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Cass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to provide of the public good. After observing 1's choice, 2 chooses a quantity 22 20 to provide. When the price of the public good is p, 1's payoff is u₁ (₁, 2) = a√√x₁ + x₂-p²₁ where a > 0 and 2's payoff is u₂(x1, x2) = √√x₁ + x₂ − px₂. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes.

Cass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to provide of the public good. After observing 1's choice, 2 chooses a quantity 22 20 to provide. When the price of the public good is p, 1's payoff is u₁ (₁, 2) = a√√x₁ + x₂-p²₁ where a > 0 and 2's payoff is u₂(x1, x2) = √√x₁ + x₂ − px₂. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
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Cass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to provide of the public good. After observing 1's choice, 2 chooses a quantity 2 ≥ 0 to provide. When the price of the public good is p, 1's payoff is u₁(₁, 2) = a√x₁ + x₂-p²₁ where a > 0 and 2's payoff is u2(x1, x2) = √√T1+T2 — px2. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes. (b) Find all subgame perfect equilibria of this game for each (positive) value of a and p. Solution: Use backward induction. If 1 contributes ₁, then 2's optimal action is to con- tribute (1) max 7-2₁,0}. Given this strategy for 2, 1's payoff to contributing ₁ is - px1 a√₁-pr This payoff is maximized by choosing r1=0 if a < 2, x₁ = 0 or ₁ = 1/p² if a = 2, and x₁ = if a > 2. Therefore, the subgame perfect equilibria are as follows: if 1 ≤² otherwise. u(x₁, x₂(x₁)) = i. If a < 2, there is a unique SPE given by x₁ = 0 and x2(x₁) = max. {+ -21,0}. ii. If a = 2, there are two SPE, one given by x₁ = 0 and x₂(x₁) = max {-₁,0}, {-1,0}. I the other given by x₁ = 1/p² and x₂(x₁) = max iii. If a > 2, there is a unique SPE given by x₁ = and x₂(x₁) = max c{₁-2₁,0}. (c) How does the total public good provision in part (b) compare to the Nash equilibrium provision when the consumers choose simultaneously? Solution: From class, total public good provision in the Nash equilibrium of the simul- taneous game is if a >1. In the sequential game, total provision is if a > 2. Total provision is never higher in the if a ≤ 1 and ifa <2, or if a = 2, and sequential game, and is strictly lower if 1 < a < 2.
Transcribed Image Text:Cass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to provide of the public good. After observing 1's choice, 2 chooses a quantity 2 ≥ 0 to provide. When the price of the public good is p, 1's payoff is u₁(₁, 2) = a√x₁ + x₂-p²₁ where a > 0 and 2's payoff is u2(x1, x2) = √√T1+T2 — px2. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes. (b) Find all subgame perfect equilibria of this game for each (positive) value of a and p. Solution: Use backward induction. If 1 contributes ₁, then 2's optimal action is to con- tribute (1) max 7-2₁,0}. Given this strategy for 2, 1's payoff to contributing ₁ is - px1 a√₁-pr This payoff is maximized by choosing r1=0 if a < 2, x₁ = 0 or ₁ = 1/p² if a = 2, and x₁ = if a > 2. Therefore, the subgame perfect equilibria are as follows: if 1 ≤² otherwise. u(x₁, x₂(x₁)) = i. If a < 2, there is a unique SPE given by x₁ = 0 and x2(x₁) = max. {+ -21,0}. ii. If a = 2, there are two SPE, one given by x₁ = 0 and x₂(x₁) = max {-₁,0}, {-1,0}. I the other given by x₁ = 1/p² and x₂(x₁) = max iii. If a > 2, there is a unique SPE given by x₁ = and x₂(x₁) = max c{₁-2₁,0}. (c) How does the total public good provision in part (b) compare to the Nash equilibrium provision when the consumers choose simultaneously? Solution: From class, total public good provision in the Nash equilibrium of the simul- taneous game is if a >1. In the sequential game, total provision is if a > 2. Total provision is never higher in the if a ≤ 1 and ifa <2, or if a = 2, and sequential game, and is strictly lower if 1 < a < 2.
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