ECONOMY

70 candidate objectives and electoral equilibrium
∆
u 3 (x ~ 3) = 0
∆
u 1 (x ~ 3)
x
~ 3
∆
u 2 (x ~ 3)
Fig. 4.2 Radial symmetry in equilibrium
Theorem 1 implies that there is a unique equilibrium, and in equilibrium the candidates
both locate at the median ideal policy. Known as the “median voter theorem,”
this connection was made by Hotelling (1929) in his model of spatial competition and
by Downs (1957) in his classic analysis of elections.
Corollary 1: (Hotelling; Downs): In the Downsian model, assume X is unidimensional
and office motivation. There is a unique equilibrium, and in equilibrium the
candidates locate at the median ideal policy.
When the policy space is multidimensional, the necessary condition of radial
symmetry becomes extremely restrictive—so restrictive that it will be violated for
almost all specifications of voter preferences, with the implication that equilibria will
almost never exist. And if there were an equilibrium, as in Figure 4.2,thenarbitrarily
small perturbations of voter preferences could break radial symmetry, implying that
existence is a “razor’s edge” phenomenon.
3.2 Policy Motivation
The objective function representing policy motivation takes the form
EU A (x A , x B )=
⎧
⎪⎨ u A (x A ) if #{i | u i (x A ) > u i (x B )} > #{i | u i (x B ) > u i (x A )}
u A (x B ) if #{i | u i (x B ) > u i (x A )} > #{i | u i (x A ) > u i (x B )}
⎪⎩ u A (x A )+u A (x B )
2
else