博弈论Chapter5解析.ppt

Consistent beliefs imply that P2’s expected payoff is 0.5(5+5)=5 from choosing up, but 0.5(2+10)=6 from down. P2 chooses down. Thus, a strategy profile including up cannot be part of a weak sequential equilibrium The assessment consisting of the behavioral strategy profile (right down) and the belief system ((o.5, o.5),(o.5,o.5)) is a weak sequential equilibrium For P1 to choose right is optimal, given P2 chooses down Example But, what about beliefs for information sets that are off the equilibrium path ? We want beliefs for information sets that are off the equilibrium path to be reasonable. But what is reasonable? Consider the NE (L, r) again. Player 2s information set will not be reached at the equilibrium, because player 1 will play L with probability 1. But assume that player 1 plays a completely mixed strategy, playing L, M, and R with probabilities 1-ε, 0.75 ε , and 0.25 ε. Then, the belief on player 2s information set is well defined. Now, if ε→0, its still well defined. NE requires two things. (1) best response to beliefs and (2) correct beliefs, in eq. In extensive form games: SPE requires credible actions and beliefs. We can extend these ideas naturally to games in general: Bayes NE :NE with payoff replaced by expected payoff. Perfect Bayes NE (PBNE): Requires actions and beliefs to be consistent at all subgames (subgame starts at a node, not an info set). Find Bayes Perfect NE Challenger can enter prepared for fight or unprepared. Incumbent only knows whether Challenger enters or not, but not whether he is prepared for fight. Harsanyi’s Transformation Harsanyi (1967/68): transform incomplete info into imperfect info What should the potential entrant do? Trick for solving this kind of game: Harsanyi transformation. Add a third player, “nature,” who moves first and chooses the incumbent’s type. The trick transforms a game of incomplete information into a game of imperfect information (which we already know how to solve). Entrant’s expected utility fr