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

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Game Theory - Mike Shor2 Review Simultaneous games Put yourself in your opponent’s shoes Iterative reasoning Sequential games Look forward and reason back Sequentially rational reasoning Repeated games

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Game Theory - Mike Shor3 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

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Game Theory - Mike Shor4 Duopoly Competition Two firms: Firm 1 and Firm 2 Two prices: low ($6) or high ($8 ) 1000 captive consumers per firm 2000 floating go to firm with lowest price

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Game Theory - Mike Shor5 Prisoner’s Dilemma Firm 2 LowHigh Firm 1 Low 12, 12 18, 8 High 8, 18 16, 16 Equilibrium: $12K Cooperation: $16K

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Game Theory - Mike Shor6 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”

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Game Theory - Mike Shor7 Finite Repetition Silly Trickery Suppose the market relationship lasts for only T periods Use backward induction (rollback) T th period: no incentive to cooperate No future loss to worry about in last period T-1 th period: no incentive to cooperate No cooperation in T th period in any case No opportunity cost to cheating in period T-1 Unraveling: logic goes back to period 1

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Game Theory - Mike Shor8 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!

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Game Theory - Mike Shor9 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

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Game Theory - Mike Shor10 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

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Game Theory - Mike Shor11 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

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Game Theory - Mike Shor12 Equilibrium in GTS: Discounting Discounting: value of future profits is less than value of current profits is the discount rate Invest: $1 today get $(1+r) tomorrow $ today, get $1 tomorrow

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Game Theory - Mike Shor13 Infinite Sums 1 + + 2 + 3 + 4 + … = Why? x= 1 + + 2 + 3 + 4 + … x= + 2 + 3 + 4 + … x- x= 1 x=

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Game Theory - Mike Shor14 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)= 16 + (16) + 2 (16) + … = (16) PV(cheat)= 18 + (12) + 2 (12) + … = 18 + (12)

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Game Theory - Mike Shor15 Equilibrium in GTS Equilibrium if: PV(collude) > PV(cheat) (16) > 18 + (12) 16 > 18 - 6 > 1/3 Cooperation is sustainable using the grim trigger strategies as long as > 1/3 Invest more than 33¢ to get $1 next year As long as firms value the future enough

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Game Theory - Mike Shor16 Payoff Stream 18 16 12 tt+1t+2t+3 collude cheat time profit

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Game Theory - Mike Shor17 Sustainability The minimum discount rate required to sustain the collusive outcome depends on the payoff structure Greater relative profits from cheating: Need larger discount rate Smaller relative profits after cheating: Need smaller discount rate

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Game Theory - Mike Shor18 Tit-for-Tat Tit-for-Tat is nicer than GTS If rival uses tit-for-tat, cooperate if: a) Colluding is better than cheating 16…16…16… >18…12…12…12…12… b) Colluding is better than cheating once 16…16…16… > 18…8…16…16…16…

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Game Theory - Mike Shor19 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

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Game Theory - Mike Shor20 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

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Game Theory - Mike Shor21 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

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Game Theory - Mike Shor22 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

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Game Theory - Mike Shor23 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

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