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Problem Problem 1: Suppose there is a unit mass of consumers who can purchase a product at some cost c. Each consumer i has a type vi drawn independently from a continuous distribution F on [0,1]. The good exhibits network effects. In particular, if a fraction x of the population purchases, then the value of purchasing to consumer i is
The value of not purchasing is zero. (a) Suppose h is nonnegative, continuous, and strictly increasing on [0, 1]. Derive the best response correspondence for this game—that is, fixing h(x), what fraction of the population xˆ wishes to purchase? Prove that an equilibrium exists. (b) Suppose h(x) = x and the cumulative distribution function F for vi takes the form
for some α > 0 and 0 < γ < β. Derive the best response correspondence. For what parameter values does there exist an equilibrium in which consumers purchase the product? Problem 2: Consider another market with network effects. There is a unit mass of potential consumers who can purchase a product at some fixed price p with 0 < p < Each consumer i has a private value vi drawn independently from a uniform distribution on [0, 1]. If a fraction x of the population purchases the product, the consumer’s payoff from purchasing is
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(a) Intuitively describe the kind of externalities g(x) is capturing. Can you give a real life example that would fit? (b) Characterize the set of equilibria. (c) Which equilibria are stable? Why? (d) Is social welfare maximized in any of the equilibria? Explain. Problem 3: Consider the local network game with strategic substitutes from the lecture slides. Each player i chooses an action xi ≥ 0 and earns the payof
Assume b is such that b0 (1) = k. Suppose G is a circle graph with four players. Compute the set of equilibria. Problem 4: Consider the following version of the Prisoner’s dilemma game:
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In an infinitely repeated version of this game, with discount rate δ, can you construct a subgame perfect equilibrium in which the players trade off cooperating and defecting on one another? That is, on the equilibrium path, in period 1 the action profile is (C, D), in period 2 it is (D, C), in period 3 it is (C, D), and so on. How high must δ be for this to be an equilibrium? How does welfare in this equilibrium compare to the equilibrium with cooperation in every period (when this is an equilibrium)? Problem 5: Consider the following game:
(a) In a one-shot play of this game, what are the pure strategy equilibria? (b) Suppose the game is played twice with no discounting ( δ = 1). What is the highest welfare that can be obtained in a subgame perfect equilibrium, and what is the equilibrum?
Solution
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Problem 1
(b) Omitted.
Problem 2 (a) It describes a good that a consumer want some people to possess but not many. For example, a party venue that gets better with more attendance, but gets worse when it is too crowded. The value vi measures how much player i likes to party. The value p is a cover charge for the club; if it is too high there is no equilibrium with postivie attendance.
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(d) Suppose consumers with values higher than 1 –x purchase the good. Then the social welfare is
This is maximized at x = 1/4. Therefore, no equilibrium attains the social optimum.
Problem 3 The circle graph with four players have the adjacency matrix of
Recall that agent i’s best response function is
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First, consider an equilibrium where everyone is active. The condition is
which is impossible. Third, consider an equilibrium with two active agents. By the same exercise, we know it is impossible to have agents 1 and 2 active. For the case with agents 1 and 4 active, we have
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This yields (x1, x2, x3, x4) = (1, 0, 0, 1) as long as δ ≥ 1/2. Note also that its rotation (0, 1, 1, 0) is also an equilibrium. Finally, we can verify that there is no equilibrium with one or zero active agent. Thus, there are two equilibria as derived above.
Problem 4 Consider the equilibrium strategy in which player 1 plays C, D, C, D, . . . as long as player 2 plays D, C, D, C, . . . , and vice versa. If the opponent deviates, then each player commits to play D forever. In period 1, player 1’s anticipated payoff along the given equilibrium path is
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Thus, we need δ ≥ 1/6. It is easy to check that player 2 in period 1 (or player 1 in period 2) has no incentive to deviate if δ ≥ 1/6. Also, it is easy to see that if either has deviated (so they are in an off-path state), then there is no incentive for either to deviate from playing D forever. Hence, the given strategies constitute an equilibrium if δ ≥ 1/6.
Problem 5 (a) The pure strategy equilibria are (B, B) and (C, C). (b) In the second period, the highest payoff attainable is 1 at (B, B) since it is the last period. In the first period, the highest possible payoff is 3 at (A, A). I argue that payoff 3 in the first stage is attainable. Consider the strategy in which a player takes A in the first period, and dependeing on the opponent’s action in the first period, the player determines the second-stage action; in particular, he takes B in the second period if the opponent took A in the first period, and takes C otherwise. The pair of this strategy earns a payoff of 4, while if one deviates, one can at most obtain 4 - 1 = 3. Therefore, there is no incentive to deviate and hence it is a subgame perfect equilibrium. Thus, the highest welfare attainable in a SPE is 2(3 + 1) = 8.