So far we assumed that all players knew all the relevant details in a game. (SUB-GAME PERFECT BAYESIAN EQUILIBRIUM) 1. I want to determine all pure-strategy Perfect Bayesian Equilibria for this task, but I cannot get very far. Nash equilibrium over and above rationalizable: correctness of beliefs about opponents’ choices. A Bayesian Nash equilibrium can be regarded as a Nash Equilibrium of some appropriately de ned strategic game. In these types of games, players do not know the state of nature (but know the set of possible states of nature). Side note: First number is payoff for A, second number payoff for player B. Game Theory: Lecture 18 Perfect Bayesian Equilibria Dynamic Games of Incomplete Information Deﬁnition A dynamic game of incomplete information consists of A set of … Perfect Bayesian equilibrium requires players to have beliefs that are consistent with the equilibrium strategies of other players. These requirements eliminate the bad subgame-perfect equilibria by requiring players to have beliefs, at each information set, about which node of the information set she has reached, conditional on being informed she is in that information set. It is based on a new consistency condition for the players’ beliefs, called plain consistency, that requires proper conditional-probability updating on inde- pendent dimensions of the strategy space. 1 Perfect Bayesian Equilibrium (PBE) The –nal type of game that we will discuss is one that is dynamic (or sequential) and where players have imperfect information. Section 4.3. Perfect Bayesian equilibrium is not a subset of Nash equilibrium. Game Theory: Lecture 18 Perfect Bayesian Equilibria Example Figure: Selten’s Horse 16 1 2 3 1, 1, 1 C D d c L R L 3, 3, 2 0, 0, 0 4, 4, 0 0, 0, 1 R Image by MIT OpenCourseWare. The Bayesian approach is most useful in dynamic games (Perfect Bayesian Equilibrium). The concept of perfect Bayesian equilibrium for extensive-form games is defined by four Bayes Requirements. Nash equilibrium is deﬁned as the set of actions chosen by the players in such way that none of them can increase its own proﬁt by individually changing its actions, thus providing most likely outcomes for the game. Sequential equilibrium are often defined as satisfying two conditions: consistency and sequential rationality. de nition in O&R). This is in reference to the Game theoretic concepts as Nash equilibrium refinements. In this simple game, both players can choose strategy A, to receive $1, or strategy B, to lose $1. • For each decision, they know exactly where they are in the tree. after histories that occur with probability zero given the equilibrium strategies. Then a mixed strategy Bayesian Nash equilibrium exists. Perfect Bayesian (Nash) Equilibria. b) The beliefs are consistent with Bayes™rule, whenever possible. 2. Bayesian-Nash equilibrium. Perfect Bayesian equilibrium Perfect Bayesian equilibrium (PBE) strengthens subgame perfection by requiring two elements: - a complete strategy for each player i (mapping from info. e.g., Bayesian Nash equilibrium [47], perfect equilib-rium [48], and perfect Bayesian equilibrium [49]. From our point of view, this new equilibrium concept provides a minimal requirement that should be imposed on equilibrium concepts that are based on Bayesian rationality. What’s Next Wednesday: A lecture on background knowledge for prediction markets Monday: Start reading research papers and student presentation I Sign up for paper presentations ASAP and no later than Wednesday! 13. • Imperfect information – When making a move, a player may not know all previous actions chosen. 15. The issue in both of the following examples is oﬀthe equilibrium path beliefs, namely I assigning positive probability to E playing a strictly dominated strategy oﬀthe equilibrium path. 2 Perfect Bayesian Equilibrium - De–nition A strategy pro–le for N players (s 1;s 2;:::;s N) and a system of beliefs over the nodes at all infor-mation sets are a PBE if: a) Each player™s strategies specify optimal actions, given the strategies of the other players, and given his beliefs. Bayesian Nash Equilibrium Comments. I One interpretation is to regard each type as a distinct player and regard the game as a strategic game among such P i jT ijplayers (cf. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. Real-World Example of the Nash Equilibrium . Game Theory: Lecture 17 Bayesian Games Existence of Bayesian Nash Equilibria Theorem Consider a ﬁnite incomplete information (Bayesian) game. 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