SPRING 2020 ECO 4400

SPRING 2020 ECO 4400
EXAM 1
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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