Consider the following sequential variant of the public goods game we studied in class. Sup- 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 x2 > 0 to provide. When the price of the public good is p, 1's payoff is u₁(x1, x2) = a√√x1 + x2−pr₁ where a > 0 and 2's payoff is u2(x1, x₂) = √√x1 + x2 - рx₂. (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 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 x1, then 2's optimal action is to con- tribute x₂(1): = max {-2₁,0}. Given this strategy for 2, 1's payoff to contributing 1 is u(x₁, x₂(x1)) = 2p-px1 a√₁ - pri and for if x1 ≤ 4² otherwise. ii. If a = 2, there are two SPE, one given by x1 the other given by x1 = 1/p² and x2(x1) = max iii. If a > 2, there is a unique SPE given by x1 = This payoff is marimized by choosing r₁ = 0 if a < 2, x₁ = 0 or x₁ = 1/p² if a = 2, and = if a > 2. Therefore, the subgame perfect equilibria are as follows: x1 = i. If a < 2, there is a unique SPE given by x1 = 0 and x2(x1) = max- : {−₁,0}. = 0 and x₂(x1) = max < {²-₁,0}, x{-x₁,0}. and x₂(x1) = max = {2-²1,0}.
Consider the following sequential variant of the public goods game we studied in class. Sup- 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 x2 > 0 to provide. When the price of the public good is p, 1's payoff is u₁(x1, x2) = a√√x1 + x2−pr₁ where a > 0 and 2's payoff is u2(x1, x₂) = √√x1 + x2 - рx₂. (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 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 x1, then 2's optimal action is to con- tribute x₂(1): = max {-2₁,0}. Given this strategy for 2, 1's payoff to contributing 1 is u(x₁, x₂(x1)) = 2p-px1 a√₁ - pri and for if x1 ≤ 4² otherwise. ii. If a = 2, there are two SPE, one given by x1 the other given by x1 = 1/p² and x2(x1) = max iii. If a > 2, there is a unique SPE given by x1 = This payoff is marimized by choosing r₁ = 0 if a < 2, x₁ = 0 or x₁ = 1/p² if a = 2, and = if a > 2. Therefore, the subgame perfect equilibria are as follows: x1 = i. If a < 2, there is a unique SPE given by x1 = 0 and x2(x1) = max- : {−₁,0}. = 0 and x₂(x1) = max < {²-₁,0}, x{-x₁,0}. and x₂(x1) = max = {2-²1,0}.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 26EQ
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Transcribed Image Text:Consider the following sequential variant of the public goods game we studied in class. Sup-
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 x2 > 0 to provide.
When the price of the public good is p, 1's payoff is u₁(x1, x2) = a√√x1 + x2−pr₁ where a > 0
and 2's payoff is u2(x1, x₂) = √√x1 + x2 - рx₂.
(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
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 x1, then 2's optimal action is to con-
tribute x₂(1): = max {-2₁,0}. Given this strategy for 2, 1's payoff to contributing
1 is
u(x₁, x₂(x1)) =
2p-px1
a√₁ - pri
and for
if x1 ≤ 4²
otherwise.
ii. If a = 2, there are two SPE, one given by x1
the other given by x1 = 1/p² and x2(x1) = max
iii. If a > 2, there is a unique SPE given by x1 =
This payoff is marimized by choosing r₁ = 0 if a < 2, x₁ = 0 or x₁ = 1/p² if a = 2, and
= if a > 2. Therefore, the subgame perfect equilibria are as follows:
x1 =
i. If a < 2, there is a unique SPE given by x1 = 0 and x2(x1) = max-
: {−₁,0}.
= 0 and x₂(x1) = max < {²-₁,0},
x{-x₁,0}.
and x₂(x1) = max = {2-²1,0}.
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