10. Electoral competition

If the members of a set of politicians choose whether to run as candidates, and if so the positions to take, then if the citizens vote sincerely, a Nash equilibrium generally does not exist. If voting is strategic and the politicians' payoff functions are strictly concave, then in an equilibrium every candidate's position is the median of the citizens' favorite positions.

If the set of politicians is the set of citizens and each candidate's position is her (known) favorite position, equilibria with a single candidate whose favorite position is the median of the citizens' favorite positions, or is close to the median, are possible. If voting is strategic, equilibria with multiple candidates whose favorite positions are the median are also possible, and whether voting is sincere or strategic, two-candidate equilibria in which the candidates' positions differ may exist.