Game Theory
“Доверяй, Но Проверяй”
- Russian Proverb
(Trust, but Verify)
- Ronald Reagan
Mike Shor
Lecture 6
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
Equilibrium: $12K
Firm 2
Low High
Low 12 , 12 18 , 8
Firm 1
High 8 , 18 16 , 16
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 Repetition
Silly 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
1
1 r
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Infinite Sums
1
1+ + 2 + 3 + 4 +…=
1
Why?
x = 1 + + 2 + 3 + 4 + …
x = + 2 + 3 + 4 + …
x- x = 1
1
x =
1
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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) + …
1
= (16)
1
PV(cheat) = 18 + (12) + 2(12) + …
= 18 + (12)
1
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Equilibrium in GTS
Equilibrium if: PV(collude) > PV(cheat)
1
1
(16) > 18 + 1 (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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Payoff Stream
profit
18
16 collude
12 cheat
t t+1 t+2 t+3 time
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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 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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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
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