Name
Discussion of Answers: Final Exam, Econ 171, March, 2013
Problem 1) The following game is based on events during World War II in
the “Battle of the Bismarck Sea.” Admiral Imamura of the Japanese Navy
was ordered to transport troops across the Bismarck Sea to New Guinea.
The fleet could travel either the northern route which is likely to be foggy or
the southern route which is likely to be clear. U.S. Admiral Kenney hopes
to bomb the troop ships. Kenny must decide whether to concentrate his
reconnaissance flights on the northern or the southern route. Once he has
found the troops he can start bombing. If Kenny guesses correctly, he can
inflict more damage. The payoffs from this game are shown below, where
Kenney chooses the row and Imamura chooses the column. The first number
in each box is Kenney’s payoff. The second number is Imamura’s.
Table 1: Battle of the Bismarck Sea
North South
North 2,-2
2,-2
South 1,-1
3,-3
A) Does this game have a strictly dominant strategy?
No
B) Do either of these strategies weakly dominate any others for either player?
Explain.
Going North weakly dominates Going South for Imamura.
C) Does this game have one or more pure strategy Nash equilibria? If so,
describe what is it (or are they).
The only Nash equilibrium is one in which Imamura goes North and
Kenney goes North.
D) Does this game have a mixed strategy Nash equilibrium that is not a
pure strategy Nash equilibrium? Explain your answer.
Since going North weakly dominates South for Imamura, if there were a
positive probability that Kenney goes South, the expected payoff from going
North would be higher for Imamura than that of going South. If Kenney
goes North with probability less than 1, then Imamura will surely go North.
But if Imamura goes North, Kenney’s best response is to go North for sure.
Therefore in any mixed strategy Nash equilibrium, Kenney must go North
for sure. If Kenney is sure to go North, then Imamura is indifferent between
North and South. If the probability that Imamura goes North is greater than
or equal to 1/2, then Kenney’s best response is to go North. So there are
Nash equilibria where Kenney goes North for sure and Imamura goes North
with probability p where 1/2 ≤ p ≤ 1.
Problem 2) Three hunters play a stag hunt game. There are two possible
strategies, Stag and Hare. Anyone who plays Hare will get a payoff of 1, no
matter what the others do. If at least two hunters play the strategy Stag,
then each player who played Stag will get a payoff of S/n where n is the
number of hunters who played Stag. If one hunter plays Stag and the other
two play Hare, then the hunter who played Stag gets 0.
A) If S > 3, find all of the pure strategy Nash equilibria for this game.
There is one equilibrium where all hunters play Hare and one where all
hunters play Stag.
B) If 2 < S < 3, find all of the pure strategy Nash equilibria for this game.
There is one Nash equilibrium in which all hunters play Hare. There are
3 other Nash equilibria. In each of these, two of the hunters play Stag and
the remaining hunter plays Hare.
C) Suppose that there is a symmetric mixed strategy equilibrium where each
hunter plays Stag with a probability p such that 0 < p < 1. Find a quadratic
equation in p that must be satisfied at such an equilibrium.
p2
2p(1 − p)
S+ S=1
2
3
which implies
2
p2 S − pS + 1 = 0
3
D) Looking at the solution to C, find the range of values of S for which there
is a symmetric mixed strategy equilibrium with 0 < p < 1.
Apply the quadratic formula to solve the equation in Part C for p.
p=
S±
q
S 2 − 83 S
4
S
3
There is a real valued solution for p in the range from 0 to 1 if and only
S > 8/3.
E) If S = 3, find a symmetric mixed strategy equilibrium that is not a pure
strategy equilibrium.
Try S = 3 in the quadratic in part D. The only solution between 0 and 1
is p = 1/2.
Problem 3) Players A and B play a game in which there are two possible
strategies for each of them. Let’s call these strategies Hawk and Dove. If
both players play Dove, they each get a payoff of 3. If both play Hawk, they
each get a payoff of 0. If one player plays Hawk and the other plays Dove,
the one who played Hawk gets a payoff of 4 and the one who played Dove
gets a payoff of 1.
