4. Voting with many alternatives

One mechanism a group of individuals may use to select an alternative from a set of many alternatives is plurality rule: each individual votes for a single alternative, and the winner is the alternative that receives the most votes (or the set of alternatives that receive the most votes, in the case of a tie). If voting is costless, each individual's voting for her least-favored alternative is dominated, but voting for any other alternative is not dominated. An equilibrium exists in which no individual's vote is weakly dominated. If the set of alternatives is an interval of numbers and the individuals' payoff functions are strictly concave, in an equilibrium at most two alternatives tie for winner; every alternative is possible in an equilibrium. In a simple model of "divided majority" with three alternatives, a majority of individuals rank alternatives a and b above c, but disagree about whether a is better or worse than b. In a model in which each individual knows her own preferences and has probabilistic beliefs about the other individuals' preferences, whether in equilibria in a large population the majority succeeds in coordinating to defeat c depends on the precise nature of the uncertainty.