Equilibrium in a discrete Downsian model given a non-minimal valence advantage and linear loss functions

This note complements Aragonès and Palfrey [Aragonés, E., Palfrey, T., 2002. Mixed strategy equilibrium in a Downsian model with a favored candidate. Journal of Economic Theory 103, 131–161.] and Hummel [Hummel, P., 2010. On the nature of equilibriums in a Downsian model with candidate valence. Games and Economic Behavior 70 (2), 425–445.] by characterizing an essentially unique mixed strategy Nash equilibrium in a two-candidate Downsian model where one candidate enjoys a non-minimal non-policy advantage over the other candidate. The policy space is unidimensional and discrete (even number of equidistant locations), the preferences of the median voter are not known to the candidates and voter’s preferences on the policy space are represented by linear loss functions. We find that if the uncertainty about the median voter’s preferences is sufficiently low, then the mixed strategy σˆA= play the two intermediate locations with probability12 for the advantaged candidate and the mixed strategy σˆD= play the least liberal location that guarantees positive probability of election givenσˆAwith probability12and the least conservative strategy that guarantees positive probability of election givenσˆAwith probability12 for the disadvantaged candidate, constitute a Nash equilibrium of the game for any admissible value of the non-policy advantage.