must have at least one solution such that

all d(x) are integers, and (dividing by 2 as necessary) we can guarantee that at least one integer

d(y) must be odd.  Then for this alternative y, we get

0 = d(y) + 

3

 d(x) + 

3

 d(z) = d(y) + 2(

3

 d(x)).

x<

 

   

z>>y

 

   

   

x<

But d(y) + 2(

3

 d(x)) is an odd integer, and 0 is even.  This contradiction proves that there

x<

cannot be any nonzero solutions for d.  Thus p = q, and so the equilibrium is unique.

The components of p must be rational numbers, because p is the unique equilibrium

strategy for a two-person zero-sum game that has payoffs in the rational numbers.  Now let p*

denote the smallest positive multiple of p that has all integer components.  This vector p*

satisfies

3

 p*(x) = 

3

 p*(x), 

x< 

   

x>>y

for every pure strategy y that is a best response to the equilibrium strategy p (which includes all y

in its support B), because all best responses give expected payoff zero.  At least one p*(z) must

be odd (or else we could divide them all by 2).  So the sum of the components

3

 p*(x) = p*(z)+ 

3

 p*(x) + 

3

 p*(x) 

x0B

 

   

 

x<

 

   

x>>z

= p*(z) + 2(

3

 p*(x))

x<

is an odd integer.  For every y that is a best response to the equilibrium strategy p, we have

p*(y) + 2(

3

 p*(x)) = 

3

 p*(x),

x< 

   

x0B

and so p*(y) is also an odd integer.  But 0 is not odd.  Thus, if y is any best response to the

equilibrium strategy p then

p(y) = p*(y)/

(

(

3

 p*(x)

)

 =/ 0.  

Q.E.D.

x0B

The set of alternatives that may be chosen with positive probability by the party leaders in

the equilibrium of this policy-positioning game is called the bipartisan set of the tournament

(Y,>>).  The bipartisan set is a subset of the uncovered set, because the covered alternatives are

the dominated strategies in this policy-positioning game, and so the bipartisan set is always

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23

contained in the top cycle.  Laffond, Laslier, and Le Breton (1993) have shown, however, that the

bipartisan set may contain alternatives that are not in the Banks set, and the Banks set may

contain alternatives that are not in the bipartisan set.

1.7  Median voter theorems

We have seen that, if a Condorcet winner exists, then we can expect it as the outcome of

rational voting in any binary agenda, or as the unique policy position that would be chosen by

party leaders in two-party competition.  But when a Condorcet winner does not exist, then

agenda manipulation with randomized alternatives can achieve virtually any outcome, and the

outcome of two-party policy positioning must have some unpredictability.  So we should be

interested in economic conditions that imply the existence of a Condorcet winner in Y.  The most

natural such condition is expressed by the median voter theorems.

There are two basic versions of the median voter theorem.  One version (from Black,

1958) assumes single-peaked preferences, and another version (from Gans and Smart, 1994,

Rothstein, 1990, 1991, and Roberts, 1977) assumes a single-crossing property.

To develop the single-crossing property, we begin by assuming that the policy

alternatives in Y are ordered completely and transitively, say from "left" to "right" in some sense. 

We may write "x < y" to mean that the alternative x is to the left of alternative y in the space of

policy alternatives.  We also assume that the voters (or their political preferences) are transitively

ordered in some political spectrum, say from "leftist" to "rightist," and we may write "i < j" to

mean that voter i is to the left of voter j in this political spectrum.  

The meaning of this ordering of voters is only that leftist voters tend to favor left policies

more than voters who are rightist in political preference.  Formally, we assume that, for any two

voters i and j such that i < j, and for any two policy alternatives x and y such that x < y,

if  u (x) < u (y)  then  u (x) < u (y),

i

   

i

  

  

j

   

j

but if  u (x) > u (y)  then  u (x) > u (y).

j

   

j

  

  

i

   

i

This assumption is called the single-crossing property.

Let us assume that the number of voters is odd and their ordering is complete and

transitive.  Then there is some median voter h, such that  #{i

0N* i < h} = #{j0N* h < j}.  For any

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24

pair of alternatives x and y such that x < y, if the median voter prefers x then all voters to the left

of the median agree with him, but if the median voter prefers y then all voters to the right of the

median agree with him.  Either way, there is a majority of voters who agree with the median

voter.  So the majority preference relation (>>) is the same as the preference of the median voter. 

Thus, the alternative that is most preferred by the median voter must be a Condorcet winner. 

That is, we have proven the following theorem.

Theorem 1.5. Suppose that there is an odd number of voters. If the alternatives in Y have

a complete transitive ordering and the voters in N have a complete transitive ordering which

together satisfy the single-crossing property, then the ideal point of the median voter is a

Condorcet winner in Y.

