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. 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. Consider a standard non-cooperative game with n players, where S i ⊆ R {displaystyle S_{i}subseteq mathbb {R} } is the strategy set of player i, S = S 1 × S 2 × … × S n {displaystyle S=S_{1} imes S_{2} imes ldots imes S_{n}} is the joint strategy set, and f i : S → R {displaystyle f_{i}:S o 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 f ~ i : S i × R → R {displaystyle { ilde {f}}_{i}:S_{i} imes mathbb {R} o mathbb {R} } such that for all s ∈ S {displaystyle sin S} : In words, payoff functions in aggregative games depend on players' own strategies and the aggregate ∑ s j {displaystyle sum s_{j}} . As an example, consider the Cournot model where firm i has payoff/profit function f i ( s ) = s i P ( ∑ s j ) − C i ( s i ) {displaystyle f_{i}(s)=s_{i}Pleft(sum s_{j} ight)-C_{i}(s_{i})} (here P {displaystyle P} and C i {displaystyle C_{i}} are, respectively, the inverse demand function and the cost function of firm i). This is an aggregative game since f i ( s ) = f ~ i ( s i , ∑ s j ) {displaystyle f_{i}(s)={ ilde {f}}_{i}left(s_{i},sum s_{j} ight)} where f ~ i ( s i , X ) = s i P ( X ) − C i ( s i ) {displaystyle { ilde {f}}_{i}(s_{i},X)=s_{i}P(X)-C_{i}(s_{i})} . A number of generalizations of the standard definition of an aggregative game have appeared in the literature. A game is generalized aggregative if there exists an additively separable function g : S → R {displaystyle g:S o mathbb {R} } (i.e., if there exist increasing functions h 0 , h 1 , … , h n : R → R {displaystyle h_{0},h_{1},ldots ,h_{n}:mathbb {R} o mathbb {R} } such that g ( s ) = h 0 ( ∑ i h i ( s i ) ) {displaystyle g(s)=h_{0}(sum _{i}h_{i}(s_{i}))} ) such that for each player i there exists a function f ~ i : S i × R → R {displaystyle { ilde {f}}_{i}:S_{i} imes mathbb {R} o mathbb {R} } such that f i ( s ) = f ~ i ( s i , g ( s 1 , … , s n ) ) {displaystyle f_{i}(s)={ ilde {f}}_{i}(s_{i},g(s_{1},ldots ,s_{n}))} for all s ∈ S {displaystyle sin S} . Obviously, any aggregative game is generalized aggregative as seen by taking g ( s 1 , … , s n ) = ∑ s i {displaystyle g(s_{1},ldots ,s_{n})=sum s_{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. 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. Aggregative games with a continuum of players are frequently studied in mean field game theory.