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程序代写案例-OLUTION 2018
时间：2022-07-04
FINAL EXAM - SOLUTION 2018
1. Consider the chicken game represented by the following payoff table.
COLUMN
SLOW SPEED
ROW SLOW 2; 2 0; 4
SPEED 4; 0 1;1
(a) Find and represent graphically the players’ best response functions, and show that the game
has three Nash equilibra:
• one in which ROW plays SPEED and COLUMN plays SLOW (let’s call it NE1),
• one in which ROW plays SLOW and COLUMN plays SPEED (let’s call it NE2),
• one in which both players play SLOW with probability 13 and SPEED with probability 23
(let’s call it NE3).
2 points
Let’s introduce the following notation: PrROW(SLOW) = p and PrCOLUMN(SLOW) = q. We
now can write the players’ expected payoffs.
EuROW(SLOW|q · SLOW + (1 q) · SPEED) = 2q
EuROW(SPEED|q · SLOW + (1 q) · SPEED) = 4q + (1) · (1 q) = 5q 1
EuCOLUMN(SLOW|p · SLOW + (1 p) · SPEED) = 2p
EuCOLUMN(SPEED|p · SLOW + (1 p) · SPEED) = 4p+ (1) · (1 p) = 5p 1
The players’ best-response function are:
p(q) =
8<: 1 if q <
1
3
0 if q > 13
[0, 1] if q = 13
, and
q(p) =
8<: 1 if p <
1
3
0 if p > 13
[0, 1] if p = 13
.
The three Nash equilibria of the game are:
NE1 = {SPEED, SLOW},
NE2 = {SLOW, SPEED}, and
NE3 =
⇢
1
3
· SLOW + 2
3
· SPEED, 1
3
· SLOW + 2
3
· SPEED
.
FINAL EXAM - SOLUTION 2018
Graphically:
(b) Compute each player’s expected payoff in each of the three Nash equilibria of the game.
1 point
In NE1, the payoffs are (4; 0).
In NE2, the payoffs are (0; 4).
In NE3, the payoffs are 13 · 13 · (2; 2) + 13 · 23 · (0; 4) + 23 · 13 · (4; 0) + 23 · 23 · (1;1) =
= 19 · [(2; 2) + 2 · (0; 4) + 2 · (4; 0) + 4 · (1;1)] = (69 ; 69) = (23 ; 23).
(c) Deduce that the game obtained by repeating the above chicken game twice has at least nine
subgame-perfect equilibria.
3 points
Note that we have a subgame-perfect Nash equilibrium in the twice repeated chicken game
if, irrespectively from the history of the game, in each stage players choose actions that con-
stitute a Nash equilibrium in the stage game.
In other words, we have a subgame-perfect Nash equilibrium in the twice repeated chicken
game if players play according to NEi in the first moment in time and play according to NEj
in each and every subgame that starts in the second moment in time (note that there are 4
such subgames).
Note that both i and j can take on three different values, so we have already constructed 9
equilibria. We can construct more by allowing players to play according different NEi in
different subgames that start in the second moment in time.
FINAL EXAM - SOLUTION 2018
2. Suppose that a class of 100 students is comparing two careers – lawyer or engineer. An engineer
gets a take-home pay of 100 thousand dollars per year, irrespective of the number of people who
choose this career. Lawyers make work for each other, so as the total number of lawyers increases,
the income of each lawyer increases – up to a point. Ultimately, the competition between them
drives down the income each earns.
Specifically, if there are N lawyers, each will get 100N N2 thousand dollars a year. The annual
cost of running a legal practice (office space, secretary, paralegals, access to online reference ser-
vices, and so forth) is 800 thousand dollars. Therefore, each lawyer takes home 100N N2 800
thousand dollars a year when there are N of them.
For simplicity, let’s assume that the class size of 100 students is large enough, so that any one per-
son makes only a very small difference on income levels.
(a) Draw a graph showing the take-home income of each lawyer and each engineer (on the verti-
cal axis) as a function of the number of lawyers (on the horizontal axis).
1 point
(b) When career choices are made in an uncoordinated way, what are the possible Nash equi-
librium outcomes? Indicate whether the Nash equilibria that you have found are stable or
unstable.
3 points
When N = 0, lawyers earn 800 thousand dollars while engineer make 100 thousand.
There are no lawyers and no one would want to become one. This points constitutes a Nash-
equilibrium outcome of the game (N1 = 0).
When N = 100, lawyers earn 800 thousand dollars while engineer make 100 thousand.
There are no engineers, but a lawyers would have incentives to switch to engineering. This
point does not constitute a Nash-equilibrium outcome of the game.
