ECONOMY

910 economic methods in political theory
and, therefore, is properly conceived as taking prices as given; in electorates, however,
while it remains true that the likelihood that any single vote tips the outcome becomes
vanishingly small as the number of voters grows, it is not true (at least for finite
electorates) that the behavior of any given voter should be conditioned on the almost
sure event that the voter’s decision is consequentially irrelevant. This is not usually a
problem for classical collective preference theory which, as the name suggests, focuses
on aggregating given preference profiles, not vote profiles. Nor is it any problem for
Nash equilibrium theory insofar as there are a huge number of equilibrium patterns
of voting in any large election (other than with unanimity rule), most of which look
empirically silly. But empirical voting patterns are not arbitrary. And once account is
taken of the fact that preferring one candidate to another in an election does not imply
voting for that candidate (individuals can abstain or vote strategically), there is clearly
a severe methodological problem with respect to analysing equilibrium behavior in
large electorates.
One line of attack has been by brute force, using combinatorial techniques to
compute the probability that a particular vote is pivotal, conditional on the specified
(undominated) votes of others (Ledyard 1984; Cox1994; Palfrey 1989). But this is
cumbersome and places considerable demands on exactly what it is that individuals
know about the behavior of others. In particular, individuals are assumed to know
the exact size of the population. Myerson (1998, 2000, 2002) relaxes this assumption
and develops a novel theory of Poisson games to analyse strategic behavior in large
populations.
Rather than assume the size of the electorate is known, suppose that the actual
number of potential voters is a random variable distributed according to a Poisson
distribution with mean n, wheren is large. Then the probability that there is any
particular number of voters in the society is easily calculated. As a statistical model
underlying the true size of any electorate, the Poisson distribution uniquely exhibits
a very useful technical property, environmental equivalence: under the Poisson distribution,
any individual in the realized electorate believes that the number of other
individuals in the electorate is also a random variable distributed according to a
Poisson distribution with the same mean. And because the number and identity of
realized individuals in the electorate is a random variable, it is enough to identify
voters by type rather than their names, where an individual’s type describes all of the
strategically relevant characteristics of the individual (for example the individual’s
preferences over the candidates seeking office in any election). If the list of possible
individual types is fixed and known, then the distribution of each type in a realized
population of any size is itself given by a Poisson distribution.
The preceding implications of modeling population size as an unobserved draw
from a Poisson distribution allow a relatively tractable and appealing strategic theory
of elections. Because only types are relevant, individual strategies are appropriately
defined as depending only on voter type rather than on voter identity. Thus individuals
know only their own types, the distribution of possible types in the population,
that the population size is a random draw from a Poisson distribution, and that all
individuals of the same type behave in the same way. Call such a strategic model a