The Bayesian foundations of solution concepts of games (original) (raw)
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The Bayesian foundations of solution concepts of games* 1
Journal of Economic Theory, 1988
We transform a non co-operati ve 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 pr iors 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 successfulIy eliminate strategies which are not best responses. This paper therefore provides the informational foundations of iteratively unàominated strategies and rationalizable strategic behavior (Bernheim (1984) and Pearce (1984». Moreover, sufficient condi tions are also found for Nash equilibrium behavior. We also provide Aumann's (1985) results on correlated equilibria. *This pape r is a substantially expanded and revised version of "The Bayesian Foundations of Rationalizable Strategic Behavior and Nash Equilibrium Behavior." We wish to acknowledge the tremendous help and useful discussions we have had with Roger Myerson who first brought our attention to the Armbruster and Bege (1979) paper. Discussion with David Hirshleifer, José Alexandre Scheinkman and Hugo Sonnenschein have also helped to clarify our ideas.
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).
Interactive epistemology in games with payoff uncertainty
Research in Economics, 2007
We adopt an interactive epistemology perspective to analyse dynamic games with partially unknown payoff functions. We consider solution procedures that iteratively delete strategies conditional on private information about the state of nature. In particular we focus on a weak and a strong version of the ∆-rationalizability solution concept, where ∆ represents given restrictions on players' beliefs about state of nature and strategies . Rationalizability in infinite, dynamic games of incomplete information. Research in Economics 57, 1-38; Battigalli, P., Siniscalchi, M., 2003. Rationalization and incomplete information. Advances in Theoretical Economics 3 (Article 3). http://www.bepress.com/bejte/advances/vol3/iss1/art3\]. We first show that weak ∆-rationalizability is characterized by initial common certainty of rationality and of the restrictions ∆, whereas strong ∆-rationalizability is characterized by common strong belief in rationality and the restrictions ∆ (cf. . Strong belief and forward induction reasoning. Journal of Economic Theory 106, 356-391]). The latter result allows us to obtain an epistemic characterization of the iterated intuitive criterion. Then we use the framework to analyse the robustness of complete-information rationalizability solution concepts to the introduction of "slight" uncertainty about payoffs. If the set of conceivable payoff functions is sufficiently large, the set of strongly rationalizable strategies with slight payoff uncertainty coincides with the set of complete-information, weakly rationalizable strategies.
Rationalizability in infinite, dynamic games with incomplete information
Research in Economics, 2003
In this paper, we analyze two nested iterative solution procedures for infinite, dynamic games of incomplete information. These procedures do not rely on the specification of a type space à la Harsanyi. Weak rationalizability is characterized by common certainty of rationality at the beginning of the game. Strong rationalizability also incorporates a notion of forward induction. The solutions may take as given some exogenous restrictions on players' conditional beliefs. In dynamic games, strong rationalizability is a refinement of weak rationalizability. Existence, regularity properties, and equivalence with the set of iteratively interim undominated strategies are proved under standard assumptions. The analysis mainly focus on two-player games with observable actions, but we show how to extend it to n-player games with imperfectly observable actions. Finally, we briefly survey some applications of the proposed approach. q
Formulation of Bayesian Analysis for Games With Incomplete Information
International Journal of Game Theory, 1985
Abstract: A formal model is given of Harsanyi's infinite hierarchies of beliefs. It is shown that the model doses with some Bayesian game with incomplete information, and that any such game can be approximated by one with a finite number of states of world.
Bayesian Games with Intentions
Electronic Proceedings in Theoretical Computer Science, 2016
We show that standard Bayesian games cannot represent the full spectrum of belief-dependent preferences. However, by introducing a fundamental distinction between intended and actual strategies, we remove this limitation. We define Bayesian games with intentions, generalizing both Bayesian games and psychological games [5], and prove that Nash equilibria in psychological games correspond to a special class of equilibria as defined in our setting.
A General Approach to Rational Learning in Games
This paper provides a general framework for analysing rational learning in strategic situations in which the players have private priors and private information. The author analyses the behaviour of Bayesian rational players both in a repeated game and in a recurrent game when they are uncertain about opponents' behaviour and the game they are playing. The aim of the paper is to explain how Bayesian rational agents learn by playing and to characterize the outcome of this learning process. By studying the concept of`conjectural equilibrium' and analysing the process of convergence of players' behaviour, the roles played by the notions of merging and of consistency are demonstrated.