Problem Set 6, page 1 of 12 Problem Set 5: Due in class on Tuesday, August 4. Test 5 on the content of this homework will be given on Wednesday, August 5 at 9:00am sharp. Please print this homework and provide your solutions on the printout. Unless stated otherwise, in the following problems assume that games considered are games of repeated Prisoner’s Dilemma with stage payoffs of 0, 1, 3 and 5, that is C D C 3 3 5 0 D 0 5 1 1 Problem 1 (6p) Consider the following strategy “Tit for Two Tats” (TF2T): Cooperate in periods 1 and 2. Thereafter defect in any period k>2 if and only if your opponent defected in k-1 and k-2. (a) Consider best response strategies to TF2T in a discounted repeated game with δ sufficiently close to zero. Is it possible to construct two strategies, j and j*, such that both of them are best responses to TF2T and (TF2T, j) is in Nash equilibrium while (TF2T, j*) is not? YES NO (circle one) If your answer is YES then give an example of two strategies like that. Whenever possible use the strategies defined in the lecture notes, otherwise define a strategy of your own. j = …………….. j* = …………….. (b) Consider best response strategies to TF2T in a discounted repeated game with δ sufficiently close to one. Is it possible to construct two strategies, j and j*, such that both of them are best responses to TF2T and (TF2T, j) is in Nash equilibrium while (TF2T, j*) is not? YES NO (circle one) Problem Set 6, page 2 of 12 If your answer is YES then give an example of two strategies like that. Whenever possible use the strategies defined in the lecture notes, otherwise define a strategy of your own. j = …………….. j* = …………….. (c) Consider best response strategies to TF2T in a repeated game without discounting (the payoff is an average per period payoff.) Is it possible to construct two strategies, j and j*, such that both of them are best responses to TF2T and (TF2T, j) is in Nash equilibrium while (TF2T, j*) is not? YES NO (circle one) If your answer is YES then give an example of two strategies like that. Whenever possible use the strategies defined in the lecture notes, otherwise define a strategy of your own. j = …………….. j* = …………….. Problem 2 (6p) Recall the strategy “Virgin” (V): V defects unconditionally in the first ten periods and then from period eleven on V cooperates unconditionally if and only if the opponent cooperated in all ten initial periods, and defects unconditionally otherwise. (a) Consider best response strategies to V in a discounted repeated game with δ sufficiently close to zero. Is it possible to construct two strategies, j and j*, such that both of them are best responses to V and (V, j) is in Nash equilibrium while (V, j*) is not? YES NO (circle one) If your answer is YES then give an ex ...

1. Problem Set 6, page 1 of 12
Problem Set 5: Due in class on Tuesday, August 4. Test 5 on
the content of this homework will be
given on Wednesday, August 5 at 9:00am sharp. Please print
this homework and provide your
solutions on the printout.
Unless stated otherwise, in the following problems assume that
games considered are games of
repeated Prisoner’s Dilemma with stage payoffs of 0, 1, 3 and
5, that is
C
D
C
3

2. 3
5
0
D
0
5
1
1
Problem 1 (6p)
Consider the following strategy “Tit for Two Tats” (TF2T):
Cooperate in periods 1 and 2.
Thereafter defect in any period k>2 if and only if your opponent
defected in k-1 and k-2.

3. (a) Consider best response strategies to TF2T in a discounted
repeated game with δ sufficiently
close to zero.
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to TF2T and (TF2T, j) is in Nash equilibrium while
(TF2T, j*) is not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..
(b) Consider best response strategies to TF2T in a discounted
repeated game with δ sufficiently

4. close to one.
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to TF2T and (TF2T, j) is in Nash equilibrium while
(TF2T, j*) is not?
YES NO (circle one)
Problem Set 6, page 2 of 12
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..
(c) Consider best response strategies to TF2T in a repeated
game without discounting (the payoff

5. is an average per period payoff.)
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to TF2T and (TF2T, j) is in Nash equilibrium while
(TF2T, j*) is not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..
Problem 2 (6p)
Recall the strategy “Virgin” (V): V defects unconditionally in
the first ten periods and then from
period eleven on V cooperates unconditionally if and only if the
opponent cooperated in all ten

6. initial periods, and defects unconditionally otherwise.
(a) Consider best response strategies to V in a discounted
repeated game with δ sufficiently close
to zero.
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to V and (V, j) is in Nash equilibrium while (V, j*) is
not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..

