(Large) finite to continuum: An approximation result for electoral competition models

We consider a model of electoral competition with two contestants where voters have single-plateaued preferences. We characterize the Nash equilibria of the electoral game for two settings: (i) finite, and (ii) continuum of voters over finitely many voter preferences. We say that the continuum model approximates the finite voters model if the Nash equilibria set in the two models is the same when the population tends to infinity. We show that approximation holds if and only if the corresponding continuum model satisfies proportion conservation at the centre (PCC) and positive mass at limit-centre (PML). PCC states that the aggregate mass of voters at the centre in the continuum model be equal to its finite (proportional) counterpart as the population tends to infinity. PML requires that the limit-centre be in the support of the limit distribution in the continuum model. Our paper provides a framework for studying approximation of equilibria in electoral competition models.