When the two professions earn the same income, we have other two Nash-equilibrium out-
comes of the game. 100N N2 800 = 100! N2 = 10 and N3 = 90.
FINAL EXAM - SOLUTION 2018
In order to determine the stability of the equilibria, note that for N < 10 and for N > 90
the expected payoff for engineers is larger than the expected payoff for lawyers. And that for
10 < N < 90, it is the other way around. These imply that from a situation with N < 10
or N > 90, the number of lawyers would tend to decrease, while from a situation with 10 <
N < 90, the number of lawyers would tend to increase. These results are illustrated with
arrows along the horizontal axis. We then can conclude that N1 = 0 and N3 = 90 are stable
equilibrium outcomes, but N2 = 10 is unstable.
(c) Now suppose that the whole class decides how many should become lawyers, aiming to
maximize the total take-home income of the whole class. What will be the number of lawyers?
• Find the mathematical expression that shows the take-home income of the whole class
as a function of the number of lawyers.
T (N) = N · (100N N2 800) + (100N) · 100 = N3 + 100N2 900N + 10 000
• If you know calculus, find the function’s maximum point.
max
N
T (N) = N3 + 100N2 900N + 10 000
FOC : 3N2 + 200N 900 = 0
SOC : 6N + 200 < 0
N⇤ = 103 (10 +
p
73) ⇡ 61.8! 62
• If you do not know calculus, show that the take-home income of the whole class is larger
when the number of lawyers is 60 (than in the stable equilibrium).
T (N) = N3 + 100N2 900N + 10 000
T (60) = 100 000
T (0) = 10 000
T (80) = 66 000
2 points
FINAL EXAM - SOLUTION 2018
3. In a simple version of the holdup problem, Alice has $3 million, which she is thinking of invest-
ing in Bob’s company. If she makes the investment, Bob can either work or slack. If he slacks, he
consumes Alice’s investment, and she gets nothing. If he works, Alice doubles her investment, and
Bob nets $2 million.
(a) Represent the game in extensive form, and find its subgame-perfect Nash equilibrium.
2 points
The subgame-perfect Nash equilibrium of the game is {DON’T INVEST; SLACK}.
(b) Represent the game in strategic form, and find all its pure-strategy Nash equilibria.
2 points
BOB
WORK SLACK
ALICE INVEST 6 ; 2 0; 3
DON’T INVEST 3; 0 3 ; 0
The game has only one Nash equilibrium in pure strategies. It is {DON’T INVEST; SLACK}.
FINAL EXAM - SOLUTION 2018
4. Suppose that the following payoff table represents the ultimate chicken game between the two
nuclear superpowers during the Cold War.
USSR
WAIT ATTACK
USA
WAIT
mutual stand-off USSR wins
0; 0 -1; 1
ATTACK
USA wins mutual destruction
1; -1 -1000; -1000
Recall the following words from the Dr. Strangelove movie.
President Merkin Muffley:
“But, how is it possible for this thing to be triggered automatically, and at the same time
impossible to untrigger?”
Strangelove:
Mr. President, it is not only possible, it is essential. That is the whole idea of this ma-
chine, you know. Deterrence is the art of producing in the mind of the enemy ... the fear
to attack. And so, because of the automated and irrevocable decision making process
which rules out human meddling, the doomsday machine is terrifying. It’s simple to
understand. And completely credible, and convincing.
• Show how such a doomsday machine changes the above game in the following two hypothet-
ical situations. Represent the game with a payoff table, explain players’ incentives and the
resulting equilibrium in each situation.
Situation (a) The USSR has a doomsday machine, but the USA does not have a doomsday machine.
2 points
In this situation an attack by the USA is going to be retaliated.
USSR
WAIT ATTACK
USA
WAIT
mutual stand-off USSR wins
0 ; 0 -1 ; 1
ATTACK
USA wins mutual destruction
-1000; -1 -1000; -1000
The players’ best responses are indicated with boxes around payoffs in the table above.
Now WAIT is a dominant strategy for the USA. The Nash equilibrium of the game is:
{WAIT; ATTACK}.
FINAL EXAM - SOLUTION 2018
Situation (b) Both the USSR and the US have a doomsday machine.
2 points
In this situation any attack by any of the players is going to be retaliated by the other.
USSR
WAIT ATTACK
USA
WAIT
mutual stand-off USSR wins
0 ; 0 -1 ; -1000
ATTACK
USA wins mutual destruction
-1000; -1 -1000; -1000
The players’ best responses are indicated with boxes around payoffs in the table above.
