**Game Theory** Topic 5Repeated Games “Доверяй, Но Проверяй” (“Trust, but Verify”) - Russian Proverb (Ronald Reagan)

**Repeated Interaction** • Review • Simultaneous games • Put yourself in your opponent’s shoes • Iterative reasoning • Sequential games • Look forward and reason back • Sequentially rational reasoning • Outline: • What if interaction is repeated? • What strategies can lead players to cooperate?

**The Prisoner’s Dilemma** Equilibrium: $54 K Cooperation: $60 K

**Prisoner’s Dilemma** • Private rationalitycollective irrationality • The equilibrium that arises from using dominant strategies is worse for every player than the outcome that would arise if every player used her dominated strategy instead • Goal: • To sustain mutually beneficial cooperative outcome overcoming incentives to cheat (A note about tacit collusion)

**Finite Interaction (Silly** Theoretical Trickery) • Suppose the market relationship lasts for only T periods • Use backward induction (rollback) • Tth period: no incentive to cooperate • No future loss to worry about in last period • T-1th period: no incentive to cooperate • No cooperation in Tth period in any case • No opportunity cost to cheating in period T-1 • Unraveling: logic goes back to period 1

**Finite Interaction** • Cooperation is impossible if the relationship between players is for a fixedandknown length of time. • But, people think forward if … • Game length uncertain • Game length unknown • Game length too long

**Long-Term Interaction** • No last period, so no rollback • Use history-dependent strategies • Trigger strategies: • Begin by cooperating • Cooperate as long as the rivals do • Upon observing a defection: immediately revert to a period of punishment of specified length in which everyone plays non-cooperatively

**Two Trigger Strategies** • Grim trigger strategy • Cooperate until a rival deviates • Once a deviation occurs, play non-cooperatively for the rest of the game • Tit-for-tat • Cooperate if your rival cooperated in the most recent period • Cheat if your rival cheated in the most recent period

**Trigger Strategy Extremes** • Grim trigger is • least forgiving • longest memory • MAD • adequate deterrence but lacks credibility Grim trigger answers: “Is cooperation possible?” • Tit-for-Tat is • most forgiving • shortest memory • proportional • credible but lacks deterrence Tit-for-tat answers: “Is cooperation easy?”

**Why Cooperate (Against GTS)?** • Cooperate if the present value of cooperation is greater than the present value of defection • Cooperate: 60 today, 60 next year, 60 … 60 • Defect: 72 today, 54 next year, 54 … 54 Firm 2

**Payoff Stream (GTS)** profit 72 cooperate 60 defect 54 t t+1 t+2 t+3 time

**Discounting** • Discounting: • $1 tomorrow is worth less than $1 today • r is the interest rate • Invest $1 today get $(1+r) next year • Want $1 next year invest $1/(1+r) today • Annuity paying $1 today and $1 every year is worth $1+1/r

**Aside: Infinite Discounting** • Why?

**Cooperate against GTS** • Cooperate if • Cooperation is sustainable using grim trigger strategies as long as r < 50% • Or … as long as $1 invested today does not return more than $1.50 next period > > > > < PV(cooperation) 60…60…60…60… 60 + 60/r 6/r r PV(defection) 72…54…54…54… 72 + 54/r 12 6/12 = 50%

**defectonce** Payoff Stream (TFT) profit 72 cooperate 60 defect 54 47 t t+1 t+2 t+3 time

**Cooperate against TFT** • Cooperate if • Much harder to sustain than grim trigger • Cooperation may not be likely PV(cooperation) PV(cooperation) 60…60…60…60… 60 + 60/(1+r) 13/(1+r) 13 r PV(defection) PV(defect once) 72…47…60…60… 72 + 47/(1+r) 12 12 + 12r 1/12 = 8.3% > and > > > > > <

**Trigger Strategies** • Grim Trigger and Tit-for-Tat are extremes • Balance two goals: Deterrence • GTS is adequate punishment • Tit-for-tat might be too little Credibility • GTS hurts the punisher too much • Tit-for-tat is credible

**Axelrod’s Simulation** • R. Axelrod, The Evolution of Cooperation • Prisoner’s Dilemma repeated 200 times • Economists submitted strategies • Pairs of strategies competed • Winner: Tit-for-Tat • Reasons: • Forgiving, Nice, Provocable, Clear

**Main Ideas from Axelrod** • Not necessarily tit-for-tat • Doesn’t always work • Don’t be envious • Don’t be the first to cheat • Reciprocate opponent’s behavior • Cooperation and defection • Don’t be too clever

**Finite Interaction (Theoretical Aside)** • Unraveling prevents cooperation if the number of periods is fixed and known • Probabilistic termination • The “game” continues to the next period with some probability p: • Equivalent to infinite game • $1 next year is worth now • Value of future = { value if there is a future } { probability of a future } • Effective interest rate:

**Subgame Perfection** • Grim trigger strategy is subgame perfect • Tit for tat is not • Consider, in the first stage, one player cooperates and the other does not • Tit for tat would have them alternating forever

**Folk Theorem** • We saw in the prisoner’s dilemma that • Always cooperate can be an equilibrium • Sufficient punishment, low interest rate • Always defect can be an equilibrium • Insufficient punishment, high interest rate

**Folk Theorem** • Folk Theorem • Almost any outcome that, on average, yields at least the mutual defection payoff for both players, can be sustained as a subgame perfect equilibrium of the infinitely repeated prisoner’s dilemma (for sufficiently low interest rates)

**The Prisoner’s Dilemma** Equilibrium: $54 K Cooperation: $60 K

**Folk Theorem** 72 Firm 1 60 All subgame perfect payoffs 54 All feasible payoffs 47 47 • 60 • 72 54 Firm 2

**Summary** • Cooperation • Struggle between high profits today and a lasting relationship into the future • Deterrence • A clear, provocable policy of punishment • Credibility • Must incorporate forgiveness • Looking ahead: • How to be credible?