Consider the version of the location game defined by the following payoff functions: U₁(x1, x2) 1- [0.25][x1 - 012 - [0.75][x2-x11², U₂(x1, x2) = 1 [0.2][1-x212 [0.8][x2 - x11². Here each player i = 1,2 chooses their own location x, E [0, 1] and receives payoff U/(x1, x2). Instead of the players' choices being game-theoretically simultaneous, suppose that player 2 chooses her location first and player 1 chooses his locatic Moreover, suppose that player 1 observes the choice of x2 before choosing x₁. This game has a unique subgame perfect Nash equilibrium in which player 2's choice of location is x*.
Consider the version of the location game defined by the following payoff functions: U₁(x1, x2) 1- [0.25][x1 - 012 - [0.75][x2-x11², U₂(x1, x2) = 1 [0.2][1-x212 [0.8][x2 - x11². Here each player i = 1,2 chooses their own location x, E [0, 1] and receives payoff U/(x1, x2). Instead of the players' choices being game-theoretically simultaneous, suppose that player 2 chooses her location first and player 1 chooses his locatic Moreover, suppose that player 1 observes the choice of x2 before choosing x₁. This game has a unique subgame perfect Nash equilibrium in which player 2's choice of location is x*.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![SUBGAME PERFECTION
Consider the version of the location game defined by the following payoff functions:
[0.25][x₁-01² - [0.75][x2 - x₁1²,
U₂(x1, x2) = 1
[0.2][1-x₂1² [0.8][x₂ - x₁]².
Here each player i = 1,2 chooses their own location x; E [0, 1] and receives payoff Ui(x1, x₂).
Instead of the players' choices being game-theoretically simultaneous, suppose that player 2 chooses her location first and player 1 chooses his location second.
Moreover, suppose that player 1 observes the choice of x2 before choosing x₁.
This game has a unique subgame perfect Nash equilibrium in which player 2's choice of location is x.
Find the value of x*. (Report your answer in decimal form, rounding if necessary to two places.)
U₁(x₁, x₂) = 1
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Transcribed Image Text:SUBGAME PERFECTION
Consider the version of the location game defined by the following payoff functions:
[0.25][x₁-01² - [0.75][x2 - x₁1²,
U₂(x1, x2) = 1
[0.2][1-x₂1² [0.8][x₂ - x₁]².
Here each player i = 1,2 chooses their own location x; E [0, 1] and receives payoff Ui(x1, x₂).
Instead of the players' choices being game-theoretically simultaneous, suppose that player 2 chooses her location first and player 1 chooses his location second.
Moreover, suppose that player 1 observes the choice of x2 before choosing x₁.
This game has a unique subgame perfect Nash equilibrium in which player 2's choice of location is x.
Find the value of x*. (Report your answer in decimal form, rounding if necessary to two places.)
U₁(x₁, x₂) = 1
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-
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