Repeated Games

• a repeated game strategy for player i specifies an action (pure or mixed) in each period t as a function of the history at t
• Payoff: discounted present value of stage-game payoffs, where the discount is the probability of repetition times the discount factor
• ADV: (1-delta) times today's payoff plus delta times ADV of continuation play
• v is a feasible average discounted payoff of the repeated stage game if it is part of the convex hull
• a payoff is individually rational if it offers a payoff at least as great as their minimax payoff
• Folk Theorem: if a payoff v is in the feasible and rational set, there exists a discount factor such that the infinitely repeated game has a Nash equilibrium where each player's ADV equals vi
• Friedman (1971) theorem: suppose v is a feasible payoff vector such that for all i, vi exceeds players i's payoff in some NE of the stage game. For discount factor sufficiently close to 1 there is a subgame perfect equilibrium of the infinitely repeated game with ADV v
• one-shot deviation principle: a proposed strategy profile is SPE if, and only if, no player has an incentive to deviate at any state while obeying the transitions denoted in an automaton
• Fudenberg and Maskin (1986) Perfect Folk Theorem: assume V* is n-dimensional. For any payoff v in V* there is a discount value such that for all discount rates lower than it, the infinitely repeated subgame has a subgame perfect equilibrium with average discounted payoffs v1 up to vn
• Criticism towards repeated game SPE with 'punishment phases': both players are worse off at the beginning of a punishment phase, so they may want to renegotiate it
• Weak renegotiation proofness: no continuation play in the 'book of plays' is Pareto dominated by any other
• Farrell and Maskin (1989) have a Folk Theorem for renegotiation-proof SPE
• Benoit and Krishna (1985) - finite repetition expands the set of SPE if the stage game has multiple Nash equilibria ranked differently by the players
• Imperfect public monitoring: past actions are unobservable, but there are public signals that are imperfectly correlated with aggregate action
• Cooperative equilibria may still be constructed under imperfect public monitoring using trigger strategies
• what makes collusion harder?

  leniency towards whistleblowers, business cycles, imperfect information, differentiation of products