In the single-peakedness version of the median-voter theorem, a complete transitive

ordering (<) is assumed on the set of alternatives Y only.  For each voter i, it is assumed that

there is some ideal point 

2  in Y such that, for every x and y in Y, 

i

if  

2  

#

 x < y  or  y < x 

#

 

2   then  u (x) > u (y).

i

                     

i

  

  

i

   

i

That is, on either side of 

2 , voter i always prefers alternatives that are closer to 2 .  This property

i

 

   

 

 

 

 

 

   

i

is called the single-peakedness assumption.  Assuming that the number of voters is odd, the

median voter's ideal point is the alternative 

2* such that 

#N/2 

$

 #{i

* 2  

<

 

2*}  and  #N/2 

$

 {i

* 2* 

<

 

2 }.

i

   

  

  

   

 

   

i

The voters who have ideal points at 

2* and to its left form a majority that prefers 2* over any

alternative to the right of 

2*, while the voters who have ideal points at 2* and to its right form a

majority that prefers 

2* over any alternative to the left of 2*.  Thus, this median voter's ideal

point 

2* is a Condorcet winner in Y. 

Single-crossing and single-peakedness are different assumptions, and neither is logically

implied by the other.  Both assumptions give us a result that says "the median voter's ideal point

is a Condorcet winner," but there is a subtle difference in the meaning of these results.  With the

single-crossing property we are speaking about the ideal point of the median voter, but with the

single-peakedness property we are speaking about the median of the voters' ideal points.  Notice

also that the majority preference relation can be guaranteed to be a full transitive ordering under

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25

the single-crossing assumption, but not under the single-peakedness assumption.  

In both versions of the median voter theorem, the set of policy alternatives must be

essentially one-dimensional, because otherwise we cannot put the alternatives in a transitive

order.  In general applications that do not have this simple one-dimensional structure, we do not

generally expect to find a Condorcet winner.

1.8  Conclusions

We have considered binary agendas and two-party competition, because they are

procedures for reducing general social choice problems with many alternatives into a simple

framework of majority voting on pairs of alternatives.  This reduction requires some decision-

making by political leaders: the chairman who sets the agenda, or the leaders who formulate

policy for the two major parties.  So it is natural ask, to what extent do the outcomes of binary

agendas or two-party competition depend on the decision-making by such political leaders, rather

than on the preferences of the voters.  The answer, we have seen, is that manipulations of an

agenda-setter or arbitrary and unpredictable positioning decisions of political leaders can

substantially affect the outcome of majority voting, except in the special case where a Condorcet

winner happens to exist.

To find ways of avoiding such dependence on an agenda setter or a couple of party

leaders, we must go on to study more general voting systems that allow voters to consider more

than two alternatives at once.  K. May's theorem (1952) assured us that majority rule is the

unique obvious way to implement the principles of democracy (anonymity, neutrality) in social

decision-making when only two alternatives are considered at a time.  In contrast, there is a wide

variety of anonymous neutral voting systems that have been proposed for choosing among more

than two alternatives (plurality voting, Borda voting, approval voting, single transferable vote,

etc.), and all of these deserve to be called democratic.  Furthermore, the impossibility theorems

of social choice theory tell us that no such voting system can guarantee a unique pure-strategy

equilibrum for all profiles of voters' preferences.  Multiplicity of equilibria means that the social

outcome can depend on any factor that focuses public attention on one equilibrium.  These focal

factors may include history, cultural tradition, and public speeches of political leaders.  (See

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26

Schelling, 1960, and Myerson and Weber, 1993.)

Our initial fable suggested that political institutions may arise out of a need to coordinate

on better equilibria in social and economic arenas, and we have found that some of this

multiplicity of equilibria may inevitably remain in any democratic political system.  But having

multiple equilibrium outcomes for some preference profiles does not imply that everything must

be an equilibrium outcome for all preference profiles.  Game-theoretic analysis of political

institutions can show substantial differences in the equilibrium outcomes under different political

institutions.  If social choice theory has not given us one perfect voting system, then it has left us

the important task of characterizing the properties and performance of the many voting systems

that we do have.  

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Author's address:  Economics Dept., University of Chicago, 1126 East 59th Street, Chicago, IL 60637.

Phone:  [phone].  Fax:  [phone].

Email:  [e-mail].  URL:  http://home.uchicago.edu/~rmyerson/

The original version of this paper was distributed (Sept 1996) as Discussion Paper #1162 of 

the Center for Mathematical Studies in Economics and Management Science, Northwestern University.

The current version is at  http://home.uchicago.edu/~rmyerson/research/schch1.pdf  

Date of this version: 1/5/2011.