We study the existence problem of Nash equilibrium as well as the patterns of equilibrium policy outcomes in an electoral competition model with mixed motivations. Each party maximizes a sum of party members' expected utility and office rent. The inclusion of office rent renders the payoff of each party discontinuous. This makes it difficult to apply usually fixed point arguments to prove the existence of Nash equilibria. By using a recently developed concept, multiple restrictional security (MR-security) we provide conditions under which a pure-strategy Nash equilibrium exists within fairly general settings, and further the analysis by presenting conditions under which various patterns of policy choices, including polarization, arise in equilibrium.

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For object reallocation problems, if preferences are strict but otherwise unrestricted, the Top Trading Cycle rule (TTC) is the leading rule: It is the only rule satisfying efficiency, the endowment lower bound, and strategy-proofness; moreover, TTC coincides with the core. However, on the subdomain of single-peaked preferences, Bade (2019a) defines a new rule, the "crawler", which also satisfies the first three properties. Our first theorem states that the crawler and a naturally defined "dual" rule are actually the same.Next, for object allocation problems, we define a probabilistic version of the crawler by choosing an endowment profile at random according to a uniform distribution, and applying the original definition. Our second theorem states that this rule is the same as the "random priority rule" which, as proved by Knuth (1996) and Abdulkadiroglu and Sönmez (1998), is equivalent to the "core from random endowments".Keywords: object reallocation problems, single-peaked preferences, the crawler, the random priority rule, the core.JEL classification: C78, D47. * We are grateful to William Thomson for his invaluable advice and support, and for his patient reviews and detailed suggestions. We thank Maciej Kotowski for his comments. We also thank seminar participants at the 2019 Ottawa Microeconomic Theory Workshop and University of Rochester. All remaining errors are our own.