Now WAIT is a dominant strategy for both players. The Nash equilibrium of the game is:
{WAIT;WAIT}.
1. Consider the chicken game represented by the following payoff table.
COLUMN
SLOW SPEED
ROW SLOW 2; 2 0; 4
SPEED 4; 0 1;1
(a) Find and represent graphically the players’ best response functions, and show that the game
has three Nash equilibra:
• one in which ROW plays SPEED and COLUMN plays SLOW (let’s call it NE1),
• one in which ROW plays SLOW and COLUMN plays SPEED (let’s call it NE2),
• one in which both players play SLOW with probability 13 and SPEED with probability 23
(let’s call it NE3).
2 points
Let’s introduce the following notation: PrROW(SLOW) = p and PrCOLUMN(SLOW) = q. We
now can write the players’ expected payoffs.
EuROW(SLOW|q · SLOW + (1 q) · SPEED) = 2q
EuROW(SPEED|q · SLOW + (1 q) · SPEED) = 4q + (1) · (1 q) = 5q 1
EuCOLUMN(SLOW|p · SLOW + (1 p) · SPEED) = 2p
EuCOLUMN(SPEED|p · SLOW + (1 p) · SPEED) = 4p+ (1) · (1 p) = 5p 1
The players’ best-response function are:
p(q) =
8<: 1 if q <
1
3
0 if q > 13
[0, 1] if q = 13
, and
q(p) =
8<: 1 if p <
1
3
0 if p > 13
[0, 1] if p = 13
.
The three Nash equilibria of the game are:
NE1 = {SPEED, SLOW},
NE2 = {SLOW, SPEED}, and
NE3 =
⇢
1
3
· SLOW + 2
3
· SPEED, 1
3
· SLOW + 2
3
· SPEED
.
FINAL EXAM - SOLUTION 2018
Graphically:
(b) Compute each player’s expected payoff in each of the three Nash equilibria of the game.
1 point
In NE1, the payoffs are (4; 0).
In NE2, the payoffs are (0; 4).
In NE3, the payoffs are 13 · 13 · (2; 2) + 13 · 23 · (0; 4) + 23 · 13 · (4; 0) + 23 · 23 · (1;1) =
= 19 · [(2; 2) + 2 · (0; 4) + 2 · (4; 0) + 4 · (1;1)] = (69 ; 69) = (23 ; 23).
(c) Deduce that the game obtained by repeating the above chicken game twice has at least nine
subgame-perfect equilibria.
3 points
Note that we have a subgame-perfect Nash equilibrium in the twice repeated chicken game
if, irrespectively from the history of the game, in each stage players choose actions that con-
stitute a Nash equilibrium in the stage game.
In other words, we have a subgame-perfect Nash equilibrium in the twice repeated chicken
game if players play according to NEi in the first moment in time and play according to NEj
in each and every subgame that starts in the second moment in time (note that there are 4
such subgames).
Note that both i and j can take on three different values, so we have already constructed 9
equilibria. We can construct more by allowing players to play according different NEi in
different subgames that start in the second moment in time.
FINAL EXAM - SOLUTION 2018
2. Suppose that a class of 100 students is comparing two careers – lawyer or engineer. An engineer
gets a take-home pay of 100 thousand dollars per year, irrespective of the number of people who
choose this career. Lawyers make work for each other, so as the total number of lawyers increases,
the income of each lawyer increases – up to a point. Ultimately, the competition between them
drives down the income each earns.
Specifically, if there are N lawyers, each will get 100N N2 thousand dollars a year. The annual
cost of running a legal practice (office space, secretary, paralegals, access to online reference ser-
vices, and so forth) is 800 thousand dollars. Therefore, each lawyer takes home 100N N2 800
thousand dollars a year when there are N of them.
For simplicity, let’s assume that the class size of 100 students is large enough, so that any one per-
son makes only a very small difference on income levels.
(a) Draw a graph showing the take-home income of each lawyer and each engineer (on the verti-
cal axis) as a function of the number of lawyers (on the horizontal axis).
1 point
(b) When career choices are made in an uncoordinated way, what are the possible Nash equi-
librium outcomes? Indicate whether the Nash equilibria that you have found are stable or
unstable.
3 points
When N = 0, lawyers earn 800 thousand dollars while engineer make 100 thousand.
There are no lawyers and no one would want to become one. This points constitutes a Nash-
equilibrium outcome of the game (N1 = 0).
When N = 100, lawyers earn 800 thousand dollars while engineer make 100 thousand.
There are no engineers, but a lawyers would have incentives to switch to engineering. This
point does not constitute a Nash-equilibrium outcome of the game.
