Conjectural variations in aggregative games: An evolutionary perspective

Abstract

Suppose that in symmetric aggregative games, in which payoffs depend only on a player's strategy and on an aggregate of all players' strategies, players have conjectures about the reaction of the aggregate to marginal changes in their strategy. The players play a conjectural variation equilibrium, which determines their fitness payoffs. The paper shows that only consistent conjectures can be evolutionarily stable in an infinite population, where a conjecture is consistent if it is equal to the marginal change in the aggregate determined by the actual best responses. In the finite population case, only zero conjectures representing aggregate-taking behavior can be evolutionarily stable.

Evolution of Consistent Conjectures in Semi-aggregative Representation of Games, with Applications to Public Good Games and Contests

“…Previous works noted this connection in simple duopoly models (Dixon and Somma, 2003;Müller and Normann, 2005), in two-player games (Possajennikov, 2009) and in symmetric aggregative games (Possajennikov, 2015). What I add in this paper is that the link between consistent and evolutionarily stable conjectures hold in more general n-player asymmetric situations.…”

Indirect Evolution and Aggregate-Taking Behavior in a Football League: Utility Maximization, Profit Maximization, and Success

“…However, aggregate-taking behavior may make the crucial difference here, as the interdependent behavioral effects between the players are much more restricted by aggregate-taking behavior. In accordance with this reasoning, Possajennikov [39] demonstrated that only so-called zero conjectures are evolutionarily stable; the latter means that players do not believe that the other players react to their actions.…”

Expectational stability in aggregative games

“…Possajennikov (2003) defines the aggregate-taking equilibrium of the game, where individual strategies do not affect the aggregate contribution, and proves its stability for games with a finite number of players. In a later work, Possajennikov (2015) considers players' beliefs about the marginal individual contribution to the aggregate strategy to define the dynamics of the game. He proves that only in games with an infinite number of players, consistent conjectures with respect to the NE are evolutionary stable.…”