game theory “Доверяй, Но Проверяй” - russian proverb (trust, but verify) -...
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Game Theory
“Доверяй, Но Проверяй”- Russian Proverb
(Trust, but Verify)- Ronald Reagan
Mike ShorLecture 6
Game Theory - Mike Shor 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 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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Prisoner’s Dilemma
Firm 2
Low High
Firm 1Low 12 , 12 18 , 8
High 8 , 18 16 , 16
Equilibrium: $12K
Cooperation: $16K
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!
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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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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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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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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
r11
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Infinite Sums 1 + + 2 + 3 + 4 + … =
Why? x = 1 + + 2 + 3 + 4 + … x = + 2 + 3 + 4 + …
x- x = 1
x =
11
11
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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)
11
1
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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
11
1
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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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Tit-for-Tat Tit-for-Tat is nicer than GTS If rival uses tit-for-tat, cooperate if:
a) Colluding is better than cheating16…16…16… > 18…12… 12…12…
12…
b) Colluding is better than cheating once16…16…16… > 18…8… 16…16…
16…
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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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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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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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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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