Final Exam Practice
Problem 1 (10 pts). Briefly define the following terms using mathematical notation as appropriate.
(a) Weakly dominant strategy
(b) Mixed strategy
(c) Information set
(d) Subgame perfect Nash equilibrium
(e) Repeated game
(f) All-pay auction
Problem 2 (20 pts). True or False. For each statement below, state whether it is true or false, and justify your answer.
(a) All games with two players are dominance solvable.
(b) A mixed strategy is a best response if and only if every pure strategy in its support is itself a best response.
(c) An extensive form. game with perfect information has a unique backward induction solution
(d) In an infinitely repeated game with a fixed discount rate δ, a forgiving trigger with a given punishment length T can support the same set of behaviors in a subgame perfect Nash equilibrium as a grim trigger strategy.
(e) If an extensive form game has exactly one pure strategy Nash equilibrium, this Nash equilibrium is subgame perfect.
(f) In a second-price auction, the only equilibrium is for all players to bid their true values.
Problem 3 (20 pts). Three profit-maximizing firms engage in Cournot competition—they produce identical goods and simultaneously choose quantities. Suppose inverse demand is given by
p(Q) = max{0, 10 - Q}.
Suppose firm 1 has a constant marginal cost of 2, while firms 2 and 3 each have a constant marginal cost of 4. Answer the following:
(a) Suppose the three firms compete in a single period. What are the Nash equilibrium production quantities and profits for each firm? Is the Nash equilibrium you found unique?
(b) Suppose the three firms compete repeatedly over T periods, for some finite T. Find a subgame perfect Nash equilibrium of the repeated game. Is it unique?
(c) In the context of the T period repeated game, suppose that firms 2 and 3 are credit constrained—if either of these two firms ever makes 0 or negative profit in a period, that firm goes bankrupt and exits the game, and firm 1 continues against either one other firm, or possibly no other firms. Is there a pure strategy subgame perfect Nash equilibrium in which firm 1 drives firms 2 and 3 to exit?
Problem 4 (20 pts). Consider the following normal form game with two players:
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L
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R
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U
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(5, 5)
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(1, 6)
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D
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(6, 1)
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(0, 0)
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Answer the following:
(a) What are the pure strategy Nash equilibria of this game? Find a mixed strategy Nash equilibrium.
(b) Suppose the game is repeated twice. Find all pure strategy subgame perfect Nash equi- libria.
(c) Suppose the game is infinitely repeated, with discount rate δ ∈ (0, 1) . Consider the following strategy profile: play (U, L) in every period, and if any player ever deviates, play the mixed strategy Nash equilibrium of the stage game (the one you found in part (a)) thereafter. Show that this profile is a subgame perfect Nash equilibrium for sufficiently high δ, and find the range of δ for which this is a subgame perfect Nash equilibrium.
Problem 5 (30 pts). There is a worker and two firms. The worker can be one of two types, high (H) or low (L) . High type workers are more productive than low type workers—a firm earns πH from hiring a high-type worker and πL from hiring a low-type worker, with πH > πL > 0. Ex-ante, the worker is a high type with probability p. The worker observes her own type, but the firms do not—the firms only know the common prior p. However, the worker can choose to go to school and earn a degree. Going to school costs effort—the high type incurs effort cost eH , the low type incurs effort cost eL , and eL > eH > 0. That is, high type workers are more productive and find going to school less costly.
The worker and the firms engage in a dynamic game. First, the worker chooses whether to incur the effort cost to earn a degree. After observing the education choice (degree or not), the two firms simultaneously make wage offers w1 and w2 to the worker—assume the worker accepts the highest wage offer, and if the two offers are equal, the worker flips a coin to decide which offer to accept. If the worker accepts wage w and incurs effort cost e, her payoff is w - e. If the firm hires a worker of type t and pays w, the firm earns πt - w. The other firm earns 0. Answer the following:
(a) Suppose at first that there is no school, and the two firms just make simultaneous wage offers. What offers do the two firms make in a Bayes-Nash equilibrium? Suppose that, after the worker’s education choice, the firms both believe the worker is a high type with probability q. What offers do the two firms make in equilibrium?
(b) In the dynamic game with a school, is there a perfect Bayesian equilibrium in which both types of worker earn degrees? Justify your answer.
(c) Find a perfect Bayesian equilibrium in which high type workers earn degrees and low type workers do not. What needs to be true about the effort costs and productivities? What wage do the firms offer to a worker with a degree? What wage do the firms offer to a worker with no degree? What do the firms believe about the worker’s type in each case? (Note: we call this a “separating equilibrium”)
(d) Find a perfect Bayesian equilibrium in which neither type of worker earns a degree. What wage do the firms offer? What do the firms believe about the worker’s type? What are the off-path beliefs for the firms (i. e. what do they infer about a worker who unexpectedly earns a degree)? (Note: we call this a “pooling equilibrium”)
(e) Suppose the school charges a fee t for earning a degree, on top of the effort cost. How does this change your answer to part (c)? How high can the tuition t grow before the only equilibrum is a pooling equilibrium?