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We study a class of location games where players want to attract as many resources as possible and pay a cost when deviating from an exogenous reference location. This class of games includes political competitions between policy-interested parties and firms' costly horizontal differentiation. We find that the introduction of reference locations simplifies the set of pure-strategy equilibrium to a unique candidate which has a strong property: at most four players, the two most-left and two most-right, deviate from their reference locations. We provide necessary and sufficient conditions for the candidate to be an equilibrium. We illustrate our results in particular cases including the duopoly competition where we moderate the principle of minimal differentiation.
We study approachability theory in the presence of constraints. Given a repeated game with vector payoffs, we study the pairs of sets (A,D) in the payoff space such that Player 1 can guarantee that the long-run average payoff converges to the set A, while the average payoff always remains in D. We provide a full characterization of these pairs when D is convex and open, and a sufficient condition when D is not convex.
A pure Hotelling game is a spatial competition between a finite number of players who simultaneously select a location in order to attract as many consumers as possible. In this paper, we study the case of a general distribution of consumers on a network generated by a metric graph. Because players do not compete on price, the continuum of consumers shop at the closest player’s location. If the number of sellers is large enough, we prove the existence of an approximate equilibrium in pure strategies, and we construct it.
We consider a game where a finite number of retailers choose a location, given that their potential consumers are distributed on a network. Retailers do not compete on price but only on location, therefore each consumer shops at the closest store. We show that when the number of retailers is large enough, the game admits a pure Nash equilibrium and we construct it. We then compare the equilibrium cost borne by the consumers with the cost that could be achieved if the retailers followed the dictate of a benevolent planner. We perform this comparison in terms of the Price of Anarchy (i.e., the ratio of the worst equilibrium cost and the optimal cost) and the Price of Stability (i.e., the ratio of the best equilibrium cost and the optimal cost). We show that, asymptotically in the number of retailers, these ratios are bounded by two and one, respectively.