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Final Exam: Solution
Date: March 25, 2009
Subject: Game Theory (ECO290E) Instructor: Yosuke YASUDA
1. Backward Induction (20 points, easy)
See the following game tree.
a) Translate this game into normal‐form and draw the corresponding payoff matrix. (6 points)
Hint: Remember that a strategy in dynamic games is a complete action plan. Answer: The payoff matrix below expresses the game in normal‐form.
1 / 2 CE CF DE DF
A 4, 1 4, 1 1, 0 1, 0
B 2, 3 3, 2 2, 3 3, 2
b) Find all pure‐strategy Nash equilibria. How many are there? (6 points) Answer: There are three Nash equilibria: (A, CE), (A, DF), (B, DE).
c) Solve this game by backward induction. (8 points)
Answer: (A, CE) is the unique outcome derived by backward induction, and the resulting payoff is (4, 1).
2. Subgame Perfect Nash Equilibrium (20 points, moderate)
See the following game tree.1
2
2 A
B
C
D E
F
(4, 1) (1, 0) (2, 3) (3, 2)
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a) How many information sets (containing two or more decision nodes) does this game have? (5 points)
Answer: There is a one information set which contains two or more decision nodes.
b) How many subgames (including the entire game) does this game have? (5 points) Answer: There are five sugbames including the entire game.
c) Find all (pure‐strategy) subgame perfect Nash equilibria. (10 points)
Answer: There are two subgame perfect Nash equilibria: (ADGI, LN) and (BDGI, KN). The resulting payoff is (2, 0) in the former and (3, 4) in the latter equilibrium.
3. Finitely Repeated Games (20 points, hard)
Consider the following two persons 3 x 3 game.1 / 2 X Y Z
A 5,5 8,4 0,0
B 4,8 7,7 1,9
C 0,0 9,1 0,0
a) Find all the pure‐strategy Nash equilibria of this game. (8 points) Answer: There are three Nash equilibria: (A, X), (B, Z), (C, Y).
1
2
2
1 1
1
A F
C
K
L
M
N
D E
G B
H
I H
I
(2, 0) (3, 4)
(1, 3)
(1, 1) (0, 4) (4, 0) (3, 3) (1, 4) (0, 2)
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b) Consider the two‐period repeated game in which the above stage game will be played twice. Suppose the payoffs are simply the sum of the payoffs in each stage game. Then, is there a subgame perfect Nash equilibrium that can achieve (B, Y) in the first period? If so, describe the equilibrium. If not, explain why. (12 points) Answer: Yes, the players can achieve (B, Y) in the first period by using the following (dynamic) strategy:
i. Play (B, Y) in the first period.
ii. If no one or both players deviate, then play (A, X) in the second period. iii. If only player 1 deviates, then play (B, Z) in the second period.
iv. If only player 2 deviates, then play (C, Y) in the second period.
Note that the all the strategies (potentially) played in the second period are Nash equilibria, so the above strategy does not rely on non‐credible threat. The key is to use (B, Z) for the punishment of player 1 and (C, Y) for that of player 2. Since the (maximum) deviation gain in the first period (2 = 9 – 7) is smaller than the punishment loss (4 = 5 – 1) in the second period, each player has no incentive to deviate from this dynamic strategy.
4. Bayesian Nash Equilibrium (20 points, tricky)
Each of two individuals receives a ticket on which there is an integer from 1 to 10 indicating the size of a prize ($) she may receive. Assume the payoff of receiving the prize $X is X. The individuals’ tickets are assigned randomly and independently; the probability of an individual receiving each possible number is 1/10. Each individual is given the option of exchanging her prize for the other individual’s prize; the individuals are given this option simultaneously. If both individuals wish to exchange, then the prizes are exchanged; otherwise each individual receives her own prize. Each individual’s objective is to maximize her expected monetary payoff.
a) Consider the above situation as a Bayesian game. Then, what are the individuals’ strategies? (6 points)
Answer: The strategy is a function from the received seize of a prize X to the binary decision (Exchange, Not). Since there are 10 possible prizes, each player has to decide whether or not to exchange in these 10 different occasions.
