Multidimensional Political Competition with Non-Common Beliefs
This paper extends a probabilistic voting model with a multidimensional policy space, allowing candidates to have different prior probability distributions of the distribution of voters' ideal policies. In this model, we show that a platform pair is a Nash equilibrium if and only if both candidates choose a common generalized median of expected ideal policies. Thus, the existence of a Nash equilibrium requires not only that each candidate's belief have an expected generalized median, which is already a knife-edge condition, but also that the two medians coincide. We also study limits of Îµ-equilibria of Radner (1980) as Îµ â†’ 0, which we call "limit equilibria." Limit equilibria are policy pairs that approximate choices by the candidates who almost perfectly optimize. We show that a policy pair is a limit equilibrium if and only if both candidates choose the same policy around which they form "opposite expectations" in a certain sense. For a limit equilibrium to exist (equivalently, for Îµ-equilibria to exist for all Îµ > 0), it is sufficient, though not necessary, that either candidate has an expected generalized median.
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