Aggregative game

In game theory, an aggregative game (AG) 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 Corchon in 1994.^[2]

Consider a standard non-cooperative game with n players, where the set of strategies (actions) available to each player i is denoted by S_i. The set of all possible strategy profiles is denoted by ${\displaystyle S=S_{1}\times S_{2}\times \ldots \times S_{n}}$. Each player i has a payoff function and ${\displaystyle u_{i}:S\to \mathbb {R} }$.

In the simplest setting studied in the context of aggregative games, S_i is assumed to be one-dimensional, ${\displaystyle S_{i}\subseteq \mathbb {R} }$. That is, the game amounts to each player choosing a number simultaneously with the others. The game is 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 ${\displaystyle \sum s_{j}}$.

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.
• Pollution game^[3] - the utility of each player depends on its own pollution level and on the total pollution level.
• Congestion game - the utility of each player depends on his own choice, and the aggregate congestion caused by others.
• See also Bertrand competition for a different competition model.

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

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 ${\displaystyle g(s_{1},\ldots ,s_{n})=\sum s_{i}}$.

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.

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.

• Generalized AGs (hence AGs) 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-AGs (hence generalized AGs, hence AGs) 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.
• AGs 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]

AGs were first studied under the assumption that the players' strategy sets are one-dimensional (each player chooses a number).

Dubey, Mas-Collel and Shubik (1980)^[12] presented the "Aggregation Axiom": each player's utility depends on the player's action and the sum of all players' actions.

Corchon (1994)^[2] introduced the term "aggregative game" to mean a game that satisfies this aggregation axiom. He studied one-dimensional AGs in which the best-response function of each player is decreasing in the sum of actions --- a situation called strategic substitutes. Moreover, he required a stronger condition: the utility function of each player should be continuously differentiable, strictly decreasing in the sum of players' actions, and strictly concave in both variables. Under these assumptions, he proved some comparative statics results, such as, when new players enter, the strategy of each player decreases, the sum of strategies increases, and the payoff of each player decreases. He proved that strategic substitution alone, without the strong concavity assumption, is insufficient.

Kukushkin (1994),^[13] refining a previous result by Novshek (1985),^[14] studied one-dimensional AGs in which the best-response correspondence of each player has a selection that is a decreasing function (a condition called weak strategic substitute). He proved that such games always have a PNE. The proof proceeds in three propositions: (1) for a special case that the strategies of each player are integers in the interval [0,m]; (2) for a more general case that the strategies are arbitrary integers; (3) for the most general case that the strategies are real numbers. He claims that Proposition 1 can be easily adapted to multi-dimensional strategy sets.

Dubey, Haimanko and Zapecheinyuk (2006)^[10] studied one-dimensional AGs with weak strategic substitutes or weak strategic complements. They proved that all these games are pseudo-potential games, and therefore have a PNE. When the strategy sets are finite, and the utilities are generic, the best response is unique. Hence, a pseudo-potential is a best-response potential; hence, every best-response sequence converges to a PNE in finite time.^{[10]: Sec.4} They show that their results can be extended to more general aggregation functions: not only sums, but also weighted sums of products.^{[10]: Sec.5} They also extend their results to some classes of discontinuous best-reply functions, which include some aggregative games with indivisibilities.^{[10]: Sec.6}

Dindos and Mezzetti (2006)^[15] studied one-dimensional AGs with quasiconcave utilities. They showed 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. They assumed that the player who deviates, as well as the improving strategy he deviates too, are determined randomly.

Cornes and Hartley (2012)^[4] generalized one-dimensional AGs by allowing the utility of each player to depend on his own action and on an arbitrary aggregation function of the others' actions (same aggregation function to all players). In addition, they assumed that the marginal utility of each player (that is, the derivative of ui w.r.t. the strategy of i) also depends on his own action and on the same aggregation function. They call such games fully aggregative. They show that every such game with three or more players is equivalent to an aggregative game in which the aggregator is a simple sum.

Jensen (2018)^[6] studied a more general notion of aggregation, but with one-dimensional strategy sets.

Jensen (2005)^[16] studied AGs with multi-dimensional strategy sets, that is, the strategy set of each agent i is a d-dimensional cube in R^d, for some d >= 1. He required the aggregator to be "separable" in the sense that each coordinate of the aggregator depends on different coordinates of strategy profile; in particular, this implies that the aggregator has at most N coordinates. He proved that, if such a game has strategic substitutes and satisfies some other technical conditions, then it has a PNE. He also analyzes the structure of the set of PNE, their comparative statics, and sufficient conditions for uniqueness and stability.

Jensen (2010)^[5] also studied AGs with multi-dimensional strategy sets. He extended the definition and named the extended notion "quasi-aggregative game". He proved that, under certain technical assumptions, such games have best-response potential functions, and thus the best-response dynamics converges to a PNE.

Cummings, Kearns, Roth and Wu (2015)^[17] study AGs with multi-dimensional strategy sets --- a general class that generalizes both anonymous games and weighted congestion games. They present an algorithm, based on a weak mediator, that converges in polynomial time to an asymptotic Nash equilibrium, when the population is large. The algorithm for computing the weak mediator runs in time polynomial in the number of players, but exponential in the dimensions of the aggregator.

Grammatico, Parise, Colombino and Lygeros (2016)^[18] present distributed algorithms for computing PNE in AGs.

Parise, Grammatico, Gentile and Lygeros (2020)^[19] study network AGs --- a variant in which the utility of each agent depends on his own strategy and on an aggregate function of his neighbors in the network. The strategy sets are multi-dimensional. They present a class of distributed algorithms that can steer the players' strategies to a PNE, both for network aggregative games and for classic aggregative games (where the utility of each player depends on the sum of all other players' actions). Their algorithms in both cases require only local communication.

• Mean field game theory
• Nash equilibrium vs. Wardrop equilibrium in aggregative games.^[20]
• Approximating Nash equilibrium in distributed aggregative games.^[21]