This thesis proposes a concept of feebly better-reply security. This concept is developed from an f-equilibrium in Carmona (2011b) and MR security in McLennan, Monteiro and Tourky (2011). Feebly better-reply security is easy to verify and applicable to nonquasiconcave and discontinuous games, which are important in the class of electoral competition models. By using this concept, we provide minimum required conditions for the existence of a pure strategy Nash equilibrium, which we refer to as a pure strategy political equilibrium (PSPE) in this thesis. In particular, we apply our existence theorem and a theorem introduced by McLennan, Monteiro and Tourky (2011) (hereafter, MMT) into two models, Roemer’s model and a model with mixed political motivations. Moreover, our existence theorem is easier to apply in those two models.
Regarding the characteristics of the two electoral competition models, in Roemer’s model, political parties are solely interested in the policy outcome after the election. In the second model, we extend Roemer’s model in the sense that parties are interested in the political power by winning the election as well as the policy outcome. In the two models, the parties’ objective functions are not either continuous or quasi-concave. Hence, in order to prove the existence of a PSPE, the MMT and our theorems are important. In the literature of pure strategy Nash equilibrium existence theorems, the MMT theorem is one of the most general theorems for games with discontinuous and/or non quasi-concave payoff functions, which extends the result of Reny (1999) in a number of ways. Feebly better-reply security generalizes many other equilibrium existence conditions, and is, in fact, a sufficient condition for MR security.