ECON C110 (POLSCI C135), Fall 2022
Problem Set 10
Problem 1
Chapter 22: Exercise 1
Consider a repeated game in which the stage game in the following figure is played in each of two
periods and there is no discounting. Fully describe a subgame perfect equilibrium in which the players
select (U, L) in the first period.
Problem 2
Chapter 22: Exercise 2
Find conditions on the discount factor under which cooperation can be supported in the infinitely
repeated games with the following stage games.
(a)
Use the grim-trigger strategy profile.
(b)
(c)
Problem 3
Chapter 22: Exercise 3
Consider the following stage game.
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ECON C110 (POLSCI C135), Fall 2022
(a) Find and report all of the (pure-strategy) Nash equilibria of this game.
(b) Consider the two-period repeated game in which this stage game is played twice and the repeatedgame payoffs are simply the sum of the payoffs in each of the two periods. Is there a subgame
perfect equilibrium of this repeated game in which (A, X) is played in the first period? If so, fully
describe the equilibrium. If not, explain why.
Problem 4
Chapter 22: Exercise 4
If its stage game has exactly one Nash equilibrium, how many subgame perfect equilibria does a twoperiod, repeated game have? Explain. Would your answer change if there were 𝑇 periods, where 𝑇 is
any finite integer?
Problem 5
Chapter 23: Exercise 1
Consider the Bertrand oligopoly model, where 𝑛 firms simultaneously and independently select their
prices, 𝑝1, 𝑝2 , …, 𝑝𝑛 , in a market. (These prices are greater than or equal to 0.) Consumers observe these
prices and only purchase from the firm (or firms) with the lowest price 𝑝, according to the demand curve
𝑄 = 110 − 𝑝, where 𝑝 = min{𝑝1 , 𝑝2 , … , 𝑝𝑛 }. That is, the firm with the lowest price gets all of the sales.
If the lowest price is offered by more than one firm, then these firms equally share the quantity demanded
𝑄. Assume that firms must supply the quantities demanded of them and that production takes place at a
constant cost of 10 per unit. (That is, the cost function for each firm is 𝑐(𝑞) = 10𝑞.) Determining the
Nash equilibrium of this game was the subject of a previous exercise.
(a) Suppose that this game is infinitely repeated. (The firms play the game each period, for an infinite
number of periods.) Define 𝛿 as the discount factor for the firms. Imagine that the firms wish to
sustain a collusive arrangement in which they all select the monopoly price 𝑝𝑀 = 60 each period.
What strategies might support this behavior in equilibrium? (Do not solve for conditions under
which equilibrium occurs. Just explain what the strategies are. Remember, this requires
specifying how the firms punish each other. Use the Nash equilibrium price as punishment.)
(b) Derive a condition on 𝑛 and 𝛿 that guarantees that collusion can be sustained.
(c) What does your answer to part (b) imply about the optimal size of cartels?
Problem 6
Chapter 23: Exercise 2
Examine the infinitely repeated tariff-setting game, where the stage game is the two-country tariff game
in Chapter 10 (see also Exercise 3 in Chapter 10).
(a) Computer the Nash equilibrium of the stage game.
(b) Find conditions on the discount factor such that zero tariffs (𝑥1 = 𝑥2 = 0) can be sustained each
period by a subgame perfect equilibrium. Use the grim-trigger strategy profile.
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ECON C110 (POLSCI C135), Fall 2022
(c) Find conditions on the discount factor such that a tariff level of 𝑥1 = 𝑥2 = 𝑘 can be sustained by
a subgame perfect equilibrium, where 𝑘 is some fixed number between 0 and 100.
Problem 7
Chapter 23: Exercise 10
Persons 1 and 2 are forming a firm. The value of their relationship depends on the effort that they each
expend. Suppose that person 𝑖’s utility from the relationship is 𝑥𝑗2 + 𝑥𝑗 − 𝑥𝑖 𝑥𝑗 , where 𝑥𝑖 ≥ 0 is person
𝑖’s effort and is the effort of the other person (𝑖 = 1, 2).
(a) Compute the partners’ best-response functions and find the Nash equilibrium of this game. Is the
Nash equilibrium efficient?
(b) Now suppose that the partners interact over time, which we model with the infinitely repeated
version of the game. Let 𝛿 denote the discount factor of the players. Under what conditions can
the partners sustain some positive effort level 𝑥 = 𝑥1 = 𝑥2 over time? (Postulate strategies and
then derive a condition under which the strategies form an equilibrium in the repeated game.)
(c) Comment on how the maximum sustainable effort depends on the partners’ patience.
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