A) Display this game in strategic form.
Table 2: Hawk and Dove
Hawk
Dove
Hawk Dove
0,0
4,1
1,4
3,3
B) Find all of the pure strategy and mixed strategy Nash equilibria for this
game.
There are two pure strategy equilibrium profiles: A plays Hawk, B plays
Dove and A plays Dove and B plays Hawk. There is one symmetric mixed
strategy in which each player plays Dove with probability 1/2.
B) Suppose that each of the two players cares about the other player’s payoff,
so that each player seeks to maximize the sum of his own payoff plus a times
the other player’s payoff. Show how the set of pure strategy and mixed
strategy Nash equilibria depends on the size of the constant a.
In this case, the payoff matrix becomes:
Table 3: Altruistic Hawk and Dove
Hawk
Dove
Hawk
0,0
4+a,1+4a
Dove 1+4a,4+a 3+3a,3+3a
The strategy profile {Dove, Dove} is the only pure strategy Nash equilibrium if 3 + 3a > 4 + a and 4a + 1 > 0. This is the case if a > 1/2. If a <
−1/4, then {Hawk, Hawk} is a pure strategy Nash equilibrium. If −1/4 <
a < 1/2, then the only pure strategy Nash equilibria are {Hawk,Dove} and
{Dove,Hawk}. If −1/4 < a < 1/2, there is a mixed strategy Nash equilibrium with
1 1 + 4a
.
p=
2 1+a
Outside this range of a, the only Nash equilibria are pure strategy Nash
equilibria.
C) Consider an infinitely repeated game in which the stage game is the game
described above and where the discount rate is δ. For what values of δ is
there a subgame perfect Nash equilibrium strategy profile that results in a
course of play where both players play Dove forever? What strategy profile
will do this?
Suppose that each player uses the grim strategy, Play Dove on the first
round. Continue to play Dove, so long as nobody has ever played Hawk. As
soon as the other player plays Hawk, Play Hawk forever. The discounted
payoff from this strategy is 3/(1 − δ). The best deviation from this strategy
would be to play Hawk on the first round, then play dove ever after. The
δ
. The grim strategy is a best
discounted payoff from this strategy is 4 + 1−δ
response if the other player is playing the grim strategy if
3
δ
>4+
1−δ
1−δ
which is the case if δ > 1/3.
Problem 4) The saga of Alice and Bob and Bob continues. . . You will recall
that Alice likes Movie A, Bob likes Movie B, and they both like each other.
Alice gets a payoff of 3 if both she and Bob go to Movie A. She gets a payoff
of 2 if they both go to Movie B. She gets a payoff of 1 from going to Movie
A without Bob and a payoff of 0 from going to Movie B without Bob. Bob
gets a payoff of 2 if both he and Alice got to Movie A, a payoff of 3 if they
both go to Movie B, a payoff of 1 from going to Movie B alone and a payoff
of 0 from going to Movie A alone. Alice and Bob both have to choose which
movie to go to without knowing which movie the other will attend.
You may also remember Carol. Carol likes Bob and she will invite him
to Movie C if she happens to be in town this weekend. (Whether she will be
in town is out of her control, so we don’t need to worry about her decision
process.) If Carol invites Bob, he has three options. He can decline Carol’s
offer and go to Movie A. He can decline Carol’s offer and go to Movie B. He
can accept Carol’s offer and go to Movie C. If Carol does not invite Bob,
then Bob can go to either Movie A or Movie B, but not to Movie C. Alice
has to decide which movie to go to without knowing whether Carol invited
Bob. Alice knows that the probability that Carol will invite Bob is c. One
thing has changed from the last time we visited them. Bob has grown fond
of Carol. Bob now gets a payoff of 4 from going to Movie C with Carol.
A. Draw an extensive-form game tree for this game, showing the payoffs
to Bob and Alice at the terminal nodes.
B. Construct a strategic form table for the game between Alice and Bob,
showing expected payoffs, given that Carol invites Bob with probability c.
