Game Theory

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Game Theory
Topic 5Repeated Games

• ???????, ?? ???????? (Trust, but Verify)

- Russian Proverb (Ronald Reagan)
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Repeated Interaction

• Review
• Simultaneous games
• Put yourself in your opponents 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 Prisoners Dilemma
Equilibrium 54 K
Firm 2 Firm 2
Low High
Firm 1 Firm 1 Low 54 , 54 72 , 47
Firm 1 Firm 1 High 47 , 72 60 , 60
Cooperation 60 K
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Prisoners 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)

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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 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 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 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
• 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

• 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?

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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

Firm 2
Low High
Firm 1 Firm 1 Low 54 , 54 72 , 47
Firm 1 Firm 1 High 47 , 72 60 , 60
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Payoff Stream (GTS)
profit
72
cooperate
60
defect
54
t
t1
t2
t3
time
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Discounting

• Discounting
• 1 tomorrow is worth less than 1 today
• r is the interest rate
• Invest 1 today ? get (1r) next year
• Want 1 next year ? invest 1/(1r) today
• Annuity paying 1 today and 1 every year
• is worth 11/r

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Aside Infinite Discounting

• Why?

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Cooperate against GTS

• Cooperate if
• Cooperation is sustainable using grim trigger
  strategies as long as r lt 50
• Or as long as 1 invested today does not return
  more than 1.50 next period

PV(defection) 72545454 72 54/r 12 6/12 50
PV(cooperation) 60606060 60 60/r 6/r r
gt gt gt gt lt
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Payoff Stream (TFT)
profit
72
cooperate
60
defect
54
47
t
t1
t2
t3
time
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Cooperate against TFT

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

PV(defection) PV(defect once) 72476060 72
47/(1r) 12 12 12r 1/12 8.3
PV(cooperation) PV(cooperation) 60606060
60 60/(1r) 13/(1r) 13 r
gt and gt gt gt gt gt lt
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Trigger Strategies

• Grim Trigger and Tit-for-Tat are
  extremes
• Balance 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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Axelrods Simulation

• R. Axelrod, The Evolution of Cooperation
• Prisoners 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-tat
• Doesnt always work
• Dont be envious
• Dont be the first to cheat
• Reciprocate opponents behavior
• Cooperation and defection
• Dont 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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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 prisoners 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 prisoners
  dilemma (for sufficiently low
  interest rates)

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The Prisoners Dilemma
Equilibrium 54 K
Firm 2 Firm 2
Low High
Firm 1 Firm 1 Low 54 , 54 72 , 47
Firm 1 Firm 1 High 47 , 72 60 , 60
Cooperation 60 K
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Folk Theorem
72
Firm 1
60
All subgame perfect payoffs
54
All feasible payoffs
47

• 72

• 60

47
54
Firm 2
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Summary

• Cooperation
• Struggle between high profits today
  and a lasting relationship into the future
• Deterrence
• A clear, provocable policy of punishment
• Credibility
• Must incorporate forgiveness
• Looking ahead
• How to be credible?