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3. Consider the following game with nature: 6, 8 3, 3 X' M 2 (P) High (1/2) Y 4, 4 8, 4 5,0 3,0 Low X' (1 -p) (1/2) (1 ) L' M'

3. Consider the following game with nature: 6, 8 3, 3 X' M 2 (P) High (1/2) Y 4, 4 8, 4 5,0 3,0 Low X' (1 -p) (1/2) (1 ) L' M'

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3. Consider the following game with nature: 6, 8 3, 3 X' M Y High (1/2) Y 4, 4 8, 4 5,0 3, 0 Low X' (1-p) (1/2) (1- 9) L' M' 1 Y Y 4, 6 8, 4 Does this game have any separating perfect Bayesian equilibrium? Show your analysis and, if there is such an equilibrium, report it (only one is required).
Transcribed Image Text:3. Consider the following game with nature: 6, 8 3, 3 X' M Y High (1/2) Y 4, 4 8, 4 5,0 3, 0 Low X' (1-p) (1/2) (1- 9) L' M' 1 Y Y 4, 6 8, 4 Does this game have any separating perfect Bayesian equilibrium? Show your analysis and, if there is such an equilibrium, report it (only one is required).
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### Game Theory: Perfect Bayesian Equilibrium **Problem Statement:** Consider the following game involving nature and two players: **Graph/Diagram Explanation:** The game tree begins with a move by nature, which decides the state of the world (High or Low) with equal probability (1/2). The players then make their moves based on the observed state of the world: - **From High State:** - Player L moves, choosing between strategies \(X\) or \(Y\): - If \(X\) is chosen, the payoffs are (6, 8). - If \(Y\) is chosen, the payoffs are (4, 4). - Player M moves, choosing between strategies \(X'\) or \(Y'\): - If \(X'\) is chosen, the payoffs are (3, 3). - If \(Y'\) is chosen, the payoffs are (10, 7). - **From Low State:** - Player L’ moves, choosing between \(X\) or \(Y\): - If \(X\) is chosen, the payoffs are (5, 0). - If \(Y\) is chosen, the payoffs are (4, 6). - Player M’ moves, choosing between \(X'\) or \(Y'\): - If \(X'\) is chosen, the payoffs are (3, 0). - If \(Y'\) is chosen, the payoffs are (8, 4). **Question:** Does this game have any separating perfect Bayesian equilibrium? Show your analysis and, if there is such an equilibrium, report it (only one is required). **Detailed Analysis: (Example for illustrative purposes)** 1. **Identify the Strategies and Beliefs:** - Players L, L’, M, and M’ choose between their respective strategies based on prior beliefs \(p\) and \(q\). 2. **Calculate Expected Payoffs:** - Calculate the payoffs for each player under both states of nature considering the mixed strategies and probabilities involved. 3. **Determine Beliefs at Each Information Set:** - Update the beliefs at each decision node based on the previous moves and the observed actions. 4. **Check Incentive Compatibility:** - Ensure that no player has an incentive to deviate from their chosen strategy given their beliefs. 5
Transcribed Image Text:### Game Theory: Perfect Bayesian Equilibrium **Problem Statement:** Consider the following game involving nature and two players: **Graph/Diagram Explanation:** The game tree begins with a move by nature, which decides the state of the world (High or Low) with equal probability (1/2). The players then make their moves based on the observed state of the world: - **From High State:** - Player L moves, choosing between strategies \(X\) or \(Y\): - If \(X\) is chosen, the payoffs are (6, 8). - If \(Y\) is chosen, the payoffs are (4, 4). - Player M moves, choosing between strategies \(X'\) or \(Y'\): - If \(X'\) is chosen, the payoffs are (3, 3). - If \(Y'\) is chosen, the payoffs are (10, 7). - **From Low State:** - Player L’ moves, choosing between \(X\) or \(Y\): - If \(X\) is chosen, the payoffs are (5, 0). - If \(Y\) is chosen, the payoffs are (4, 6). - Player M’ moves, choosing between \(X'\) or \(Y'\): - If \(X'\) is chosen, the payoffs are (3, 0). - If \(Y'\) is chosen, the payoffs are (8, 4). **Question:** Does this game have any separating perfect Bayesian equilibrium? Show your analysis and, if there is such an equilibrium, report it (only one is required). **Detailed Analysis: (Example for illustrative purposes)** 1. **Identify the Strategies and Beliefs:** - Players L, L’, M, and M’ choose between their respective strategies based on prior beliefs \(p\) and \(q\). 2. **Calculate Expected Payoffs:** - Calculate the payoffs for each player under both states of nature considering the mixed strategies and probabilities involved. 3. **Determine Beliefs at Each Information Set:** - Update the beliefs at each decision node based on the previous moves and the observed actions. 4. **Check Incentive Compatibility:** - Ensure that no player has an incentive to deviate from their chosen strategy given their beliefs. 5
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