ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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A4
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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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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Follow-up Questions
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Follow-up Question
### 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
expand button
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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Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
### 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
expand button
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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