Generalizing Location Games to a Graph

Abstract

We study two-firm location games on graphs. Earlier work analyzes twofirm location games on a line or a circle, and all examples given possess pure Nash equilibria. We produce an example of a graph with no pure Nash equilibria and also a general class of graphs that do possess pure Nash equilibria. CONSIDER a small region with a population distributed uniformly along a network of roads. Two competing firms locate in the region. Each inhabitant of the region is a consumer who patronizes the firm nearest his home, where the distance from consumer to firm is the length of the shortest drive betwen them. There may be stretches of road along which each consumer is equidistant from the two firms. Such a stretch is divided into two stretches of equal length, and one of these is assigned to each firm. A firm's payoff is the sum of the lengths of the stretches of road lined by its patrons. The above is an example of a two-firm location game (Hotelling [1929]) on a graph. We are interested in finding pure Nash equilibria for such a game. The original statement of the two firm location game was Hotelling's model of two firms that locate on a line segment on which customers are uniformly distributed. The unique Nash equilibrium is paired location in the center. This result has been dubbed the Principle of Minimum Differentiation, and the Principle has been used to explain spatial (location) and nonspatial (for example, product characteristics) similarities between firms and the products they produce. Researchers have generalized the original statement of the location game by altering the basic assumptions of Hotelling's model. The problem has been investigated (for two or more firms, pure and mixed strategies) in the special cases where the graph is a line interval or circle (Eaton and Lipsey [1975], Shaked [1982]). There have also been studies involving geometric objects other than finite graphs (Dasgupta and Maskin [1986] and Simon [1987]). These various location games possess a variety of configurations of Nash equilibrium. For example, in the case of a two firm game on a circle, every strategy pair is a Nash equilibrium. Hence, the well ingrained Principle of Minimum Differentiation need not hold when variations are made in the original statement of the problem.