Shino Takayama, Yuki Tamura, School of Economics Discussion Paper No. 546 May 2015, School of Economics, The University of Queensland.

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Since the introduction of better-reply security by Reny (1999), the literature studying the existence of a pure strategy Nash equilibrium (PSNE) in discontinuous games has grown substantially. In this paper, we introduce a weak notion of better-reply security, which is applicable to both quasiconcave and nonquasiconcave games. Our conditions for feeble better-reply security are simple, easy to verify and particularly useful in electoral competition games. We also provide necessary and sufficient conditions for the existence of a PSNE in a canonical electoral competition game. Finally, this paper demonstrates why a PSNE fails to exist when a particular type of discontinuity exists in a model.