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Find the backwards-induction outcome for the following game. L R 2 R'

Find the backwards-induction outcome for the following game. L R 2 R'

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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### Transcription for Educational Content on Backward Induction in Game Theory **Title: Understanding Backward Induction in Sequential Games** **Section: Game Analysis** **Problem 2: Backward-Induction Outcome** Find the backward-induction outcome for the following game. **Diagram Explanation:** The diagram in question is a decision tree representing a sequential game. It consists of nodes and branches that depict possible actions and their respective outcomes: 1. **Initial Node (Root):** The first player can choose between two actions: L (Left) or R (Right). - **If L is chosen:** The game ends with the outcomes (1, 2), representing payoffs for the two players involved. 2. **After choosing R:** - Leads to a second decision point where the second player can choose between L' (Left Prime) and R' (Right Prime). - **Choosing L':** Ends the game with payoffs (2, 3). - **Choosing R':** Leads to another decision point for the first player, who can choose between L" (Left Double Prime) and R" (Right Double Prime). - **Choosing L":** Ends with payoffs (3, 4). - **Choosing R":** Concludes with payoffs (4, 1). **Backward Induction Process:** - Begin from the last decision point, where the first player, after R', chooses between L" or R". The player will choose R" because the payoff (4, 1) gives them a higher payoff than L" with (3, 4). - Given that at R', the outcome is determined to be (4, 1), the second player will analyze their choices at reaching R and choose between L' (2, 3) and R', which results in (4, 1). Here, L' offers the second player a better payoff. - Thus, knowing that the second player would choose L', the first player's initial decision should be between L for (1, 2) or R, which ultimately results in (2, 3). Here, R offers the first player a better payoff. **Conclusion:** By applying backward induction, the optimal path in the game is initial choice R, leading the second player to choose L'. The final outcome through backward induction is (2, 3). This approach systematically leads to the subgame perfect equilibrium, ensuring
Transcribed Image Text:### Transcription for Educational Content on Backward Induction in Game Theory **Title: Understanding Backward Induction in Sequential Games** **Section: Game Analysis** **Problem 2: Backward-Induction Outcome** Find the backward-induction outcome for the following game. **Diagram Explanation:** The diagram in question is a decision tree representing a sequential game. It consists of nodes and branches that depict possible actions and their respective outcomes: 1. **Initial Node (Root):** The first player can choose between two actions: L (Left) or R (Right). - **If L is chosen:** The game ends with the outcomes (1, 2), representing payoffs for the two players involved. 2. **After choosing R:** - Leads to a second decision point where the second player can choose between L' (Left Prime) and R' (Right Prime). - **Choosing L':** Ends the game with payoffs (2, 3). - **Choosing R':** Leads to another decision point for the first player, who can choose between L" (Left Double Prime) and R" (Right Double Prime). - **Choosing L":** Ends with payoffs (3, 4). - **Choosing R":** Concludes with payoffs (4, 1). **Backward Induction Process:** - Begin from the last decision point, where the first player, after R', chooses between L" or R". The player will choose R" because the payoff (4, 1) gives them a higher payoff than L" with (3, 4). - Given that at R', the outcome is determined to be (4, 1), the second player will analyze their choices at reaching R and choose between L' (2, 3) and R', which results in (4, 1). Here, L' offers the second player a better payoff. - Thus, knowing that the second player would choose L', the first player's initial decision should be between L for (1, 2) or R, which ultimately results in (2, 3). Here, R offers the first player a better payoff. **Conclusion:** By applying backward induction, the optimal path in the game is initial choice R, leading the second player to choose L'. The final outcome through backward induction is (2, 3). This approach systematically leads to the subgame perfect equilibrium, ensuring
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