Aggregative game

In game theory, an aggregative game is a game in which every player’s payoff is a function of the player’s own strategy and the aggregate of all players’ strategies. The concept was first proposed by Nobel laureate Reinhard Selten in 1970 who considered the case where the aggregate is the sum of the players' strategies.

Definition

Consider a standard non-cooperative game with n players, where  S_i \subseteq \mathbb{R} is the strategy set of player i,  S=S_1 \times S_2 \times \ldots \times S_n is the joint strategy set, and f_i:S \to \mathbb{R} is the payoff function of player i. The game is then called an aggregative game if for each player i there exists a function \tilde{f}_i:S_i \times \mathbb{R} \to \mathbb{R} such that for all  s \in S :

 f_i(s)=\tilde{f}_i \left( s_i,\sum_{j=1}^n s_j \right)

In words, payoff functions in aggregative games depend on players' own strategies and the aggregate \sum s_j. As an example, consider the Cournot model where firm i has payoff/profit function f_i(s)=s_i P\left(\sum s_j\right)-C_i(s_i) (here  P and  C_i are, respectively, the inverse demand function and the cost function of firm i). This is an aggregative game since f_i(s)=\tilde{f}_i\left(s_i,\sum s_j\right) where \tilde{f}_i(s_i,X)=s_i P(X)-C_i(s_i) .

Generalizations

A number of generalizations of the standard definition of an aggregative game have appeared in the literature. A game is generalized aggregative[1] if there exists an additively separable function  g:S \to \mathbb{R} (i.e., if there exist increasing functions  h_0,h_1,\ldots,h_n:\mathbb{R} \to \mathbb{R} such that  g(s)=h_0(\sum_i h_i(s_i)) ) such that for each player i there exists a function \tilde{f}_i:S_i \times \mathbb{R} \to \mathbb{R} such that  f_i(s)=\tilde{f}_i(s_i,g(s_1,\ldots,s_n)) for all  s \in S . Obviously, any aggregative game is generalized aggregative as seen by taking  g(s_1,\ldots,s_n)=\sum g_i . A more general definition still is that of quasi-aggregative games where agents' payoff functions are allowed to depend on different functions of opponents' strategies.[2] Aggregative games can also be generalized to allow for infinitely many players in which case the aggregator will typically be an integral rather than a linear sum.[3] Aggregative games with a continuum of players are frequently studied in mean field game theory.

Properties

See also

Notes

  1. 1 2 Cornes, R. and Harley, R. (2012). "Fully Aggregative Games". Economic Letters 116. pp. 631–633.
  2. 1 2 Jensen, M.K. (2010). "Aggregative Games and Best-Reply Potentials". Economic Theory 43. pp. 45–66.
  3. Acemoglu, D. and Jensen, M.K. (2010). "Robust Comparative Statics in Large Static Games". IEEE Proceedings on Decision and Control 49. pp. 3133–3139.
  4. 1 2 Novshek, W. (1985). "On the Existence of Cournot Equilibrium". Review of Economic Studies 52. pp. 86–98.
  5. Dubey, P., Haimanko, O., Zapechelnyuk, A. (2006). "Strategic Complements and Substitutes, and Potential Games". Games and Economic Behavior 54. pp. 77–94.
  6. Corchon, L. (1994). "Comparative Statics for Aggregative Games. The Strong Concavity Case". Mathematical Social Sciences 28. pp. 151–165.
  7. Acemoglu, D. and Jensen, M.K. (2013). "Aggregate Comparative Statics". Games and Economic Behavior 81. pp. 27–49.

References

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