Aggregative games and best-reply potentials

Abstract

This paper introduces quasi-aggregative games and establishes conditions under which such games admit a best-reply potential. This implies existence of a pure strategy Nash equilibrium without any convexity or quasi-concavity assumptions. It also implies convergence of best-reply dynamics under some additional assumptions. Most of the existing literature's aggregation concepts are special cases of quasi-aggregative games, and many new situations are allowed for. An example is payoff functions that depend on own strategies as well as a linear combination of the mean and the variance of players' strategies.

Cournot Tatonnement in Aggregative Games with Monotone Best Responses

“…Thus, if we assumed that each R i is upper hemicontinuous, the difference between our Theorem 1 and the main result of Jensen (2010) would be quite minor. However, we do not impose that assumption.…”

“…Our conditions (6), (7), and (9) are exactly the same as in Jensen (2010). There was no need for explicit conditions like (4) there since all strategy sets were assumed to be subsets of R m .…”

“…We follow the same logic as in Kukushkin (2005) and Jensen (2010): each R i may be perceived either as the total best response correspondence or as an increasing selection from it.…”

“…Dubey et al (2006), having modified a trick invented by Huang (2002) for different purposes, developed a tool applicable to a broader class of aggregation rules. Kukushkin (2005) and, especially, Jensen (2010) extended its sphere of applicability much further. It looks plausible that the latter paper describes the most general class of aggregation rules for which this approach can still work.…”

“…The point is that the best response correspondences in the main results of Jensen (2010) were assumed upper hemicontinuous. Although one can plausibly argue that the upper hemiconinuity of the best responses holds in "most" of important economic models, "most" cannot be replaced with "all."…”