7. Problem Set 6, page 3 of 12
(b) Consider best response strategies to V in a discounted
repeated game with δ sufficiently close
to one.
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to V and (V, j) is in Nash equilibrium while (V, j*) is
not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..

8. (c) Consider best response strategies to V in a repeated game
without discounting (the payoff is
an average per period payoff.)
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to V and (V, j) is in Nash equilibrium while (V, j*) is
not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..
Problem 3 (6p)

9. Recall the strategy “Casanova” (CA): C cooperates
unconditionally in the first fifteen periods
and then from period sixteen on it defects unconditionally.
(a) Consider best response strategies to CA in a discounted
repeated game with δ sufficiently
close to zero.
Problem Set 6, page 4 of 12
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to CA and (CA, j) is in Nash equilibrium while (CA,
j*) is not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.

10. j = ……………..
j* = ……………..
(b) Consider best response strategies to CA in a discounted
repeated game with δ sufficiently
close to one.
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to CA and (CA, j) is in Nash equilibrium while (CA,
j*) is not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..

11. (c) Consider best response strategies to CA in a repeated game
without discounting (the payoff is
an average per period payoff.)
Is it possible to construct two strategies, j and j*, such that both
of them are best
responses to CA and (CA, j) is in Nash equilibrium while (CA,
j*) is not?
YES NO (circle one)
If your answer is YES then give an example of two strategies
like that. Whenever
possible use the strategies defined in the lecture notes,
otherwise define a strategy of your
own.
j = ……………..
j* = ……………..
Problem Set 6, page 5 of 12

12. Problem 4 (6p)
In the following problem assume that the game is that of
infinitely repeated Prisoner’s Dilemma
with one-shot payoffs of 0, 1, 3 and 5 and without discounting
(payoffs are calculated as per-
period average.)
BR below stands for Best Response.
Are the following statements true of false? Circle the correct
answer.
(1) If a pair of strategies (i,j) is such that i=BR(j) then u(i,j) ≥
3. TRUE FALSE
(2) If a pair of strategies (i,j) is such that j=BR(i) then u(i,j) ≥
3. TRUE FALSE
(3) If a pair of strategies (i,j) is such that i=BR(j) then u(i,j) ≥
1. TRUE FALSE
(4) If a pair of strategies (i,j) is such that j=BR(i) then u(i,j) ≥
1. TRUE FALSE

13. (5) If a pair of strategies (i,j) is such that i=BR(j) then u(i,j) <
5. TRUE FALSE
(6) If a pair of strategies (i,j) is such that j=BR(i) then u(i,j) <
5. TRUE FALSE
(7) If a pair of strategies (i,j) is in Nash equilibrium then u(i,j)
≥ 3. TRUE FALSE
(8) If a pair of strategies (i,j) is in Nash equilibrium and
u(i,j)>3 then u(j,i)<3. TRUE FALSE
(9) If a pair of strategies (i,j) is in Nash equilibrium and
u(i,j)<3 then u(j,i)>3. TRUE FALSE
(10) If a pair of strategies (i,j) is in Nash equilibrium then u(i,j)
≥ 1. TRUE FALSE
(11) If a pair of strategies (i,j) is in Nash equilibrium then u(i,j)
≤ 3. TRUE FALSE
(12) If a pair of strategies (i,j) is in Nash equilibrium then u(i,j)
< 5. TRUE FALSE
Consider now the following strategies in an infinitely repeated

