Correlated Equilibria and Communication in Games

Abstract

GlossaryBayesian game: an interactive decision problem consisting of a set of n players, a set of types for every player, a probability distribution which accounts for the players' beliefs over each others' types, a set of actions for every player and a von Neumann-Morgenstern utility function de…ned over n-tuples of types and actions for every player.Nash equilibrium: in an n-person strategic form game, a strategy n-tuple from which unilateral deviations are not pro…table.von Neumann-Morgenstern utility function: a utility function which re ‡ects the individual's preferences over lotteries. Such a utility function is de…ned over outcomes and can be extended to any lottery by taking expectation with respect to . 1 Pure strategy (or simply strategy): a mapping which, in an interactive decision problem, associates an action with the information of a player whenever this player can make a choice.Sequential equilibrium: a re…nement of the Nash equilibrium for n-person multistage interactive decision problems, which can be loosely de…ned as a strategy n-tuple together with beliefs over past information for every player, such that every player maximizes his expected utility given his beliefs and the others' strategies, with the additional condition that the beliefs satisfy (possibly sophisticated) Bayes updating given the strategies.Strategic (or normal) form game: an interactive decision problem consisting of a set of n players, a set of strategies for every player and a (typically, von Neumann-Morgenstern) utility function de…ned over n-tuples of strategies for every player.Utility function: a real valued mapping over a set of outcomes which re‡ects the preferences of an individual by associating a utility level ( a "payo¤") with every outcome.

Correlated Equilibria in Wireless Power Control Games

“…Algorithm 2 Algorithm leading to the optimal communication equilibrium in power control game 1: The mediator simulates the sequence of reports that could receive from the users, that correspond to channel gain profiles, and the power profiles that could be received by the users given a channel gain profile. 2: Using a method to solve the linear program constituted by the objective function (8) and the constraints (9)- (11), to find the optimal probability distributions p(.|t) for all type profile t. 3: For each player i, the Nature randomly chooses type, that corresponds to the channel gain profile (g ii , . .…”

Exchangeable Equilibria, Part I: Symmetric Bimatrix Games

“…These are exactly the outcomes which could conceivably arise when rational players make independent environmental measurements, then interpret and act on this information in the same way. This is a version of the "revelation principle" [15] for exchangeable equilibria (compare Proposition 4.6 with Proposition 4.1). Exchangeable equilibria can achieve higher payoffs and social welfare than symmetric Nash equilibria (Example 5.4).…”

Bounding equilibrium payoffs in repeated games with private monitoring

“…The proof is long and is deferred to the Supplement. The main idea is as in the literature on implementing correlated equilibria without a mediator (see Forges (2009) for a survey). More specifically, Proposition 2 is similar to Theorem 9 of Heller et al (2012), which shows that communication equilibria in repeated games with perfect monitoring can always be implemented by ex ante correlation and cheap talk.…”