Don’t bother to fill in the payoffs for strategies that are strictly dominated
for Bob.
Alice has only two possible strategies, A or B. Bob has 6 possible strategies. If Carol invites him, there are three things he can do and if Carol
doesn’t invite him there are two things. We write x/y to represent ”Go to
x if Carol invites him and go to y if she doesn’t.” In the table below, Alice
chooses the row and Bob chooses the column. I haven’t filled in the payoffs
in the first four columns because the strategies of going to A or B if invited
by Carol are dominated for Bob by going to C.
Table 4: Alice, Bob, and Carol
A/A
A/B B/A B/B
A
B
C/A
C/B
c+3(1-c),4c+2(1-c)
1,4c+(1-c)
0,4c
2(1-c),4c+3(1-c)
Simplifying the terms in this table, we have
Table 5: Alice, Bob, and Carol (simplified)
A/A
A
B
A/B B/A
B/B
C/A
3-2c,2+2c
0,4c
C/B
1,1+3c
2-2c,3+c
C. Find all of the subgame-perfect Nash equilibria in which Alice and Bob
use pure strategies if c < 1/2.
If c < 1/2, then there are two subgame perfect Nash equilibria. In one
of them, Alice goes to A and Bob goes to C if Carol invites him and to A
if Carol does not. In the other, Alice goes to B and Bob goes to C if Carol
invites him and to B if she does not.
D. Find all of the pure strategy subgame-perfect Nash equilibria in which
Alice and Bob use pure strategies if c > 1/2.
If c > 1/2, there is only one subgame-perfect Nash equilibrium. In this
equilibrium Alice goes to A and Bob goes to C if Carol invites him and to A
if she does not.
5. Two firms are located in the same shopping center. Advertising expenditure by either firm attracts customers to the shopping center and helps both
firms. Each firm can choose any positive level of advertising expenditure.
Where a1 is advertising expenditure of firm 1 and a2 is advertising expenditure of firm 2, the profits of firm 1 are given by the function a1 (100 + a2 − a1 )
and profits of firm 2 are given by a2 (100 + a1 − a2 ).
A. Suppose that the two firms choose their advertising levels simultaneously. What is the Nash equilibrium level of advertising for each firm? What
is the profit of each firm?
In Nash equilibrium, each firm chooses the expenditure that maximizes its
profits given what the other has chosen. Profits of firm 1 are 100a1 +a2 a1 −a21 .
This is maximized when its derivative with respect to a1 is zero. (Note that
the second derivative is negative.) This happens when
100 + a2 − 2a1 = 0.
This implies that a1 = 50 + a2 /2. Profits of firm 2 are 100a2 + a1 a2 − a22 . Set
the derivative equal to zero and solve for a2 . We find that a2 = 50 + a1 /2.
Now we have two equations in the two unknowns a1 and a2 . Solve these for
a1 and a2 . Solving these, we find a1 = a2 = 100. Then profits of each firm
are 100 × 100 = 10, 000.
B. Suppose that instead of moving simultaneously, firm 1 moves first and
chooses its advertising expenditure a1 . Firm 2 observes firm 1’s choice of
a1 and then chooses a2 . Describe firm 2’s best response function to firm 1’s
strategy.
If firm 1 chooses a1 , then firm 2’s best response is to choose a2 = 50+a1 /2.
C. Find a subgame perfect Nash equilibrium when firm 1 moves first and
chooses its advertising expenditure a1 and firm 2 observes firm 1’s choice of
a1 and then chooses a2 . What are the profits of each firm in this case?
If firm 1 moves first and chooses a1 , then in a subgame perfect Nash
equilibrium, firm 2 does its best response to firm 1’s action. Profits of firm
1 will then be
a1 (100 + (50 + a1 /2) − a1 ) = 150a1 − a21 /2.
These are maximized when 150 = a1 . In this case, a2 = 50 + a1 /2 = 125.