14. Prisoner’s Dilemma game:
ALL D: defect unconditionally in all iterations of the game;
ALL C: cooperate unconditionally in
all iterations of the game; TFT (tit for tat): cooperate in the first
interaction and then cooperate if
the other player cooperated on the previous interaction and
defect if he defected; STFT
(suspicious tit for tat): defect in the first interaction and then
cooperate if the other player
cooperated on the previous interaction and defect if he defected;
MACHO defect in the first
iteration; in second iteration cooperate if the other player
defected in the first iteration and then
play “tit for tat”; if he cooperated in the first iteration, defect
unconditionally in all periods of the
game from iteration 2 onwards.
Problem Set 6, page 6 of 12
Problem 5 (8p)
Consider an infinitely repeated Prisoner’s Dilemma game with
values of δ sufficiently close to

15. (but not equal to) 0. Which of the following are true?
(1) ALL D is a best response strategy to TFT TRUE FALSE
(2) MACHO is a best response strategy to MACHO TRUE
FALSE
(3) TFT is a best response strategy to TFT TRUE FALSE
(4) ALL D is a best response strategy to MACHO TRUE
FALSE
(5) ALL D is a best response strategy to ALL C TRUE FALSE
(6) ALL C is a best response strategy to TFT TRUE FALSE
(7) STFT is a best response strategy to ALL D TRUE FALSE
(8) STFT is a best response strategy to STFT TRUE FALSE
Consider now an infinitely repeated Prisoner’s Dilemma game
with values of δ sufficiently close
to 1. Which of the following are true?

16. (9) ALL D is a best response strategy to TFT TRUE FALSE
(10) MACHO is a best response strategy to MACHO TRUE
FALSE
(11) TFT is a best response strategy to TFT TRUE FALSE
(12) ALL D is a best response strategy to MACHO TRUE
FALSE
(13) ALL D is a best response strategy to ALL C TRUE
FALSE
(14) ALL C is a best response strategy to TFT TRUE FALSE
(15) STFT is a best response strategy to ALL D TRUE FALSE
(16) STFT is a best response strategy to STFT TRUE FALSE
Problem Set 6, page 7 of 12

17. Problem 6 (4p)
Which of the following pairs of strategies are in Nash
equilibrium in the repeated Prisoner’s
Dilemma game without discounting? Circle YES if a pair is in
Nash equilibrium and NO
otherwise.
(MACHO, MACHO) YES NO
(TFT, STFT) YES NO
(ALL D, ALL D) YES NO
(TFT, TFT) YES NO
Problem 7 (4p)
The game below is similar to the game of Chicken (aka Hawk
and Dove.) Suppose it is a stage
game of an infinitely repeated game without discounting (the
payoff is an average per period
payoff.)
Using the coordinate system below draw a “folk theorem
diagram” i.e., a diagram that shows
which pairs of payoffs are possible in the game and which pairs
of payoffs are possible in Nash

18. equilibria of this game. Mark all necessary numerical values in
the coordinate system.
A
B

19. A
1
1
3
1
B
1
3
0
0
Problem Set 6, page 8 of 12

20. Problem 8 (4p)
The game below is similar to that of Prisoner’s Dilemma.
Suppose it is a stage game of an
infinitely repeated game without discounting (the payoff is an
average per period payoff.)
Using the coordinate system below draw a “folk theorem
diagram” i.e., a diagram that shows
which pairs of payoffs are possible in the game and which pairs
of payoffs are possible in Nash
equilibria of this game. Mark all necessary numerical values in
the coordinate system.

21. Problem 9 (4p)
The game below is similar to the Pareto Coordination game.
Suppose it is a stage game of an
infinitely repeated game without discounting (the payoff is an
average per period payoff.)
Using the coordinate system below draw a “folk theorem
diagram” i.e., a diagram that shows
which pairs of payoffs are possible in the game and which pairs
of payoffs are possible in Nash
equilibria of this game. Mark all necessary numerical values in
the coordinate system.
C

