Fundamentals of social choice theory

11

To see the general impossibility of constructing social choice functions that are both

anonymous and neutral, it suffices to consider a simple example with three alternatives Y =

{a,b,c} and three voters N = {1,2,3}.  Consider the preference profile u such that

u (a)  >  u (b)  >  u (c),

1

   

1

   

1

u (b)  >  u (c)  >  u (a),

2

   

2

   

2

u (c)  >  u (a)  >  u (b).

3

   

3

   

3

We may call this example the ABC paradox (where "ABC" stands for Arrow, Black, and

Condorcet, who drew attention to such examples); it is also known as the Condorcet cycle.  An

example like this appeared at the heart of the proof of the impossibility theorem in the preceding

section.  Any alternative in this example can be mapped to any other alternative by a permutation

of Y such that an appropriate permutation of N can then return the original preference profile. 

Thus, an anonymous neutral social choice correspondence must choose either the empty set or

the set of all three alternatives for this ABC paradox, and so an anonymous neutral social choice

function cannot be defined.

This argument could be also formulated as a statement about implementation by Nash

equilibria.  Under any voting procedure that treats the voters anonymously and is neutral to the

various alternatives, the set of equilibrium outcomes for this example must be symmetric around

the three alternatives {a,b,c}.  Thus, an anonymous neutral voting game cannot have a unique

pure-strategy equilibrium that selects only one out of the three alternatives for the ABC paradox.

This argument does not generalize to randomized-strategy equilibria.  The symmetry of

this example could be satisfied by a unique equilibrium in randomized strategies such that each

alternative is selected with probability 1/3.  The Muller-Satterthwaite theorem does not consider

randomized social choice functions, but Gibbard (1978) has obtained related results on

dominant-strategy implementation with randomization.  (Gibbard characterizes the dominant-

strategy-implementable randomized social choice functions as probabilistic mixtures of unilateral

and duple functions, which are generalizations of dictatorship and binary voting.)

Randomization confronts democratic theory with the same difficulty as multiple

equilibria, however.  In both cases, the social choice ultimately depends on factors that are

unrelated to the individual voters' preferences (private randomizing factors in one case, public

---

a

c

b

c

Figure 1.1

12

focal factors in the other).  As Riker (1982) has emphasized, such dependence on extraneous

factors implies that the outcome chosen by a democratic process cannot be characterized as a

pure expression of the voters' will.

1.4  Tournaments and binary agendas

When there are only two alternatives, majority rule is a simple and compelling social

choice procedure.  K. May (1952) showed that, when #Y = 2 and #N is odd, choosing the

alternative that is preferred by a majority of the voters is the unique social choice function that

satisfies anonymity, neutrality, and monotonicity.

When there are more than two alternatives, we might still try to apply the principle of

majority voting by dividing the decision problem into a sequence of binary questions.  For

example, one simple binary agenda for choosing among three alternatives {a,b,c} is as follows. 

At the first stage, there is a vote on the question of whether to eliminate alternative a or

alternative b from further consideration.  Then, at the second stage, there is a vote between

alternative c and the alternative among {a,b} that survived the first vote.  The winner of this

second vote is the implemented social choice.

This binary agenda is represented graphically in Figure 1.1.  The agenda begins at the top,

and at each stage the voters must choose to move down the agenda tree along the branch to the

left or to the right.  The labels at the bottom of the agenda tree indicate the social choice for each

possible outcome at the end of the agenda.  Thus, at the top of Figure 1.1, the left branch

represents eliminating b at the first vote, and the right branch represents eliminating a.  Then at

each of the lower nodes, the right branch represents choosing c and the left branch represents

choosing the other alternative that was not eliminated at the first stage.

---

(b)

(c)

(b)

a

c

b

c

Figure 1.2

13

Now suppose that the voters have preferences as in the ABC paradox example (described

in the preceding section).  Then there is a majority (voters 1 and 3) who prefer alternative a over

b, there is a majority (voters 1 and 2) who prefer alternative b over c, and there is a majority

(voters 2 and 3) who prefer alternative c over a.  Let us use the notation  x >> y (or equivalently 

y << x) to denote the statement that a majority of the voters prefer x over y.  Then we may

summarize the majority preference for this example as follows:

a >> b,  b >> c,  c >> a.

(This cycle, of course, is what makes this example paradoxical.)

Given these voters' preferences, what will be the outcome of the binary agenda in Figure

1.1?  At the second stage, a majority would choose alternative b against c if alternative a were

eliminated at the first stage, but a majority would choose alternative c against a if alternative b

were eliminated at the first stage.  So a majority of voters should vote to eliminate alternative a at

the first stage (even though a majority prefers a over b), because they should anticipate that the

ultimate result will be to implement b rather than c, and a majority prefers alternative b over c. 

This backwards analysis is shown in Figure 1.2 which displays, in parentheses above each

decision node, the ultimate outcome that would be chosen by sophisticated majority voting if the

process reached this node.

In general, given any finite set of alternatives Y,  a binary agenda on Y is a rooted tree

that has two branches coming out of each nonterminal node, together with a labelling that assigns

an outcome in Y to every terminal node, such that each alternative in Y appears as the outcome

---