Then profits of firm 1 will be 150(100 + 125 − 150) = 150 × 75 = 11, 250 and
profits of firm 2 will be 125(100 + 150 − 125) = 125 × 125 = 15, 625.
6. Consider an infinitely repeated version of the stage game shown In Figure
6 below. In the stage game, each player can take one of three actions, A, B,
and C. Assume that each player’s payoff is the present value of her payoff
stream and the discount factor is δ.
Table 6: Figure 6
A
B
C
A 9,9 15,2 1,1
B 2,15 10,10 2,17
C 1,1 17,2 3,3
A. For what values of δ (if any) is there a subgame perfect Nash equilibrium
in which each player plays the strategy:
Play B on every round. Explain your answer.
There is no Nash equilibrium in which each player plays B on every round,
regardless of the size of δ. If one player is playing B on every round, the
other player’s best response would be to play C on every round.
B. For what values of δ (if any) is there a subgame perfect Nash equilibrium
in which each player plays the strategy:
Play C on every round. Explain your answer.
For any non-negative value of δ, there is a subgame perfect Nash equilibrium in which both players play C on every round. We see that the stage
game has a Nash equilibrium with both players playing C. If the other player
is playing C on every round, no matter what you do, your best response is
to play C on every round.
C. For what values of δ (if any) is there a subgame perfect Nash equilibrium
in which each player plays the strategy:
Play B on the first round and continue to play B so long as on all previous
rounds, both players have always played B. If the other player ever plays
anything other than B, then play C on all remaining rounds.
This works for all δ > 1/2. To see this, note that if the other player
is playing this strategy, then if you play the same strategy, you will both
play B forever and your discounted payoff will be 10/(1 − δ). The best
deviation from this strategy that you could do would be to play C on the
first round and then continue to play C because the other guy has moved to
C forever. The payoff from this deviation is 17 + 3δ/(1 − δ). We find that
10/(1 − δ) > 17 + 3δ/(1 − δ) if and only if δ > 1/2.
D. For what values of δ (if any) is there a subgame perfect Nash equilibrium
in which both players play the strategy:
Play B on the first round. Continue to play B so long as on all previous
rounds, both players have played only B. If the other player ever plays anything other than B, then play A forever if the deviation was to A and Play
C forever if the first deviation was to C.
For this we need δ > 5/6. Otherwise it would pay to deviate to A and
continue to play A forever. (Note that the strategy profile where each plays
A forever, no matter what, is a Nash equilibrium.)
7. Consider a finitely repeated version of the stage game shown in Figure
6 of the previous problem.
A. Is there a subgame perfect Nash equilibrium in which Player 1 plays
A and player 2 plays A on every round? Is there a subgame perfect Nash
equilibrium in which Player 1 plays B and player 2 plays B on every round?
Is there a subgame perfect Nash equilibrium in which Player 1 plays C and
player 2 plays C on every round? Explain your answers.
Since the best response to A is A, if the other player plays A on every
round, the best response is to play A on every round, so there is a subgame
perfect Nash equilibrium in which Player 1 plays A and player 2 plays A on
every round. c If the other player plays B on every round, the best response
on the last round must be B. So in a finitely repeated game, there can’t be
a Nash equilibrium where everyone plays B on every round.
Since the best response to C is C, if the other player plays C on every
round, the best response is to play C on every round, so there is a subgame
perfect Nash equilibrium in which Player 1 plays C and player 2 plays C on
every round.
B. Find a strategy profile that is a subgame perfect Nash equilibrium and
results in an outcome path in which each player chooses the strategy B on
all rounds before the last two rounds and where each plays A on the last two
rounds.
Each player plays the strategy “Play B on the first round and on all
rounds before the last two rounds, continue to play B so long as nobody has
ever played anything other than B. If somebody plays something other than
B before the last two rounds, play C for all remaining rounds. If nobody has
played anything other than B before the last two rounds, then on the last
two rounds, play A.
