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.^[1] He considered the case where the aggregate is the sum of the players' strategies. The term "aggregative game" was introduced by Cochron in 1994.^[2]

Definition

Consider a standard non-cooperative game with n players, where ${\displaystyle S_{i}\subseteq \mathbb {R} }$ is the strategy set of player i, ${\displaystyle S=S_{1}\times S_{2}\times \ldots \times S_{n}}$ is the set of all strategy profiles, and ${\displaystyle u_{i}:S\to \mathbb {R} }$ is the payoff function of player i. The game is then called an aggregative game if for each player i there exists a function ${\displaystyle {\tilde {u}}_{i}:S_{i}\times \mathbb {R} \to \mathbb {R} }$ such that for all ${\displaystyle s\in S}$:

${\displaystyle u_{i}(s)={\tilde {u}}_{i}\left(s_{i},\sum _{j=1}^{n}s_{j}\right)}$

In words, payoff functions in aggregative games depend on players' own strategies and the aggregate ${\displaystyle \sum s_{j}}$.

Example

Consider the Cournot competition, where firm i has payoff/profit function ${\displaystyle u_{i}(s)=s_{i}P\left(\sum s_{j}\right)-C_{i}(s_{i})}$ , where:

• ${\displaystyle P}$ is the inverse demand function - it maps the total supplied amount to the price of the product;
• ${\displaystyle C_{i}}$ is the cost function of firm i.

This is an aggregative game since ${\displaystyle u_{i}(s)={\tilde {u}}_{i}\left(s_{i},\sum s_{j}\right)}$ where ${\displaystyle {\tilde {u}}_{i}(s_{i},X)=s_{i}P(X)-C_{i}(s_{i})}$.

Some other natural examples of aggregative games are:

• Public goods game - the utility of each player depends only on his own contribution to the public good, and on the total amount contributed by all players.^{[clarification needed]}
• Pollution game^[3] - the utility of each player depends on its own pollution level and on the total pollution level.^{[clarification needed]}
• Congestion game - the utility of each player depends on his own choice, and the aggregate congestion caused by others.^{[clarification needed]}
• See also Bertrand competition for a different competition model.^{[clarification needed]}

Generalizations

A number of generalizations of the standard definition of an aggregative game have appeared in the literature.

Generalized aggregative game

Comes and Harley (2012)^[4] define a game as generalized aggregative if there exists an additively separable function ${\displaystyle g:S\to \mathbb {R} }$ (i.e., if there exist increasing functions ${\displaystyle h_{0},h_{1},\ldots ,h_{n}:\mathbb {R} \to \mathbb {R} }$ such that ${\displaystyle g(s)=h_{0}(\sum _{i}h_{i}(s_{i}))}$) such that for each player i there exists a function ${\displaystyle {\tilde {u}}_{i}:S_{i}\times \mathbb {R} \to \mathbb {R} }$ such that ${\displaystyle u_{i}(s)={\tilde {u}}_{i}(s_{i},g(s_{1},\ldots ,s_{n}))}$ for all ${\displaystyle s\in S}$. Obviously, any aggregative game is generalized aggregative as seen by taking ${\displaystyle g(s_{1},\ldots ,s_{n})=\sum s_{i}}$.

Quasi-aggregative game

Jensen (2010)^[5] defined an even more general definition of quasi-aggregative games, where payoff functions of different players are allowed to depend on different functions of opponents' strategies.

Jensen (2018)^[6] introduced an even more general definition, but still for one-dimensional aggregators.

Jensen (2005)^[7] introduced a definition that allows the output of the aggregation to be multi-dimensional.

Infinitely many players

Acemoglu and Jensen (2010)^[8] generalized aggregative games 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.

Properties

• Generalized aggregative games (hence aggregative games) admit backward reply correspondences and in fact, is the most general class to do so.^[4] Backward reply correspondences, as well as the closely related share correspondences, are powerful analytical tools in game theory. For example, backward reply correspondences were used to give the first general proof of the existence of a Nash equilibrium in the Cournot model without assuming quasiconcavity of firms' profit functions.^[9] Backward reply correspondences also play a crucial role for comparative statics analysis (see below).
• Quasi-aggregative games (hence generalized aggregative games, hence aggregative games) are best-response potential games if best-response correspondences are either increasing or decreasing.^[10][5] Precisely as games with strategic complementarities, such games therefore have a pure strategy Nash equilibrium regardless of whether payoff functions are quasiconcave and/or strategy sets are convex. The existence proof of Novshek for a Cournot equilibrium^[9] is a special case of such more general existence results.
• Aggregative games have strong comparative statics properties. Under very general conditions one can predict how a change in exogenous parameters will affect the Nash equilibria.^[2][11]

Dindos and Mezzetti^[12] study games with quasiconcave utilities that depend only on the player's action and the sum of all actions. They show that the better-reply dynamics converges globally to a Nash equilibrium if actions are either strategic substitutes or strategic complements for all players around each asymptotically-stable equilibrium. In contrast, if the derivatives of the best-reply functions have different signs, then the better-reply dynamics might not converge even with two players.

See also

• Mean field game theory
• Nash equilibrium vs. Wardrop equilibrium in aggregative games.^[13]
• Approximating Nash equilibrium in distributed aggregative games.^[14]