THE GEOMETRY OF AGGREGATIVE GAMES

We study aggregative games in which players’ strategy sets areconvex intervals of the real line and (not necessarily differentiable)payoffs depend only on a player’s own strategy and the sum of allplayers’ strategies. We give sufficient conditions on each player’s payofffunction to ensure the existence of a unique Nash equilibrium inpure strategies, emphasizing the geometric nature of these conditions.These conditions are almost best possible in the sense that the requirementson one player can be slightly weakened, but any furtherweakening may lead to multiple equilibria. The same conditions alsopermit 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 allrestrictions on payoffs in aggregative games that guarantee the existenceof a unique equilibrium of which we are aware are covered bythese conditions. When payoffs are sufficiently smooth, these conditionscan be tested using derivatives of the marginal payoff and weillustrate these tests in the applications introduced earlier. We alsoinvestigate conditions under which the unique equilibrium is locallystable...