Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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A4
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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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.
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