22. D
C
5
5
5
0
D
0
5
1
1

23. Problem Set 6, page 9 of 12
Problem 10 (6p)
(1) Consider the following game.

24. A
B
A
1
1
2
3
B
3
2
0

25. 0
Suppose that what evolves in an evolutionary game are
strategies (both pure and mixed) in a one-
shot game depicted above.
Is pure strategy A an ESS? (Circle one.) YES NO
Is pure strategy B an ESS? (Circle one.) YES NO
A
B
A
2
2

26. 1
1
B
0
0
1
1
Problem Set 6, page 10 of 12
(2) Consider the following coordination game:
A

27. B
A
2
2
0
0
B
0
0
1
1

28. Suppose that what evolves in an evolutionary game are
strategies (both pure and mixed) in a one-
shot game depicted above.
Is pure strategy A an ESS? (Circle one.) YES NO
Is pure strategy B an ESS? (Circle one.) YES NO
(3) Consider the following Chicken aka Hawk-Dove game
(strategies denote choosing
“aggress” (B) or “not aggress” (A)):
A
B
A
2
2
5

29. 1
B
1
5
0
0
Suppose that what evolves in an evolutionary game are
strategies (both pure and mixed) in a one-
shot game depicted above.
Is pure strategy A an ESS? (Circle one.) YES NO
Is pure strategy B an ESS? (Circle one.) YES NO

30. Problem Set 6, page 11 of 12
Problem 11 (3p)
A
B
A
1
1
2
3
B

31. 3
2
0
0
Consider now an evolutionary game where what evolves are not
strategies in a one-shot game
but strategies in a repeated game without discounting. The
stage game of the repeated game is
given above. Consider only pure strategies in this repeated
game. (WESS stands for weak ESS.)
(a) Is strategy ALL A (“play A unconditionally in all iterations
of the game”) an WESS?
(Circle one.) YES NO
(b) Is strategy ALL B (“play B unconditionally in all iterations
of the game”) an WESS?
(Circle one.) YES NO
(c) Is strategy RECIPROCATE (“play A in period 1 and then
play what the other player

32. played in the previous period”) an WESS? (Circle one.) YES
NO
Problem 12 (3p)
A
B
A
2
2
0
0
B

33. 0
0
1
1
(a) Is strategy ALL A (“play A unconditionally in all iterations
of the game”) an WESS?
(Circle one.) YES NO
(b) Is strategy ALL B (“play B unconditionally in all iterations
of the game”) an WESS?
(Circle one.) YES NO
(c) Is strategy RECIPROCATE (“play A in period 1 and then
play what the other player
played in the previous period”) an WESS? (Circle one.) YES
NO
Problem Set 6, page 12 of 12
In the problems below assume that the evolutionary games

34. involve an infinitely repeated
Prisoner’s Dilemma with one-shot payoffs of 0, 1, 3 and 5 and
without discounting (payoffs are
calculated as per-period average.)
Problem 13 (4p)
Consider now the following strategies in an infinitely repeated
Prisoner’s Dilemma game:
ALL D: defect unconditionally in all iterations of the game;
STFT (suspicious tit for tat): defect
in the first interaction and then cooperate if the other player
cooperated on the previous
interaction and defect if he defected; MACHO: defect in the
first iteration; in second iteration
cooperate if the other player defected in the first iteration and
then play “tit for tat”; if he
cooperated in the first iteration, defect unconditionally in all
periods of the game from iteration 2
onwards; TF2T (tit for two tats): cooperate in periods 1 and 2.
Thereafter defect in any period
k>2 if and only if your opponent defected in k-1 and k-2.
Which strategies from the set above are weak ESS (WESS)?
List all strategies that are weak

35. WESS:
…………………………………………………………………………
……………………………
Problem 14 (2p)
What is the relationship between Nash equilibrium and WESS?
More specifically,
1. Can you find a strategy s such that (s, s) is in Nash
equilibrium and s is not WESS?
(Circle one.) YES NO
2. Can you find a strategy s which is WESS and (s, s) is not a
Nash equilibrium? (Circle
one.) YES NO
Problem 15 (4 extra credit points)
Can you find a strategy s such that u(s, s) = 3, (s, s) is in Nash
equilibrium, and s is not WESS?
Prove your claim.