When the two professions earn the same income, we have other two Nash-equilibrium out-
comes of the game. 100N N2 800 = 100! N2 = 10 and N3 = 90.
FINAL EXAM - SOLUTION 2018
In order to determine the stability of the equilibria, note that for N < 10 and for N > 90
the expected payoff for engineers is larger than the expected payoff for lawyers. And that for
10 < N < 90, it is the other way around. These imply that from a situation with N < 10
or N > 90, the number of lawyers would tend to decrease, while from a situation with 10 <
N < 90, the number of lawyers would tend to increase. These results are illustrated with
arrows along the horizontal axis. We then can conclude that N1 = 0 and N3 = 90 are stable
equilibrium outcomes, but N2 = 10 is unstable.
(c) Now suppose that the whole class decides how many should become lawyers, aiming to
maximize the total take-home income of the whole class. What will be the number of lawyers?
• Find the mathematical expression that shows the take-home income of the whole class
as a function of the number of lawyers.
T (N) = N · (100N N2 800) + (100N) · 100 = N3 + 100N2 900N + 10 000
• If you know calculus, find the function’s maximum point.
max
N
T (N) = N3 + 100N2 900N + 10 000
FOC : 3N2 + 200N 900 = 0
SOC : 6N + 200 < 0
N⇤ = 103 (10 +
p
73) ⇡ 61.8! 62
• If you do not know calculus, show that the take-home income of the whole class is larger
when the number of lawyers is 60 (than in the stable equilibrium).
T (N) = N3 + 100N2 900N + 10 000
T (60) = 100 000
T (0) = 10 000
T (80) = 66 000
2 points
FINAL EXAM - SOLUTION 2018
3. In a simple version of the holdup problem, Alice has $3 million, which she is thinking of invest-
ing in Bob’s company. If she makes the investment, Bob can either work or slack. If he slacks, he
consumes Alice’s investment, and she gets nothing. If he works, Alice doubles her investment, and
Bob nets $2 million.
(a) Represent the game in extensive form, and find its subgame-perfect Nash equilibrium.
2 points
The subgame-perfect Nash equilibrium of the game is {DON’T INVEST; SLACK}.
(b) Represent the game in strategic form, and find all its pure-strategy Nash equilibria.
2 points
BOB
WORK SLACK
ALICE INVEST 6 ; 2 0; 3
DON’T INVEST 3; 0 3 ; 0
The game has only one Nash equilibrium in pure strategies. It is {DON’T INVEST; SLACK}.
FINAL EXAM - SOLUTION 2018
4. Suppose that the following payoff table represents the ultimate chicken game between the two
nuclear superpowers during the Cold War.
USSR
WAIT ATTACK
USA
WAIT
mutual stand-off USSR wins
0; 0 -1; 1
ATTACK
USA wins mutual destruction
1; -1 -1000; -1000
Recall the following words from the Dr. Strangelove movie.
President Merkin Muffley:
“But, how is it possible for this thing to be triggered automatically, and at the same time
impossible to untrigger?”
Strangelove:
Mr. President, it is not only possible, it is essential. That is the whole idea of this ma-
chine, you know. Deterrence is the art of producing in the mind of the enemy ... the fear
to attack. And so, because of the automated and irrevocable decision making process
which rules out human meddling, the doomsday machine is terrifying. It’s simple to
understand. And completely credible, and convincing.
• Show how such a doomsday machine changes the above game in the following two hypothet-
ical situations. Represent the game with a payoff table, explain players’ incentives and the
resulting equilibrium in each situation.
Situation (a) The USSR has a doomsday machine, but the USA does not have a doomsday machine.
2 points
In this situation an attack by the USA is going to be retaliated.
USSR
WAIT ATTACK
USA
WAIT
mutual stand-off USSR wins
0 ; 0 -1 ; 1
ATTACK
USA wins mutual destruction
-1000; -1 -1000; -1000
The players’ best responses are indicated with boxes around payoffs in the table above.
Now WAIT is a dominant strategy for the USA. The Nash equilibrium of the game is:
{WAIT; ATTACK}.
FINAL EXAM - SOLUTION 2018
Situation (b) Both the USSR and the US have a doomsday machine.
2 points
In this situation any attack by any of the players is going to be retaliated by the other.
USSR
WAIT ATTACK
USA
WAIT
mutual stand-off USSR wins
0 ; 0 -1 ; -1000
ATTACK
USA wins mutual destruction
-1000; -1 -1000; -1000
The players’ best responses are indicated with boxes around payoffs in the table above.
Now WAIT is a dominant strategy for both players. The Nash equilibrium of the game is:
{WAIT;WAIT}.
