8. Electoral competition

Two candidates compete in an election. Each candidate chooses a position in a given set, and prefers to win than to tie than to lose. Each member of a finite set of citizens has preferences over positions, and votes for the candidate whose position she prefers. The candidates know the citizens' preferences. In any Nash equilibrium of the strategic game in which the candidates are the players, each candidate's position is a Condorcet winner of the collective choice problem in which the individuals are the citizens. So if the citizens' preferences are single-peaked, each candidate's position in a Nash equilibrium is the median of the individuals' favorite positions, and if the preferences are single-crossing, it is the favorite position of the median citizen. If the candidates choose their positions sequentially, these positions are also Condorcet winners if such winners exist.

Suppose that the candidates are uncertain of the citizens' preferences and the set of positions is an interval of real numbers. If the candidates share the belief that the median of the citizens' favorite positions has a given distribution, in an equilibrium they both choose the median of that distribution. If the candidates are privately informed about the median, the equilibrium position of a candidate with a given signal is the median of the distribution when the other candidate's signal is the same.

The chapter explores also a model in which voting is costly and one in which citizens have preferences over candidates independent of the positions chosen by the candidates.