Section 8: Redux

1. Section 8: Redux

2. 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

3. Overview • Nash Equilibrium • Mixed-strategy Nash Equilibrium • Subgame Perfect Equilibrium • Bayesian Equilibrium • Perfect Bayesian Equilibrium

4. 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 player’s information set • In Bayes Nash equilibrium, a belief is a probability distribution over the nodes in nature’s information set and the corresponding type of other player • In perfect Bayesian equilibrium, a belief is a probability distribution over the nodes in one’s own information set.

5. 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 player’s strategy is the single best-response to the strategies of the other players • Note that there is no strategic uncertainty • Each player’s belief about another’s strategy is concentrated on actual strategy the other player uses

6. 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 otherplayers 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

7. 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

8. Bayesian Equilibrium • Bayesian equilibrium consists of • Each player’s 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 • Nature’s 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

9. Perfect Bayesian Equilibrium • A perfect Bayesian equilibrium is: • a belief-strategy pairing, over the nodes at all information sets, such that • each player’s strategy specifies optimal actions, given everyone’s 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 same—updated beliefs=old beliefs • Separating: the types of behave differently • Useful for sequential games of incomplete information

10. Perfect Bayesian Equilibrium Nuclear Deterrence Example • 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, there’s 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

11. Perfect Bayesian Equilibrium Nuclear Deterrence Example (-a, -r) A 1 (-r, -a) a ½ 2 D N d (0, 0) ½ (-a, -r) a 2 (-a, -r) A d Such that 0 > -a > -r 1 D (0, 0)

12. Perfect Bayesian Equilibrium Nuclear Deterrence Example 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 (-a, -r) A 1 (-r, -a) a ½ 2 D N d (0, 0) ½ (-a, -r) a 2 (-a, -r) A d Such that 0 > -a > -r 1 D (0, 0) Bayes Rule P( |at info set) = P(at info set | )*P( ) / P(at info set) PBE: (A, a, 1, 1)…though Nature goes (½, ½). Another: (D, d, ½, ½)… again, by symmetry, and because beliefs are consistent w/ play. PBE outcomes are consistent w/ players’ PBE beliefs Only 1 more PBE: When m*(1-q)*(-r) = -a and m = (½) / (½+½*q) -----This is Bayes’ rule! -a = -r*(1-q) / (1+q), q* = (r–a)/(r+a) = p*, m* = (r–a)/2*(r+a) = n*: PBE: (p*, q*, m*, n*)