Answers to practice problems,
chapter 11, repeated games.
Related to Figure 11.14 (p. 432).
Three Questions
Chapter 11: Using Figure 11.4 on p. 432 as your stage
game, determine the values of δ for which the following
strategy pairs are in equilibrium in infinite play.
1. (GT, GT)
2. (TFT, Alt), where “Alt” alternates D,C,D,C,… infinitely.
3. (TFTD,TFT), where “TFTD” is TFT using D in the first round.
Question 1
Grim Trigger (GT, GT)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
Both Strategies:
C in first period.
C as long as other plays C.
D forever if other plays D in any round.
• Step 1
–
–
–
–
Player 1:
Player 2:
Payoffs 1:
Payoffs 2;
C, C, C, …
C, C, C, …
-1, -1δ, -1δ2, …
-1, -1δ, -1δ2, …
• Step 2
– Sum of payoffs: c / (1 - δ) = -1 / (1 - δ).
3
Question 1
Stage game
Grim Trigger (GT, GT)
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
Both Strategies:
C in first period.
C as long as other plays C.
D forever if other plays D in any round.
• Step 3
– If player 1 deviates to “always D” (or identically grim
trigger with D in the first round), then the two will get:
• Player 1: D, D, D, …
• Player 2: C, D, D, …
4
Question 1
Stage game
Grim Trigger (GT, GT)
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
Both Strategies:
C in first period.
C as long as other plays C.
D forever if other plays D in any round.
• Step 4: It is rational for player 1 to deviate to “always D” iff:
EU1(always D, GT)
> EU1(GT, GT)
Conclude: If δ < 1/12, then this
deviation is rational.
If δ ≥ 1/12, then (GT, GT) is a Nash
Equilibrium, generating the outcome
(C,C) in every period.
Why? Because deviating to “always
defect,” in a later period produces the
same condition, as argued in class.
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Question 2
(TFT, Alt)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
Alt: D,C,D,C...
• Step 1
–
–
–
–
Player 1:
Player 2:
Payoffs 1:
Payoffs 2:
C, D, C, D…
D, C, D, C…
-20, 0δ, -20δ2, 0δ3 …
0, -20δ, 0δ2, -20δ3 …
• Step 2
– Payoff for player 1: -20 / (1 - δ2)
– Payoff for player 2: -20δ / (1 - δ2)
6
Question 2
(TFT, Alt)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
• Step 3
– If player 1 deviates to “always D”, then the two will get:
• Player 1: D, D, D, …
• Player 2: D, C, D, …
– Player 1’s payoff from deviating:
7
Question 2
(TFT, Alt)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
• Step 4: It is rational for player 1 to deviate to “always D” iff:
EU1(always D, Alt)
> EU1(TFT, Alt)
Conclude: It is always rational for
player 1 to deviate to “always D”
for all values of δ. Hence, (TFT,
Alt) is never in equilibrium –
regardless of the interests of
player 2.
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Question 3
(TFTD, TFT)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
TFTD: start with D.
play C if opponent played C in previous round.
play D if opponent played D in previous round.
• Step 1
–
–
–
–
Player 1:
Player 2:
Payoffs 1:
Payoffs 2:
D, C, D, C…
C, D, C, D…
0, -20δ, 0δ2, -20δ3 …
-20, 0δ, -20δ2, 0δ3 …
• Step 2
– Payoff for player 1: -20δ / (1 - δ2)
– Payoff for player 2: -20 / (1 - δ2)
9
Question 3
(TFTD, TFT)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
• Step 3
– If player 1 deviates to “TFT”, then the two will get:
• Player 1: C, C, C, …
• Player 2: C, C, C, …
– Player 1’s payoff from deviating:
10
Question 3
(TFTD, TFT)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
• Step 4: It is rational for player 1 to deviate to “TFT” iff:
EU1(TFT, TFT)
> EU1(TFTD, TFT)
Using the quadratic formula …
δ > .0525
Note: It is rational for player 1 to
deviate to “always D” for larger
values of δ (which is worth
checking). Hence, (TFTD, TFT)
appears only to be in equilibrium
for very small values of δ, namely
δ ≤ .0525.
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Question 3
(TFTD, TFT)
Stage game
C
D
C
‒1, ‒1
‒20, 0
D
0, ‒20
‒12, ‒12
• Note: We should also check whether it is rational for player 2 to deviate
from (TFTD,TFT) to (TFTD,Always D). This is true iff:
EU2(TFTD, Always D) > EU2(TFTD, TFT)
Solving and using the quadratic
formula …
δ > 2/3
Note: It is rational for player 2 to
deviate to “always D” for larger
values of δ than the values for
player 1. Hence, if we use a
common δ, then (TFTD, TFT) is only
in equilibrium for very small values
of δ, namely δ ≤ .0525 – the
condition on the previous slide.
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