The Geometry of Aggregative Games

We study aggregative games in which players’ strategy sets are convex intervals of the real line and (not necessarily differentiable) payoffs depend only on a player’s own strategy and the sum of all players’ strategies. We give sufficient conditions on each player’s payoff function to ensure the existence of a unique Nash equilibrium in pure strategies, emphasizing the geometric nature of these conditions. These conditions are almost best possible in the sense that the requirements on one player can be slightly weakened, but any further weakening may lead to multiple equilibria. The same conditions also permit the analysis of comparative statics and the competitive limit. We discuss the application of these conditions in a range of examples, chosen to illustrate various aspects their use. We also show that all restrictions on payoffs in aggregative games that guarantee the existence of a unique equilibrium of which we are aware are covered by these conditions. When payoffs are sufficiently smooth, these conditions can be tested using derivatives of the marginal payoff and we illustrate these tests in the applications introduced earlier. We also investigate conditions under which the unique equilibrium is locally stable. These hold in particular in a symmetric game under the same conditions required to ensure the existence of a unique equilibrium.

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