Game Theory in Economics Exam
DIRECTIONS: Answer each question to the best of your ability. This exam has two parts. Part I
consists of 4 common questions. Part II consists of a pool of 2 questions – please choose and
complete one. All point values are indicated below, and partial credit is available throughout. It
is your responsibility to indicate your final answer to each question.
Name:
N#:
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PART I: Answer all 4 questions (70 points)
1. (20 points) Answer each of the following. Your responses should be no longer than a brief
paragraph.
(a) Distinguish between a “first-mover advantage” and a “second-mover advantage.” Be sure to
explain what is meant by an ‘advantage.’
(b) What is the most important difference between simultaneous-move and sequential-move
games?
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(c) What makes the Prisoners’ Dilemma so famous, important, and widely studied?
(d) What is the role of patience in sustaining cooperation in repeated games?
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FIGURE 1 Joy
Stag Hare
Suzanne
Stag (5, 5) (0, 4)
Hare (4, 0) (4, 4)
2. (20 points) Use FIGURE 1 to answer the questions below.
Consider this simultaneous-move hunting game between Suzanne and Joy (first presented by
Jean-Jacques Rousseau). Each player must choose independently whether to target the Stag or to
target the Hare. Payoffs represent player utilities, and they reflect the fact that a single hunter can
successfully harvest a hare, but a successful stag hunt requires two hunters. Payoffs are
(Suzanne, Joy).
(a) Find all Nash equilibria in pure strategies. If there are none, write “none.”
(b) For each equilibrium you identify in (a), determine whether it is consistent with (i)
distributive equality, (ii) Kaldor-Hicks efficiency, and (iii) Pareto efficiency. If you did not find
any Nash equilibria in (a), then write “none” here.
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(c) In explaining socially suboptimal outcomes, Watson (2013) defines the “First Strategic
Tension” as the clash between individual and group interests. To what extent is this tension
present in this game? Explain. (Hint: if Suzanne and Joy are stuck with hares, is it due to a
fundamental conflict of interest?)
(d) In explaining socially suboptimal outcomes, Watson (2013) defines the “Second Strategic
Tension” as strategic uncertainty – that is, players’ uncertainty about other players’ strategy
choices. To what extent is this tension present in this game? Explain. (Hint: if Suzanne and Joy
are stuck with hares, is it due to the game being simultaneous rather than sequential?)
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FIGURE 2 Player Beta
Payoffs are (Alpha, Beta)
X Y Z
Player
Alpha
A (5, 6) (3, 7) (0, 4)
B (8, 3) (3, 1) (5, 2)
C (7, 5) (4, 4) (5, 6)
D (3, 4) (7, 5) (3, 3)
3. (15 points) Use FIGURE 2 to answer the questions below.
(a) Is this a constant-sum or variable-sum game? How do you know?
(b) Find all IEDS equilibria. If there are none, write “none.”
(c) Find all Nash equilibria in pure strategies. If there are none, write “none.”
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(d) If you found zero or one Nash equilibrium in part (c), explain why this solution is unique. If
you found multiple Nash equilibria in part (c), make a case for which equilibrium you think is
most likely to occur.
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4. (15 points) Use FIGURE 3 (next page) to answer the questions below.
(a) Write down all strategies available to each player.
(b) Find the rollback (SPNE) equilibrium. What are the equilibrium payoffs?
(c) Find the rollback (SPNE) equilibrium. What is the equilibrium path?
(d) Find the rollback (SPNE) equilibrium. What are the equilibrium strategies?
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Tom
Jerry
Tom
Jerry
(2, 5, 0)
UP
MIDDLE
(5, 4, 3)
(0, 4, 0)
(6, 2, 2)
(2, 6, 4)
NO
YES
NO
YES
RIGHT
LEFT
DOWN
(1, 1, 2)
Butch
(4, 2, 5)
EVEN
ODD
FIGURE 3 – payoffs are (Tom, Jerry, Butch)
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PART II: Choose ONE of the following 2 questions (30 points)
FIGURE 4a Jasmine
Payoffs are (Rose, Jasmine)
Search Enjoy Distractions
Rose
Support (4, 2) (-1, 4)
No support (-1, 1) (0, 0)
5. Having enjoyed a magical study abroad experience, Jasmine has decided to return to Europe
after graduation to pursue her job search. Rose wishes to support Jasmine’s efforts, financially
and otherwise, but she knows that Europe has many distractions that can undermine Jasmine’s
search. Consider the resulting simultaneous-move game presented in FIGURE 4a where
payoffs are utility scores.
(a) Find all Nash equilibria in pure strategies.
(b) Find all Nash equilibria in mixed strategies.
(c) Calculate the expected payoff of each player using your solution in part (b).
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FIGURE 4b Jasmine
Payoffs are (Rose, Jasmine)
Search Enjoy Distractions
Rose
Support (4, 2) (x , 4)
No support (-1, 1) (0, 0)
Now consider a more general version of the game in which x represents Rose’s payoff when her
support is wasted while Jasmine enjoys distractions. All other details of the game remain the
same. Use the resulting game in FIGURE 4b to answer the questions below.
(d) Find all Nash equilibria in mixed strategies. Write your answers as functions of x as needed.
(e) Calculate the expected payoff of each player using your solutions in part (d). Write your
answers as functions of x as needed.
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(f) What is the relationship between the value of x and the probability that Jasmine will choose to
search? Explain completely.
(g) For what value(s) of x does this game have a Nash equilibrium in pure strategies?
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6. Consider (again) the tutoring problem where graduate students Caroline and Mason act as
Cournot duopolists within the market for tutoring at their university. Weekly consumer demand
is given byQ D = 100 p− 2 . Caroline has cost function ??(????) = 15???? and Mason has cost
function ??(????) = 10????.
(a) Set up and solve the game in which Caroline and Mason choose quantities simultaneously.
Calculate the values of ????, ????, ??, ???? , ???? in the Nash equilibrium.
(b) This is a game with asymmetric players and so your solutions should be asymmetric. Are
your solutions consistent with your economic intuition? Explain.
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(c) If Caroline and Mason could effectively collude and agreed to split profits evenly, then it can
be shown that the equilibrium values would be p = 30, ???? + ???? = 800, ???? = 0 and ???? = 40.
This means that Mason would pay Caroline 400 to do nothing, literally. Would Mason agree to
this contract? Explain completely.
(d) As with any sibling, Mason might wonder why he should necessarily split the profits evenly.
Identify the range of possible payments from Mason to Caroline that would support this collusive
equilibrium. Be sure to justify your answer.
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