|
Article Excerpt
Summary. In a (generalized) symmetric aggregative game, payoffs depend only on individual strategy and an aggregate of all strategies. Players behaving as if they were negligible would optimize taking the aggregate as given. We provide evolutionary and dynamic foundations for such behavior when the game satisfies supermodularity conditions. The results obtained are also useful to characterize evolutionarily stable strategies in a finite population.
Keywords and Phrases: Aggregative games, Evolutionarily stable strategy, Price-taking behavior, Stochastic stability, Supermodularity.
JEL Classification Numbers: C72, D41, D43.
1 Introduction
In perfectly competitive markets, price-taking behavior is often justified by assuming that agents are small relative to market size. The implication of this assumption is that prices are almost insensitive to individual actions. Hence, even if agents behave strategically, equilibrium behavior corresponds to price-taking optimization as the economy becomes large. The crucial axioms underlying this non-cooperative foundation of competitive equilibrium are anonymity--the names of the agents are irrelevant to the market--and aggregation--individual actions affect market price only through the average of all actions (Dubey et al. [8]).
Following Corchon [4], we say that a game is a (generalized) aggregative game if payoffs depend only on individual strategies and an aggregate of all strategies. (1) A prominent example is a Cournot oligopoly, where profits depend exclusively on individual and total output. If, additionally, payoffs do not depend on the names of the agents, the game is symmetric. Aggregate-taking optimization--the natural generalization of price-taking behavior--is then still well defined even if agents are not negligible, although it does not correspond to strategic, rational behavior. An optimal aggregate-taking strategy (ATS) is one that is individually optimal given the value of the aggregate that results when all players adopt it. In an ATS, players who are not negligible behave as if they were.
Instead of absolute payoffs, evolutionary game theory proposes relative performance as the important criterion for the survival of a strategy. The underlying assumption is that if a strategy earns higher payoffs than opponent strategies, it tends to be copied more frequently and propagates faster at the expense of worse performing strategies. We then say that a strategy is evolutionarily stable (ESS) if, once adopted by all players, it will not be discarded due to the appearance of a small fraction (2) of experimenters choosing a competing different strategy. If an ESS resists the appearance of any fraction of such experimenters, we say that it is globally stable. Evolutionary stability thus implies maximization of the difference between own and opponents' payoffs. (3)
In this context, Schaffer [20] observed that, in a Cournot duopoly, the output corresponding to a competitive equilibrium--the output level that maximizes profits at the market-clearing price--is evolutionarily stable. That is, a firm deviating from the competitive equilibrium will earn lower profits than its competitor after deviation. (4) This result was extended to a general oligopoly by Vega-Redondo [26], who additionally showed that the competitive equilibrium would be the only long-run outcome of a learning dynamics based on imitative behavior. The evolutionary approach, hence, provides foundations for competitive equilibrium dispensing with the assumption of negligible agents.
In the present work, we identify the structural characteristics of the Cournot oligopoly which underlie these results. The first is the fact that it is an aggregative game. The second is the strategic substitutability between individual and total output. Since the incentive to increase individual output decreases the higher the total output in the market, the Cournot oligopoly has a submodular structure. (5)
Indeed, we find that the results for the Cournot oligopoly are but an instance of a general phenomenon. An ATS is evolutionarily stable in any aggregative game with a submodular structure. This has a natural counterpart in the supermodular case, where any ESS corresponds to aggregate-taking optimization.
Possajennikov [18] already observed a relation between optimal aggregate-taking strategies and evolutionarily stable strategies in aggregative games. Under differentiability, he finds that the first-order conditions of their defining optimization problems are identical. Careful examination of the second-order conditions allows to determine conditions under which both concepts coincide. In contrast, our approach relies exclusively on the structure of the game and provides an intuitive and direct way of relating both concepts.
In the submodular case, we obtain even stronger results. Any ATS is weakly globally stable, i. e. weakly better in relative terms independently of the fraction of opponents behaving differently. If the game has a strict ATS, then this is strictly globally stable and the unique ESS.
Furthermore, we show that a strictly globally stable ESS is always the long-run outcome of a learning dynamics based on imitation and experimentation. This result, which is of independent interest, is proven for arbitrary (not necessarily aggregative) symmetric games. As a corollary, this will also hold for any strict ATS of a submodular aggregative game. In short, the dynamic stability result of price-taking behavior quoted above generalizes for aggregate-taking optimization to arbitrary submodular aggregative games.
In our view, these results might be taken to provide an alternative, evolutionary foundation for the perfect competition paradigm. In contrast to the large-population approach, this foundation does not rely on agents being negligible. In fact, the evolutionary success of behaving as if they were negligible is due precisely to the fact that they are not. When an agent optimizes assuming that she will not affect the aggregate, the latter will actually change, but in such a way that it is her opponents who will be more harmed. A key new insight is that this property derives directly from the supermodular or submodular structure of the game.
These results are also of interest for evolutionary game theory, since they provide either necessary or sufficient conditions to obtain ESS for a class of aggregative games. In the submodular case, we actually provide shortcuts for the computation of an ESS and the long-run outcomes of imitative learning dynamics. Further, our result on imitative dynamics is, to our knowledge, the first general result on the dynamic properties of finite-population ESS.
The paper is organized as follows. Section 2 introduces the notion of (generalized) aggregative games and presents examples beyond the Cournot oligopoly. Section 3 presents the concepts of evolutionary and global stability for n-player games and particularizes them for aggregative games. Section 4 discusses aggregate-taking behavior. Section 5 presents the results relating aggregate-taking behavior and evolutionary stability. Section 6 contains the dynamic results. Section 7 concludes.
2 Generalized symmetric aggregative games
A game is called aggregative if the payoffs to any player depend only on that player's strategy and the sum of all strategies chosen. If the sum is replaced by an arbitrary aggregate g, we refer to a generalized aggregative game (Corchon [4]).
In the present work we will consider symmetric games with a strategy space S common to all players, assumed to be a subset of a totally ordered space X. For our purposes it will be enough to let S [??] X = R. Further we will assume the aggregate g to be a symmetric and monotone increasing function. (6) For the sake of expositional simplicity we will drop the qualifiers generalized, symmetric, and monotone, referring to such games simply as aggregative games.
Definition 1 A (generalized) symmetric aggregative game with aggregate g is a tuple [GAMMA] [equivalent to] (N, S, [pi]) where N is the number of players, the strategy set S, common to all players, is a subset of a totally ordered space X, [pi] : S x X [right arrow] R is a real-valued function, and g : [S.sup.N] [right arrow] X is a symmetric and monotone increasing function, such that individual payoff functions are given by [[pi].sub.i](s) [equivalent to] [pi]([s.sub.i], g(s)) for all s = ([s.sub.1],..., [s.sub.N]) [member of] [S.sup.N] and i = 1,..., N.
2.1 Families of aggregative games
Existence of a monotone aggregate function is the only requirement for a game to be representable as an aggregative game. Hence, this class of games may be rather large. Actually, in the examples we consider the aggregate is a functional form that can be extended to...
|