If the other player is using this strategy, the best deviation from this
strategy would be to play C on the last 3 rounds. In this case, you would get
a payoffs of 17, 3, and 3 on the last 3 rounds for a total of 23. If you stuck
with the proposed strategy, your payoffs on the last 3 rounds would be 10,
9, and 9, for a total of 28.
C. Is there a strategy profile that is a subgame perfect Nash equilibrium
and results in an outcome path in which each player chooses the strategy
B on all rounds before the last round and where each plays A on the last
round?
No. Whatever strategy the other guy is using, if he plays B on the next
to the last round, you could increase your payoff from 10 to 17 on the next
to the last round by playing C on the next to the last round. In a subgame
perfect equilibrium, the actions of the two players in the last round must
constitute a Nash equilibrium for the stage game. This means that in the
last round, the two of you must each play C or each play A. The difference in
payoffs between these two equilibria is only 6. This is not enough to induce
anyone to play B rather than C when the other guy is playing B.
8. (This one is probably more challenging than the others. Attempt it only
after you are satisfied with your answers to the other questions.) Consider
the stage game described in Problem 3 and suppose that this stage game is
played exactly 3 times and that payoffs to each player are the sum of the
payoffs in the three rounds.
A. Is the following strategy profile a subgame perfect Nash equilibrium?
Explain your answer.
Player 1’s strategy: Player 1 plays Dove on the first round. No matter
what Player 2 does in the first two rounds, Player 1 will play Hawk on the
third round. If Player 2 played Dove on the first round, Player 1 will play
Dove on the second round. If Player 2 plays Hawk on the first round, Player
1 will play Hawk on the second round.
Player 2’s strategy: Player 2 plays Dove on the first round. If Player 1
plays Dove on the first round, Player 2 will play Hawk on the second round
and Dove on the third round. If Player 1 plays Hawk on the first round, Player
2 will play Hawk on both round 2 and round 3, no matter what Player 1 does
in round 2.
Yes, this is a subgame perfect equilibrium profile. First we note that
in the subgame consisting of the last two rounds, these strategies prescribe
Nash equilibrium actions. Each is doing a best response to the strategy of the
other. How about the first round? If the players play these strategies, then
Player 1’s total payoff is 3+4+1=8 and Player 2’s total payoff is 3+1+4=8.
Suppose Player 1 deviates by playing Hawk on the first round. Then Player
2 will play Hawk on both remaining rounds. Then best Player 1 can do then
is to play Dove on the last two rounds. Player 1’s toal payoff would then be
4+1+1=6¡8. So it doesn’t pay Player 1 to deviate, given Player 2’s strategy.
Similarly if Player 2 deviates, the best he can do is to get 4+1+1=6¡8. So it
doesn’t pay either of them to deviate from the proposed strategy profile
B. If the game is repeated 20 times, can you find a subgame perfect Nash
equilibrium strategy profile such that the outcome path has the two players
both playing dove for the first 18 times? Can you find a subgame perfect
Nash equilibrium strategy profile such that the outcome path has the two
players both playing dove for the first 19 times?
Here is a subgame perfect strategy profile that generates a course of play
in which both play dove on the first 18 rounds.
The strategy for Player 1: Play Dove on the first round and for the first 18
times, continue to play Dove if the other player has played dove every time.
If the other player ever plays Hawk in the first 18 plays, move to playing
Hawk every time after this. If the other player does not deviate on the first
18 plays, then play Dove on the 19th round and Hawk on the 20th round.
The strategy for Player 2: Play Dove on the first round and for the first 18
times, continue to play Dove if the other player has played dove every time.
If the other player ever plays Hawk in the first 18 plays, move to playing
Hawk every time after this. If the other player does not deviate on the first
18 plays, then play Hawk on the 19th round and Dove on the 20th round.
There is no subgame perfect Nash equilibrium profile that would generate
a course of play in which each plays dove on the first 19 rounds. On the 20th
round, in a subgame perfect Nash equilibrium, one player must play Hawk
and the other must play Dove. But if this is the case, the player who is to
play Dove on the 20th round, could improve his total score by playing Hawk
on the 19th round.