哈佛博弈论课件section.ppt

Section 9: Redux,Alexis Diamond ,Outline,Overview of equilibrium concepts so far Example of perfect Bayesian equilibrium Example of monetary policy (see handout from last section) Theories of delegation Time-consistency,Overview,Nash Equilibrium Mixed-strategy Nash Equilibrium Subgame Perfect Equilibrium Bayesian Equilibrium Perfect Bayesian Equilibrium,Key Terms,Equilibrium path: a node to node path observed in equilibrium A strategy off the equilibrium path is never observed Any branch on which there is positive support (probability) for play forms part of the equilibrium path There may be multiple equilibrium paths if there are multiple equilibria In Nash equilibrium, the credibility of best responses judged along equilibrium path; off the path, NE may include incredible strategies In subgame perfect NE, best replies are judged in every subgame In perfect Bayesian equilibrium, best replies judged at each information set Beliefs: a probability distribution for a particular information set In mixed-strategy NE, a belief is a probability distribution over the nodes in the other players information set In Bayes Nash equilibrium, a belief is a probability distribution over the nodes in natures information set and the corresponding type of other player In perfect Bayesian equilibrium, a belief is a probability distribution over the nodes in ones own information set.,Nash Equilibrium,A strategy profile S is a Nash equilibrium if and only if each player is a playing a best response to the strategies of the other players A strategy profile S is a strict Nash equilibrium if and only if each players strategy is the single best-response to the strategies of the other players Note that there is no strategic uncertainty Each players belief about anothers strategy is concentrated on actual strategy the other player uses,Mixed-Strategy Nash Equilibrium,A mixed-strategy Nash equilibrium is a mixed-strategy profile whereby each player is playing a best response to the others strategies Consists of a probability distribution over the set of strategies, for each player Of course, probability density may be 0 for some strategies To solve, look for a mixed strategy for one player that makes the other players indifferent between a subset of their pure strategies If a player mixes over a set of strategies, it must be the case that each of those strategies yields the same expected payoff Thus the player is indifferent about which strategy is played This indifference will occur when other players are mixing over their own strategies in the appropriate way Useful for simultaneous games,Subgame Perfect Equilibrium,A strategy profile is called a subgame perfect Nash equilibrium if it specifies a Nash equilibrium in every subgame The SPNE is the “no bluffing” equilibrium. Here, all strategies are credible Think of the pirate game: (99,0,1) is SPE, but there are many not-credible NE of the form (100-x, 0, x) Grim trigger strategies are supported as subgame perfect equilibria Player 1 cooperates, given that player 2 will punish Cooperation is a best-response if the future matters Grim trigger is credible: NE in every subgame Useful for sequential games,Bayesian Equilibrium,Bayesian equilibrium consists of Each players strategy, which is a best response, given the strategies of the other players, and given players beliefs (prior probabilities) about the probability distribution over moves by nature Natures moves determine the type of player, where type corresponds with the payoffs associated with that player Useful for simultaneous games of incomplete information Typically, transform such games into games of imperfect information and different types, modeled as a simultaneous game Helpful to make a normal form (given beliefs about nature) and solve,Perfect Bayesian Equilibrium,A perfect Bayesian equilibrium is: a belief-strategy pairing, over the nodes at all information sets, such that each players strategy specifies optimal actions, given everyones strategies and beliefs, and beliefs are consistent with Bayes rule whenever possible Beliefs mean something new here! Probability distribution over location in the information set. Strategies corresponding to events off the equilibrium path may be paired with any beliefs, because these strategies are not consistent with Bayes Rule (and occur with probability zero.) Two major types of equilibria: Pooling: the types behave the sameupdated beliefs=old beliefs Separating: the types of behave differently Useful for sequential games of incomplete information,Perfect Bayesian Equilibrium,2 players, each contemplating nuclear war Neither knows if the other is about to strike Each can attack (A, a) or delay (D, d) If war, theres a 1st strike advantage Outcomes: No war results in (0, 0) The outcome of being the first-striker in war is a The outcome of NOT being the first-striker in war is r Assume 0 -a -r,Nuclear Deterrence Example,A,A,N,1,D,a,1,d,2,2,a,d,D,(-a, -r),(-r, -a),(0, 0),(0, 0),(-a, -r),(-a, -r),Such that 0 -a -r,Perfect Bayesian Equilibrium,Nuclear Deterrence Example,Perfect Bayesian Equilibrium,Nuclear Deterrence Example,A,A,N,1,D,a,1,d,2,2,a,d,D,(-a, -r),(-r, -a),(0, 0),(0, 0),(-a, -r),(-a, -r),Such that 0 -a -r,Player1 forms beliefs over and Player 2 forms beliefs over and Equilibrium will look like: p = Prob(Player 1 plays D), m = Prob( ) q = Prob(Player 2 plays d), n = Prob( ) 1 Play D: m*q*0+(1-q)*(-r)+(1-m)*0 = m*(1-q)*(-r) 1 Play A: m*(-a)+ (1-m)*(-a) = -a Always play A if m = 1, q = 0, else play D,PBE: (A, a, 1, 1)though Nature goes (, ). Another: (D, d, , ) again, by symmetry, and because beliefs are consistent w/