On the existence of Nash equilibrium in electoral competition

This paper generalizes previous existence results on unidimensional electoral competition, by extending the traditional two-party electoral game to the case where parties have mixed motivations, in the sense that they are interested in winning the election, but also in the policy implemented after the contest. Although this game has discontinuous payoffs, it satisfies payoff security and reciprocally upper semi- continuity. However, conditional payoffs might violate quasi-concavity. Hence, our first result shows that the existence of a pure-strategy Nash equilibrium can be guaranteed only if parties' interests are symmetric. Instead, we prove that the mixed extension satisfies better reply security and, therefore, that a mixed-strategy equilibrium always exists. We also characterize the set of equilibria for a tractable version of the model. This shows that the interaction between the electoral uncertainty, the aggregate level of opportunism and its distribution among parties shape the equilibrium strategies. In particular, when the opportunism is large and asymmetrically distributed, the support of each mixed-strategy equilibrium is a closed interval located on one side of the median. Further, as the uncertainty increases, the probability distributions concentrate on the extremes of the support. And the mixed-strategy equilibrium vanishes above a critical level, over which each party plays a pure strategy in its own ideological side.