b) If an individual receives the ticket with $10, will she have an incentive to exchange or not? Explain why. (6 points)
Answer: No, she does not have such incentive. Suppose player 1 has an incentive to
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exchange. Then, it must be the case that player 2 is going to exchange only if he receives $10 (otherwise, player 1’s expected value from exchange becomes strictly less than $10). However, given that player 1 with $10 will exchange, player 2 with $1 always has a strict incentive to exchange since the expected value from exchange is strictly more than $1 for him, which lead to a contradiction. Therefore, in Bayesian Nash equilibria, players with $10 never exchange.
c) Solve for the Bayesian Nash equilibrium. Can the exchange happen in equilibrium? (8 points)
Answer: By (b), we know that players with $10 never exchange. Given that, it is straightforward to see that players with $9 never exchange, since the expected value from exchange is strictly lower than $9 (the same argument as in (b). This logic goes through until players with $2. That is, players with $2 or higher never exchange in Bayesian Nash equilibria. Players with $1 are indifferent between “exchange” and “not exchange” since the expected value from exchange is $1. Thus, trade can happen only when both players happen to receive $1. The Bayesian Nash equilibrium is:
Not exchange if a player receives $2 or higher prize, and either exchange or not exchange if she receives $1 (for each player).
5. Cournot Model (40points, moderate)
Suppose two firms produce an identical good. The (inverse) demand function for the good is given as P = 130 – Q, where Q is the total quantity produced by the two firms. Each firm has a constant marginal cost 10 of producing the good.
a) Suppose that firms compete as quantity setting duopolists. Find the Cournot Nash equilibrium of this game. What quantities will they produce, what is the market price and how much profit does each firm earn? (8 points)
Answer: q1 = q2 = 40, p = 50, π1 = π2 = 1600.
b) What happens if firm 2’s marginal cost becomes private information and takes either 0 or 20 with probability 1/2 each? Note that only firm 2 knows the true cost while firm 1 cannot observe it. Assume firm 1’s cost remains to be 10 and this information is common knowledge. Find a Bayesian Nash equilibrium of this game. What quantities will they produce? (8 points)
Hint: Note that firm 2’s strategy is a function (or, a complete plan of actions depending on realized costs) from each possible cost to its output.
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Answer: q1 = 40, q2(0) = 45, q2(20) = 35.
c) Now suppose firm 2’s marginal cost becomes common knowledge and is 10. Thus, as in (a), there is no asymmetric information (we assume this in (d) and (e) as well). Suppose that firm 1 decides how much to produce first; firm 2 chooses only after observing firm 1’s choice. Find the subgame perferct Nash equilibrium (Stackelberg equilibrium) of this game. What quantities will they produce, what is the market price and how much profit does each firm earn? (8 points)
Answer: q1 = 60, q2 = 30, p = 40, π1 = 1800, π2 = 900.
d) Suppose the firms form a cartel: each firm produced the same output and maximizes their joint profit. What quantity would each firm produce? What would be the market price? What would be the profit of each firm? (8 points)
Answer: q1 = q2 = 30, p = 70, π1 = π2 = 1800.
e) Now suppose that the firms play this Cournot game infinitely many times, and discount future profits at a discount factor δ. For what value of δ is there an equilibrium where firms follow the “trigger strategies” discussed in class? (8 points) Hint: In trigger strategies, each firm produces the cartel output (your answer in (d)) as long as long as no firm has deviated before, and starts producing the Cournot output forever after someone deviates.
Answer: Suppose firm 1 deviates. Since firm 2 produces q2 = 30 in a collusive path, firm 1’s most profitable deviation is achieved when q1 = 60 ‐ q2/2 = 45 (by the first order condition or best reply of firm 1). The firm 1’s profit becomes 452 = 2025. Now, we can write the payoff streams in both the collusive and the deviation paths as follows:
Collusion 1800 1800 1800 1800 …
Deviation 2025 1600 1600 1600 …
Difference ‐225 200 200 200 …
The collusion can be sustained (by the trigger strategy) if and only if: 225 200δ/1‐δ δ 9/17.
