IE 443 Economic Models for Decision and Policy Analysis
Repeated Games
MIDDLE EAST TECHNICAL UNIVERSITY
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Today:
➢ Repeated Games
Related Sections from reference Textbooks.
1) Chapter 10 from:
Dixit, Avinash K., and Susan Skeath. Games of strategy: Fourth international student edition. WW Norton &
Company, 2015.
Refer the following if you need further details
2) Chapter 14&15 (only the subsections compliant with this lecture notes) from:
Osborne, Martin J. An introduction to game theory. Vol. 3. No. 3. New York: Oxford university press, 2004.
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Prisoners’ Dilemma Revisited
• Rational Self-Interest Can Lead to Suboptimal Outcomes: Even though mutual cooperation would yield
a better overall outcome, individual incentives can lead to both players acting in their own self-interest and
ending up worse off. This can be seen in many real-world situations such as pollution, overfishing, or
nuclear arms races, where individual or national self-interest leads to collective harm.
P-1
S1
P-1
S1
S2
A, A
D, C
P-2
S1
S2
S1
P-2
S2
C, D
B, B
S2
C>A>B>D
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Repeated Games
• How can we promote Pareto Efficient solution?
Through Cooperation
• Communication and Trust Are Crucial for Cooperation:
In the Prisoner's Dilemma, the inability of the prisoners to communicate and trust each other leads to the
suboptimal outcome. In real-world situations, creating mechanisms for communication, building trust, and
ensuring enforceable agreements can help overcome these challenges and lead to more cooperative outcomes.
• How can we sustain cooperation?
The promise of future rewards (carrots) and the threat of future punishments (sticks) may provide incentives
for good* behavior today.
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Finitely Repeated Games
Consider that the following game is played twice. In each period moves are simultaneous. First-period outcome
can be publicly observed.
Assume that your objective is to promote cooperation.
Do we use first stage outcome to coordinate future actions?
Let’s find Backward Inductions solution.
In stage 2 N.E. Solution is (D,D)
Eq. Payoffs : (1,1)
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Finitely Repeated Games
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Finitely Repeated Games
Payoff in the first stage
• Unique equilibrium in period 2
• First-period play cannot credibly affect the future
• This is True for all finitely-repeated PD
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Finitely Repeated Games
• Let us consider the case where the Prisoner’s Dilemma is played k times and let’s assume both players
know k.
• Then in the last period (i.e. period k) there is no point in cooperating since there is no future to worry
about(No future punishment).
• Therefore both players will defect in period k. Then in period k-1, knowing that both of them will defect in
period k anyway, they can defect as well. This reasoning continuous up to the first period and both players
shirk at each period.
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Finitely Repeated Games
Question: But in real life cooperation is sustained in finitely repeated games; how come?
We can give two answers to this question:
Answer 1: If agents do not know k, that is if they do not know when the last period is, then at each period
there is a potential future to worry about and hence cooperation may be sustained.
Answer 2: If there are “nice” players who are non-strategic and who cooperate no-matter what, we may want
to give the impression that we are one of them and cooperate at least at the beginning.
Answer 3 :
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Example: Suppose there are two similar companies sharing the market equally and competing on price to
increase market shares.
Company-2
Low
High
Low
10,10
20, 5
High
5, 20
15, 15
Company-1
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Example: Suppose there are two similar companies sharing the market equally and competing on price to
increase market shares.
Company-2
Low
High
Low
10,10
20, 5
High
5, 20
15, 15
Much Higher
3, 25
7, 30
Company-1
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Example: Suppose there are two similar companies sharing the market equally and competing on price to
increase market shares.
Company-2
Low
High
Much Higher
Low
10,10
20, 5
25, 3
High
5, 20
15, 15
30, 7
Much Higher
3, 25
7, 30
20, 20
Company-1
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Question: Any other possible cooperation in finitely repeated games?
Possible if the game is changed
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Finitely Repeated Games
• Consider the following two stage game
Stage-1
Stage-2
• Consider the strategy: “Play Cooperate in Stage 1. If the other player also chose Cooperate, play Trust in
Stage 2. If the other player did not choose Cooperate, play Don’t.”
• Is the threat of «Don’t Trust» credible? Can it induce cooperation in the first stage?
• Key point: How many N.E. Are the in stage 2?
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Finitely Repeated Games
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Finitely Repeated Games
Observations:
1. History-independent play → guaranteed defect
2. Future play must be variable (condition on the past)
3. Mutual defection (and distrust) may still be an equilibrium
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Infinitely Repeated Games
• We saw that end-game effects were crucial in finitely repeated games.
• What if no end game (or I don’t know it)?
• Consider infinite repetition of this game:
• How many strategies are there?
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Infinitely Repeated Games
Two famous strategies:
Grim-trigger:
Play Cooperate in the 1st period
Play Cooperate if no-one has ever Defected
Play Defect otherwise
Tit-for-tat:
Play Cooperate in the 1st period
Play Cooperate if your opponent Cooperated in previous period
Play Defect otherwise
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Infinitely Repeated Games
Grim-trigger:
Play Cooperate in the 1st period
Play Cooperate if no-one has ever Defected
Play Defect otherwise
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Infinitely Repeated Games
• Payoffs in Grim trigger strategy: Assume that one player deviated from playing C to D, the player gets 5 in
this round and then gets 1 forever since he will be punished by the other player and D,D will be played
forever. On the other hand, the player would get 4 forever by sticking the agreed strategy.
• Which is better? Do you think about defect?
• (5 once and 1 forever) vs (4 forever)
• The decision depends on the discount factor for the future payoff
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Infinitely Repeated Games
• Discount is applied in repeated games: Time preference / opportunity cost / Probability of breakdown etc.
In all these cases, future payoffs matter less
1
• Discount Rate (or interest rate): r, therefore the weight on tomorrow’s outcome


1
to save notation 
 or we can use  =

1
+
r
( )


(1 + r )
Which payoff do you prefer:
1
1
 r +1
5+
1+
1 + .... = 5 + 1 

2
1
+
r
r
( ) (1 + r )


4+
1
1
 r +1
4+
4
+
..
.
.
=
4


(1 + r ) (1 + r )2
 r 
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Infinitely Repeated Games
One-shot deviation principle:
In order to find Subgame Perfect N.E., we need to check whether is there any profitable deviation for the
player in every subgame. How can we do this in infinite horizon games? There are infinitely many subgames.
Remember that a set of strategies is a subgame perfect equilibrium if and only if no player can profitably
deviate from his strategy at a single stage and maintain his strategy.
In an infinitely repeated game, players may deviate from the a strategy in any given round. Since it is infinite,
when we consider the game from that round we still have an infinite horizon ahead.
For such games, compare «deviating now and move on according to defined strategy» with «not deviating at
all» to find SPNE.
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Infinitely Repeated Games
Example: Assume that the following game is played infinitely. what is the discount factor to sustain (C,C)
with grim trigger punishment strategy (play (C,C), if the other player plays D, then play D forever)
remember that 1 +  + 2 + .... =
1
1− 
Compare payoffs:
1 
3
Not deviate: 3,3,3,3,3,3….→ present value with   u = 3 + 3 + 32 + .... = 3 (1 +  + 2 + ....) = 3 
=

1−  
Deviate:




5,1,1,1,1,1…. → present value with   u = 5 + 1 + 12 + 13.... = 5 +  1 + 1 + 12 ...  = 5 +
1− 
1


=
1−


Does not deviate if
1− 
3

 5+
1− 
1− 
Thus (C,C) with grim trigger strategy is SPNE for
1

2
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