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Up Down Player 1 in the game above, what is/are the sub-game perfect Nash equilibrium? (up,up) (up,down) (down, up) (down, down) No equilibrium exists Up Down Up Down Player 2 P1 gets $100 P2 gets $100 P1 gets $10 P2 gets $10 P1 gets $15 P2 gets $5 P1 gets $110 P2 gets $10

Up Down Player 1 in the game above, what is/are the sub-game perfect Nash equilibrium? (up,up) (up,down) (down, up) (down, down) No equilibrium exists Up Down Up Down Player 2 P1 gets $100 P2 gets $100 P1 gets $10 P2 gets $10 P1 gets $15 P2 gets $5 P1 gets $110 P2 gets $10

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
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**Game Theory Exercise: Sub-game Perfect Nash Equilibrium** This diagram represents a strategic game between two players, Player 1 and Player 2. In this game: - Player 1 has two choices: "Up" or "Down." - If Player 1 chooses "Up," Player 2 then gets to choose between "Up" and "Down." - If Player 1 chooses "Down," Player 2 again gets to choose between "Up" and "Down." The resulting payoffs for Player 1 (P1) and Player 2 (P2) are shown at the end of each branch. The payoffs are denoted in dollar amounts that each player will receive based on the decisions made. Let's break down the game to determine the sub-game perfect Nash equilibrium. **Decision Tree and Payoffs:** 1. If Player 1 chooses "Up": - If Player 2 then chooses "Up": - Player 1 gets $100 - Player 2 gets $100 - If Player 2 then chooses "Down": - Player 1 gets $10 - Player 2 gets $10 2. If Player 1 chooses "Down": - If Player 2 then chooses "Up": - Player 1 gets $15 - Player 2 gets $5 - If Player 2 then chooses "Down": - Player 1 gets $110 - Player 2 gets $10 **Question:** In the game above, what is/are the sub-game perfect Nash equilibrium? - ☐ (up, up) - ☐ (up, down) - ☐ (down, up) - ☑ (down, down) - ☐ No equilibrium exists By analyzing Player 2's decisions in each possible sub-game, the best responses for Player 2 can be determined. Player 2 will choose the action that maximizes their payoff given Player 1's choice. By tracing back these decisions to Player 1's initial choice, we determine that the sub-game perfect Nash equilibrium is (down, down). This means Player 1 will choose "Down," and Player 2 will respond with "Down."
Transcribed Image Text:**Game Theory Exercise: Sub-game Perfect Nash Equilibrium** This diagram represents a strategic game between two players, Player 1 and Player 2. In this game: - Player 1 has two choices: "Up" or "Down." - If Player 1 chooses "Up," Player 2 then gets to choose between "Up" and "Down." - If Player 1 chooses "Down," Player 2 again gets to choose between "Up" and "Down." The resulting payoffs for Player 1 (P1) and Player 2 (P2) are shown at the end of each branch. The payoffs are denoted in dollar amounts that each player will receive based on the decisions made. Let's break down the game to determine the sub-game perfect Nash equilibrium. **Decision Tree and Payoffs:** 1. If Player 1 chooses "Up": - If Player 2 then chooses "Up": - Player 1 gets $100 - Player 2 gets $100 - If Player 2 then chooses "Down": - Player 1 gets $10 - Player 2 gets $10 2. If Player 1 chooses "Down": - If Player 2 then chooses "Up": - Player 1 gets $15 - Player 2 gets $5 - If Player 2 then chooses "Down": - Player 1 gets $110 - Player 2 gets $10 **Question:** In the game above, what is/are the sub-game perfect Nash equilibrium? - ☐ (up, up) - ☐ (up, down) - ☐ (down, up) - ☑ (down, down) - ☐ No equilibrium exists By analyzing Player 2's decisions in each possible sub-game, the best responses for Player 2 can be determined. Player 2 will choose the action that maximizes their payoff given Player 1's choice. By tracing back these decisions to Player 1's initial choice, we determine that the sub-game perfect Nash equilibrium is (down, down). This means Player 1 will choose "Down," and Player 2 will respond with "Down."
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In the game above, what is/are the EFFICIENT sub-game perfect Nash equilibrium? 

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### Educational Content: Understanding Efficient Sub-Game Perfect Nash Equilibrium #### Game Theory Diagram Explanation The above diagram represents a sequential game involving two players, Player 1 and Player 2. It is depicted in a tree structure with decision nodes and terminal nodes indicating the payoffs for both players. Here is the breakdown: 1. **Initial Decision Node (Player 1)** - **Up**: Leads to a decision node for Player 2. - **Down**: Leads to another decision node for Player 2. 2. **Player 2’s Decision Nodes** - Following Player 1's **Up**: - **Up**: Results in a payoff of P1 gets \$100 and P2 gets \$100. - **Down**: Results in a payoff of P1 gets \$10 and P2 gets \$10. - Following Player 1's **Down**: - **Up**: Results in a payoff of P1 gets \$15 and P2 gets \$5. - **Down**: Results in a payoff of P1 gets \$110 and P2 gets \$10. #### Question on Efficient Sub-Game Perfect Nash Equilibrium **In the game above, what is/are the EFFICIENT sub-game perfect Nash equilibrium?** - [ ] (up,up) - [ ] (up,down) - [ ] (down, up) - [ ] (down, down) - [ ] No EFFICIENT equilibrium exists **Explanation:** The equilibrium points need to be analyzed to determine which choice combinations yield the highest combined payoffs and are stable given the strategies of rational players at each decision node. This includes evaluating payoffs for each combination and confirming the rational choices that maximize their respective utilities. --- This educational content is designed to help students understand the concept of Nash equilibrium in extensive form games and sub-game perfect equilibrium by analyzing a simple binary choice sequential game.
Transcribed Image Text:### Educational Content: Understanding Efficient Sub-Game Perfect Nash Equilibrium #### Game Theory Diagram Explanation The above diagram represents a sequential game involving two players, Player 1 and Player 2. It is depicted in a tree structure with decision nodes and terminal nodes indicating the payoffs for both players. Here is the breakdown: 1. **Initial Decision Node (Player 1)** - **Up**: Leads to a decision node for Player 2. - **Down**: Leads to another decision node for Player 2. 2. **Player 2’s Decision Nodes** - Following Player 1's **Up**: - **Up**: Results in a payoff of P1 gets \$100 and P2 gets \$100. - **Down**: Results in a payoff of P1 gets \$10 and P2 gets \$10. - Following Player 1's **Down**: - **Up**: Results in a payoff of P1 gets \$15 and P2 gets \$5. - **Down**: Results in a payoff of P1 gets \$110 and P2 gets \$10. #### Question on Efficient Sub-Game Perfect Nash Equilibrium **In the game above, what is/are the EFFICIENT sub-game perfect Nash equilibrium?** - [ ] (up,up) - [ ] (up,down) - [ ] (down, up) - [ ] (down, down) - [ ] No EFFICIENT equilibrium exists **Explanation:** The equilibrium points need to be analyzed to determine which choice combinations yield the highest combined payoffs and are stable given the strategies of rational players at each decision node. This includes evaluating payoffs for each combination and confirming the rational choices that maximize their respective utilities. --- This educational content is designed to help students understand the concept of Nash equilibrium in extensive form games and sub-game perfect equilibrium by analyzing a simple binary choice sequential game.
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