Strategic complements and substitutes, and potential games

Aggregative games and best-reply potentials

“…The results can be seen as a direct continuation of Dubey et al (2006), who define pseudo-potential games and show that any game with linear aggregation and an increasing or decreasing continuous best-reply selection, belongs to this class. The authors also extend this observation to aggregation rules which are only required to be symmetric and linear in any single player's strategy.…”

“…1 This paper's two main objectives are, firstly, to define a new and very general notion of aggregation, called quasi-aggregation, and secondly, to prove that if all best-reply selections are either increasing or decreasing, quasi-aggregative games are best-reply potential games (Voorneveld (2000)). If it is only assumed that some selection has this property, the games are pseudo-potential games (Dubey et al (2006)). A direct consequence of these results is that such games have a pure strategy Nash equilibrium (PSNE) irrespective of whether strategy sets are convex or payoff functions quasi-concave.…”

Cournot tatonnement and potentials

“…Kukushkin (2004) proved the impossibility of Cournot cycles in both Novshek' case and when σ i (x −i ) = j =i x j . Dubey et al (2006) modified a trick developed by Huang (2002) for different purposes, providing a tool for the construction of a continuous partial Cournot potential. A rather broad class of aggregative games where the trick works is described in Jensen (2010); the class may be the broadest possible although it is unclear how such a claim could be proven.…”

Cournot Tatonnement in Aggregative Games with Monotone Best Responses

“…Kukushkin (2004) showed that monotonicity conditions in games with additive aggregation ensure the acyclicity of Cournot tâtonnement rather than the mere existence of an equilibrium. 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.…”