The Bayesian foundations of solution concepts of games* 1 (original) (raw)
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The Bayesian foundations of solution concepts of games
Journal of Economic Theory, 1988
We transform a noncooperative game into a Bayesian decision problem for each player where the uncertainty faced by a player is the strategy choices of the other players, the priors of other players on the choice of other players, the priors over priors, and so on. We provide a complete characterization between the extent of knowledge about the rationality of players and their ability to successively eliminate strategies which are not best responses. This paper therefore provides the informational foundations of iteratively undominated strategies and rationalizable strategic behavior
Bayesian Theory of Games: A Statistical Decision Theoretic Based Analysis of Strategic Interactions
2010
Bayesian rational prior equilibrium requires agent to make rational statistical predictions and decisions, starting with first order non informative prior and keeps updating with statistical decision theoretic and game theoretic reasoning until a convergence of conjectures is achieved. The main difference between the Bayesian theory of games and the current games theory are: I. It analyzes a larger set of games, including noisy games, games with unstable equilibrium and games with double or multiple sided incomplete information games which are not analyzed or hardly analyzed under the current games theory. II. For the set of games analyzed by the current games theory, it generates far fewer equilibria and normally generates only a unique equilibrium and therefore functions as an equilibrium selection and deletion criterion and, selects the most common sensible and statistically sound equilibrium among equilibria and eliminates insensible and statistically unsound equilibria. III. It...
SSRN Electronic Journal, 2012
This paper introduces a new game theoretic equilibrium which is based upon the Bayesian subjective view of probability, BEIC (Bayesian equilibrium iterative conjectures). It requires players to make predictions, starting from first order uninformative predictive distribution functions (or conjectures) and keep updating with statistical decision theoretic and game theoretic reasoning until a convergence of conjectures is achieved. Information known by the players such as the reaction functions are thereby incorporated into their higher order conjectures and help to determine the convergent conjectures and the equilibrium. In a BEIC, conjectures are consistent with the equilibrium or equilibriums they supported and so rationality is achieved for actions, strategies and conjectures. The BEIC approach is capable of analyzing a larger set of games than current Nash Equilibrium based games theory, including games with inaccurate observations, games with unstable equilibrium and games with double or multiple sided incomplete information games. On the other hand, for the set of games analyzed by the current games theory, it generates far lesser equilibriums and normally generates only a unique equilibrium. It treats games with complete and perfect information as special cases of games with incomplete information and noisy observation whereby the variance of the prior distribution function on type and the variance of the observation noise term tend to zero. Consequently, there is the issue of indeterminacy in statistical inference and decision making in these games as the equilibrium solution depends on which variances tends to zero first. It therefore identifies equilibriums in these games that have so far eluded the classical theory of games. Finally, it also resolves inconsistencies in equilibrium results by different solution concepts in current games theory such as that between Nash Equilibrium and iterative elimination of dominated strategies and that between Perfect Bayesian Equilibrium and backward induction (Subgame Perfect Equilibrium).
Correlated Equilibrium as an Expression of Bayesian Rationality
Econometrica, 1987
CORRELATED EQUILIBRIUM AS AN EXPRESSION OF BAYESIAN RATIONALITY BY ROBERT J. AUMANNI Correlated equilibrium is formulated in a manner that does away with the dichotomy usually perceived between the "Bayesian" and the "game-theoretic" view of the world. From the Bayesian viewpoint, probabilities should be assignable to everything, including the prospect of a player choosing a certain strategy in a certain game. The so-called "game-theoretic" viewpoint holds that probabilities can only be assigned to events not governed by rational decision makers; for the latter, one must substitute an equilibrium (or other game-theoretic) notion. The current formulation synthesizes the two viewpoints: Correlated equilibrium is viewed as the result of Bayesian rationality; the equilibrium condition appears as a simple maximization of utility on the part of each player, given his information. A feature of this approach is that it does not require explicit randomization on the part of the players. Each player always chooses a definite pure strategy, with no attempt to randomize; the probabilistic nature of the strategies reflects the uncertainty of other players about his choice. Examples are given.
