Game Theory

“Доверяй, Но Проверяй” (“Trust, but Verify”)

- Russian Proverb (Ronald Reagan)

Topic 5Repeated Games

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?

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The Prisoner’s Dilemma

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Firm 2

Low High

Firm 1Low 54 , 54 72 , 47

High 47 , 72 60 , 60

Equilibrium: $54 K

Cooperation: $60 K

Prisoner’s Dilemma

Private rationality collective 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)

Mike Shor4

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 cooperateNo cooperation in Tth period in any caseNo opportunity cost to cheating in period T-1

Unraveling: logic goes back to period 1

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Finite Interaction

Cooperation is impossible if the relationship between players is for a fixed and known length of time.

But, people think forward if … Game length uncertain Game length unknown Game length too long

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Long-Term Interaction

No last period, so no rollback Use history-dependent strategies Trigger strategies:

Begin by cooperatingCooperate as long as the rivals doUpon 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 Cooperate if your rival cooperated

in the most recent period Cheat if your rival cheated

in the most recent period

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Trigger Strategy Extremes

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Tit-for-Tat is most forgiving shortest memory proportional credible

but lacks deterrence

Tit-for-tat answers:

“Is cooperation easy?”

Grim trigger is least forgiving longest memory MAD adequate

deterrence but lacks credibility

Grim trigger answers:

“Is cooperation possible?”

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

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Low High

Firm 1Low 54 , 54 72 , 47

High 47 , 72 60 , 60

Firm 2

Payoff Stream (GTS)

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72

54

t t+1 t+2 t+3

defect

time

profit

60 cooperate

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 yearis worth $1+1/r

Mike Shor

Aside: Infinite Discounting

Why?

1 xz z

...xxxx xz

...xxxx 1 z

xDefine

432

432

r1

1

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r1

1r

r11

1x1

1 z

r)(11

r

11...

r)(1

1

r)(1

1

r)(1

1

r)(1

11 432

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

Mike Shor14

PV(defection)

72…54…54…54…

72 + 54/r

12

6/12 = 50%

PV(cooperation)

60…60…60…60…

60 + 60/r

6/r

r

>

>

>

>

<

Payoff Stream (TFT)

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72

47

t t+1 t+2 t+3 time

profit

60 cooperatedefect oncedefect54

Cooperate against TFT Cooperate if

Much harder to sustain than grim trigger Cooperation may not be likely

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PV(defection)

PV(defect once)72…47…60…60…

72 + 47/(1+r)12

12 + 12r1/12 = 8.3%

PV(cooperation)

PV(cooperation)60…60…60…60…

60 + 60/(1+r)13/(1+r)

13r

>and

>>>>><

Trigger Strategies

Grim Trigger and Tit-for-Tat are extremes

Balance two goals:Deterrence

GTS is adequate punishmentTit-for-tat might be too little

CredibilityGTS hurts the punisher too muchTit-for-tat is credible

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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 from Axelrod

Not necessarily tit-for-tatDoesn’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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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:

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pr1

1

prp

r 1

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

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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

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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)

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The Prisoner’s Dilemma

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Firm 2

Low High

Firm 1Low 54 , 54 72 , 47

High 47 , 72 60 , 60

Equilibrium: $54 K

Cooperation: $60 K

Folk Theorem

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72

60

54

726047

Firm 1

Firm 2

47

54

All feasible payoffs

All subgame perfect payoffs

Summary

CooperationStruggle between high profits today

and a lasting relationship into the future

DeterrenceA clear, provocable policy of punishment

CredibilityMust incorporate forgiveness

Looking ahead: How to be credible?

Mike Shor26