Fully aggregative games

Abstract

We study aggregative games in which players' strategy sets are convex intervals of the real line and (not necessarily differentiable) payoffs depend only on a player's own strategy and the sum of all players' strategies. We give sufficient conditions on each player's payoff function to ensure the existence of a unique Nash equilibrium in pure strategies, emphasizing the geometric nature of these conditions. These conditions are almost best possible in the sense that the requirements on one player can be slightly weakened, but any further weakening may lead to multiple equilibria. The same conditions also permit the analysis of comparative statics and the competitive limit. We discuss the application of these conditions in a range of examples, chosen to illustrate various aspects their use. We also show that all restrictions on payoffs in aggregative games that guarantee the existence of a unique equilibrium of which we are aware are covered by these conditions. When payoffs are sufficiently smooth, these conditions can be tested using derivatives of the marginal payoff and we illustrate these tests in the applications introduced earlier. We also investigate conditions under which the unique equilibrium is locally stable. These hold in particular in a symmetric game under the same conditions required to ensure the existence of a unique equilibrium.

Multi-Leader Multi-Follower Model with Aggregative Uncertainty

“…In order to prove the existence and uniqueness of the follower Nash equilibrium and the existence of a multi-leader multi-follower equilibrium with aggregate uncertainty, we will use the following assumptions, already presented in literature (see [12,28]). …”

“…In this section, based on the results obtained by Acemoglu and Jensen in [12] and by Cornes and Hartley in [28], we give theorems that prove first of all the existence and uniqueness of the followers' Nash equilibrium and then the existence of a multi-leader multi-follower equilibrium with aggregative uncertainty.…”

“…Note that (i) corresponds to the law of diminishing marginal utility, while (ii) is assumed in order to obtain that the share functions are strictly decreasing and so to ensure the uniqueness of the followers' Nash equilibrium (see [28]). …”

Forecast Dispersion in Finite-Player Forecasting Games

“…4 See Ben McClure, "Earnings Forecasts: A Primer," accessed August 22, 2017, http://www.investopedia.com/articles/stocks/06/earningsforecasts.asp. 5 Aggregate games with additively separable aggregators, such as the mean, are studied in Acemoglu and Jensen (2013) and Cornes and Hartley (2012). The general definition of aggregate games with a linear aggregate is treated in Martimort and Stole (2012).…”

Evolution of Consistent Conjectures in Semi-aggregative Representation of Games, with Applications to Public Good Games and Contests

“…An early example of a work on aggregative games is Corchón (1994), but in a more recent series of works Richard Cornes and Roger Hartley elucidated the usefulness of studying the mathematical structure of these games for establishing equilibrium existence and for finding equilibria in situations going beyond textbook symmetric examples. They applied this methodology to such classic examples of economic analysis as public good games (Cornes and Hartley, 2007) and contests Hartley, 2003, 2005), as well as listing further examples (Cornes and Hartley, 2011) and studying the structure of aggregate games further (Cornes and Hartley, 2012).…”