Non-Probabilistic Correlated Equilibrium as an Expression of Non-Bayesian Rationality
2013
We answer the question of what players can commonly know about the strategy profile they play when they commonly know that they are rational. We first study four notions of dominance rationality, which we refer to as non-Bayesian because they do not involve probabilistic beliefs of the players. A strategy can be weakly or strictly dominated and dominance can be by either a pure or a mixed strategy. Letting d vary over these four types of dominance, we say that a player is d-dominance rational when she does not play a strategy that is d-dominated relative to what she knows. Analogously to the definition of correlated equilibrium, we define, for each d, a family of sets of strategy profiles, sets that we call d-correlated equilibria. We study the structure of these families and show that players who commonly know that they are d-rational can commonly know that the profile played is in a given set of profiles if and only if this set contains a d-correlated equilibrium. We also show that when Bayesian rationality is commonly known then players can commonly know that the profile played is in a given set if and only if this set contains a d-correlated equilibrium for dominance d defined by strict domination by a mixed strategy.
Behavioral Perfect equilibrium in Bayesian Games 1 1
2013
We develop the notion of perfect Bayesian Nash Equilibrium—perfect BNE—in general Bayesian games. We test perfect BNE against the criteria laid out by Kohlberg and Mertens [15]. We show that, for a focal class of Bayesian games, perfect BNE exists. Moreover, when payoffs are continuous, perfect BNE is limit undominated for almost every type. We illustrate the use of perfect BNE in the context of a second-price auction with interdependent values. Perfect BNE selects the unique pure strategy equilibrium in continuous strategies that separates types. Moreover, when valuations become independent, the equilibrium converges to the classical truthful dominant strategy equilibrium. We also show that less intuitive equilibria in which types are pooled are ruled out by our selection criterion. We further argue that standard selection criteria for second-price auctions have no bite here. Bidders have no dominant strategies, and the separating equilibrium is not sincere. JEL Codes. C72.
Behavioral Perfect Equilibrium in Bayesian Games
We develop the notion of perfect Bayesian Nash Equilibrium-perfect BNE-in general Bayesian games. We test perfect BNE against the criteria laid out by Kohlberg and Mertens [15]. We show that, for a focal class of Bayesian games, perfect BNE exists. Moreover, when payoffs are continuous, perfect BNE is limit undominated for almost every type. We illustrate the use of perfect BNE in the context of a second-price auction with interdependent values. Perfect BNE selects the unique pure strategy equilibrium in continuous strategies that separates types. Moreover, when valuations become independent, the equilibrium converges to the classical truthful dominant strategy equilibrium. We also show that less intuitive equilibria in which types are pooled are ruled out by our selection criterion. We further argue that standard selection criteria for second-price auctions have no bite here. Bidders have no dominant strategies, and the separating equilibrium is not sincere. JEL Codes. C72.
Strategic independence and perfect Bayesian equilibria
Journal of Economic Theory, 1996
This paper evaluates different refinements of subgame perfection, which rely on different restrictions on players' assessments, using a simple and intuitive independence property for conditional probability systems on the space of strategy profiles. This independence property is necessary for full consistency of assessments, and it is equivalent to full consistency in games with observable deviators. Furthermore, while every conditional system on the strategies satisfying the independence property corresponds to a generally reasonable extended assessment as defined by Fudenberg and Tirole [J. Econ. Theory 53 (1991), 236 260], such extended assessments may violate independence, full consistency, and invariance with respect to interchanging of essentially simultaneous moves.
On purification of equilibrium in Bayesian games and expost Nash equilibrium
International Journal of Game Theory, 2008
Treating games of incomplete information, we demonstrate that the existence of an ex post stable strategy vector implies the existence of an approximate Bayesian equilibrium in pure strategies that is also ex post stable. Through examples we demonstrate the 'bounds obtained on the approximation' are tight.