Game Theory

“Доверяй, Но Проверяй”- Russian Proverb

(Trust, but Verify)- Ronald Reagan

Mike ShorLecture 6

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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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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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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

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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”

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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

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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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Payoff Stream

18

16

12

t t+1 t+2 t+3

collude

cheat

time

profit

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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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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