ECONOMY

john duggan 79
When the policy space is multidimensional, the task of equilibrium characterization
is more difficult, and we simplify matters by specializing to the representative
voter stochastic preference model. We say a policy x is a generalized median in all
directions if, compared to every other policy y, thevoterismorelikelytopreferx to
y than the converse: for every policy y, wehave
Pr({Ë | u(y, Ë) > u(x, Ë)}) ≤ 1 2 .
By strict concavity and our dispersion condition, if there is a generalized median in all
directions, then there is exactly one, which we denote x „ . In the quadratic version of
the model, x „ is equivalent to a median in all directions, in the usual sense. 11 When the
policy space is multidimensional, such a policy exists, for example, if G has a radially
symmetric density function, such as the normal distribution. This can be weakened,
but existence of a median in all directions is quite restrictive when the policy space
has dimension at least two.
The next result provides a characterization of equilibria in the multidimensional
stochastic preference model: in equilibrium, the candidates must locate at the generalized
median in all directions. Under our maintained assumptions, a generalized
median in all directions is essentially an “estimated median,” as in Calvert (1985), so
the next result is close to a result contained in that paper.
Theorem 9:(Calvert): In the representative voter stochastic preference model, assume
vote motivation. There is an equilibrium (x ∗ A , x∗ B
) if and only if there is a generalized
median in all directions. In this case, the equilibrium is unique, and the candidates
locate at the generalized median: x ∗ A =x∗ B =x „.
As we have seen before, strategic incentives drive the candidates to take identical
positions in equilibrium, but the implication for equilibrium existence in multidimensional
policy spaces is negative, as existence of a generalized median in all
directions is extremely restrictive.
5.2 Win Motivation
Under win motivation in the stochastic preference model, we again have discontinuities
along the diagonal. Despite this, there is a unique equilibrium when the policy
space is unidimensional, as with vote motivation, but now the characterization is
changed. Let H Ï denote the distribution of the median ideal policy, i.e. H Ï (x) is
the probability that the median voter’s ideal policy is less than or equal to x. Byour
dispersion assumption, H Ï is strictly increasing and has a unique median, denoted x Ï .
The next result, due to Calvert (1985), establishes that in equilibrium the candidates
must locate at the same policy, the median of x Ï . Thus, in the unidimensional version
¹¹ That is, every hyperplane through x „ divides the space in half: the probability that the
representative voter’s ideal policy is to one side of the hyperplane is equal to one half.