Conjectural Variations in Aggregative Games: An Evolutionary Perspective

Abstract

Suppose that in aggregative games, in which a player's payoff depends only on this player's strategy and on an aggregate of all players' strategies, the players are endowed with constant conjectures about the reaction of the aggregate to marginal changes in the player's strategy. The players play the equilibrium determined by their conjectures and equilibrium strategies determine the players' payoffs, which can be different for players with different conjectures. It is shown that with random matching in an infinite population, only consistent conjectures can be evolutionarily stable, where a conjecture is consistent if it is equal to the marginal change in the aggregate at equilibrium, determined by the players' actual best responses. In the finite population case in which relative payoffs matter, only zero conjectures representing aggregate-taking behavior can be evolutionarily stable. The results are illustrated with the examples of a linear-quadratic game (that includes a Cournot oligopoly) and a rent-seeking game.