Game Theory

1. Game Theory “Доверяй, Но Проверяй” - Russian Proverb (Trust, but Verify) - Ronald Reagan Mike Shor Lecture 6

2. Review • Simultaneous games • Put yourself in your opponent’s shoes • Iterative reasoning • Sequential games • Look forward and reason back • Sequentially rational reasoning • Repeated games Game Theory - Mike Shor

3. Prisoner’s Dilemma • Each player has a dominant strategy • 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 • Private rationalitycollective irrationality • Goal: • To sustain mutually beneficial cooperative outcome overcoming incentives to cheat Game Theory - Mike Shor

4. Duopoly Competition • Tourists & Natives Bar competition: • Two firms: Firm 1 and Firm 2 • Two prices: low ($4) or high ($5 ) • 3000 captive consumers per firm (tourists) • 4000 floating go to firm with lowest price (natives) Game Theory - Mike Shor

5. Prisoner’s Dilemma Equilibrium: $20K Cooperation: $25K Game Theory - Mike Shor

6. Repeated Interaction • Repeated Interaction • Ongoing relationship between players • Current action affects future interactions • History-Dependent Strategies • Choose an action today dependent on the history of interaction • Can history-dependent strategies help enforce mutual cooperation? • Sayeth the Economist: “It depends” Game Theory - Mike Shor

7. Finite RepetitionSilly 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 Game Theory - Mike Shor

8. Finite Repetition • Cooperation is impossible if the relationship between players is for a fixed and known length of time. • Why do people cooperate even though they don’t live forever? • More on this next time! Game Theory - Mike Shor

9. Infinite Repetition • 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 Game Theory - Mike Shor

10. 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 Strategy • Cooperate if your rival cooperated in the most recent period • Cheat if your rival cheated in the most recent period Game Theory - Mike Shor

11. Grim Trigger Strategy • In any period t, a firm faces one of two histories of play: • Zero deviations up to that point • Charge the high price in the next period • One or more deviations up to that point • Charge the low price from that point on in every period • Since { low, low } is the Nash equilibrium, each firm is doing the best it can Game Theory - Mike Shor

12. Equilibrium in GTS: Discounting • Discounting: value of future profits is less than value of current profits • r is the interest rate • is the discount rate • Invest: • $1 today get $(1+r) tomorrow • $ today, get $1 tomorrow Game Theory - Mike Shor

13. Infinite Sums • 1+ x+ x2 + x3 + x4 + … = • Why? z = 1+ x+ x2 + x3 + x4 + … zx = x+ x2 + x3 + x4 + … z- zx = 1 z = Game Theory - Mike Shor

14. Equilibrium in GTS • For GTS to be an equilibrium, the present value of colluding must be greater than the present value of cheating • PV(collude) = 25 + 25/(1+r) + 25/(1+r)2 + … = 25 + 25/r • PV(cheat) = 28 + 20/(1+r) + 20/(1+r)2 + … = 28 + 20/r Game Theory - Mike Shor

15. Equilibrium in GTS • Equilibrium if: PV(collude) > PV(cheat) • 25 + 25/r > 28 + 20/r • 5/r > 3 • r < 5/3 • Cooperation is sustainable using the grim trigger strategies as long as r < 5/3 • Or… as long as  > 3/8 • Invest more than 37¢ to get $1 next year • As long as firms value the future enough Game Theory - Mike Shor

16. Payoff Stream profit 28 collude 25 cheat 20 t t+1 t+2 t+3 time Game Theory - Mike Shor

17. Sustainability • The minimum discount rate required to sustain the collusive outcome depends on the payoff structure • Greater relative profits from cheating: • Need lower interest (larger discount) rate • Smaller relative profits after cheating: • Need higher interest (smaller discount) rate Game Theory - Mike Shor

18. Tit-for-Tat • Tit-for-Tat is nicer than GTS • If rival uses tit-for-tat, cooperate if: a) Colluding is better than cheating 25…25…25… > 28… 20… 20…20…20… b) Colluding is better than cheating once 25…25…25… > 28… 15… 25…25…25… Game Theory - Mike Shor

19. 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 Game Theory - Mike Shor

20. Main Ideas • 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 Game Theory - Mike Shor

21. Trigger Strategies • GTS and Tit-for-Tat are extremes • 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 Game Theory - Mike Shor

22. Inducing Cooperation • Trigger strategies revisited: • Announce the trigger • Announce the punishment COMMANDMENT In announcing a punishment strategy: Punish enough to deter your opponent Temper punishment to remain credible Game Theory - Mike Shor

23. Conclusion • Cooperation • Struggle between high profits today and a lasting relationship into the future • Deterrence • A clear, provocable policy of punishment • Credibility • Must incorporate forgiveness Game Theory - Mike Shor