Theorem 2.2, the equilibrium existence theorem, has laid the groundwork for the analysis in this chapter. Specifically, with Theorem 2.2 established, we can now analyze electoral equilibria. In particular, we derive the most important properties of the redistributional equilibria whose existence is assured by Theorem 2.2.

Section 3.1 starts the analysis by deriving the following five properties: (i) In an equilibrium, each candidate's strategy is the unique best response to the strategy being used by the other candidate (Theorem 3.1); (ii) for any given electorate, there is a unique electoral equilibrium (Theorem 3.2); (iii) in the unique equilibrium, both candidates have the same redistributional reputations (Theorem 3.3); (iv) in the unique equilibrium, the expected plurality for each candidate is zero (corollary to Theorem 3.3); (v) the common redistributional reputation of the candidates in the unique equilibrium can be viewed as the solution of a particular optimization problem that involves candidate 1's expected plurality function (Theorem 3.4).

Two theorems in Section 3.2 identify the locations of the electoral equilibria in all of the special cases that can arise. In particular, Theorem 3.5 covers special cases in which the location can be identified by using a Lagrangian, and Theorem 3.6 covers all remaining cases. These location theorems identify the exact redistributional reputation that each candidate wants to have, for each possible electorate. Section 3.3 provides a different perspective on the redistributional equilibria being studied, by establishing that, for each possible electorate, the location of the electoral equilibrium is also the maximum of the sum of the voters' utility functions (analogous to the conclusion in theorem 1 in Ledyard 1984).