Dynamic Games

• backward induction results in a subgame perfect equilibrium in perfect-information dynamic games
• the number of a players strategies equals the product of the numbers of actions available at each of her information sets
• a Nash equilibrium may fail time consistency: strategies may fail to best respond in some out-of-equilibrium continuation
• a subgame is a continuation of the game after a specific history (node) such that no information set is 'broken up' (no info set with nodes both inside and outside subgame)
• A SPE of a game is a strategy profile that is Nash equilibrium in every subgame
• Kuhn Theorem (1953) - every finite game in which players have perfect recall has a subgame perfect equilibrium
• Perfect Bayesian Equilibrium is a SPE strategy-profile and beliefs for all players at their information sets satisfying: 1. sequential rationality - each player chooses optimally given her beliefs at each information set and the others' eqm strategies and 2. Bayesian beliefs - beliefs are computed based on equilibrium strategies via Bayes' rule whenever possible. No restrictions at 'unreached' information sets in two-player games
• Fudenberg and Tirole (1991) - every finite game has a PBE
• Kreps and WIlson (1982) - every finite game has a sequential equilibrium. All SE are PBE and all PBE are SPE
• When converting to reduced normal form, remember that strategies are complete contingent plans, but remove duplicate strategies
• Sequential equilibrium: off-path beliefs must be the limit of Bayesian beliefs generated by fully-mixed strategies ('trembles') otherwise same as PBE
• Cho-Kreps Intuitive Criterion: upon observing a deviation, put zero weight on types whose equilibrium payoff already exceeds any possible payoff from deviating, assuming the opponent's reaction to a deviation is rational for some beliefs
• Divinity D1 (Cho and Sobel, 1991) - upon observing the sender's out-of-equilibrium action, the receiver must assign 0 beliefs to a sender type if there is another type that gains compared to its eqm payoff for a larger set of sequentially rational receiver responses
• with two types PBE and the intuitive criterion yield the minimal-cost separating outcome