Game Theory II – Repeated Games Managerial Economics Kyle Anderson

Repeated Games Does the outcome of a game change if the game is played multiple times? Repeated games: Multiple players Simultaneous moves Perfect information Game to be repeated: Finite number of times Infinite number of times Kyle J. Anderson

Repeated Pricing Game 0, 0 40, -30 -30, 40 10, 10 Perry low price Perry high price Anderson low price 0, 0 40, -30 Anderson high price -30, 40 10, 10 What if this game were to be played 5 times?

Repeated Prisoners’ Dilemma Perry low price Perry high price Anderson low price 0, 0 40, -30 Anderson high price -30, 40 10, 10 The NE in a one-time game is low-low (Prisoners’ Dilemma). Repeat game five times. Backward induction: what will happen in the fifth game? Work backwards In any finitely repeated game, the NE of each playing is the same as the NE of the one time game.

Infinitely Repeated Prisoners’ Dilemma If a Prisoners’ Dilemma game is repeated an infinite number of times, is there another Nash Equilibrium? No, the Nash Eq. of the one time game is just repeated forever. Yes, firms should always cooperate. Depends on the Federal Reserve.

Infinitely Repeated Prisoners’ Dilemma Perry low price Perry high price Anderson low price 0, 0 40, -30 Anderson high price -30, 40 10, 10 Trigger Strategy: Anderson will cooperate (high) in the first period. In the second period, cooperate if Perry cooperated previously. If Perry plays low, then play low forever after. Should Perry play the same strategy?

Infinitely Repeated Prisoners’ Dilemma Perry low price Perry high price A low price 0, 0 40, -30 A high price -30, 40 10, 10 Trigger Strategy: Anderson will cooperate (high) in the first period. In the second period, cooperate if Perry cooperated previously. If Perry plays low, then play low forever after. Perry’s Payouts: Cheat (low) first period: 40 + 0 + 0 + 0 … Trigger Strategy: 10 + 10 + 10 + 10 … Is 10 + 10 + 10 + 10 … > 40

Infinitely Repeated Prisoners’ Dilemma Perry low price Perry high price A low price 0, 0 40, -30 A high price -30, 40 10, 10 Trigger Strategy: Perry’s Payouts: (40-10)/(10-0)<1/i 3 < 1/i If the interest rate is less than 33.3%, the Trigger Strategy is a Nash Equilibrium.

Franchising: an application Business Traveler trying to find a hotel: Hotel – decent quality Hotel – low quality Stay 5, 5 -10, 10 Don’t stay 0, -5 0, 0 Cheat: 10, Coop: 5, NE: 0 If the Interest rate is less than 100% then it is better to provide decent quality. Kyle J. Anderson

Can Cooperative Pricing Occur: Standardized pricing High market concentration Reaction speed and detection lags Information about transactions Number of buyers Volatility of demand High firm symmetries Price sensitivity of buyers Kyle J. Anderson

Big Finish Finitely repeated games Infinitely repeated games have other NE if the interest rate is sufficiently low. In practice, cooperation is harder due to a number of reasons. Kyle J. Anderson