How many pure Nash equilibria does this game have? 0 1 O 2 O 4
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![**Payoff Matrix Analysis for a Game with Players A and B**
The image depicts a payoff matrix for a game involving two players, A and B. The matrix shows the payoffs for each combination of strategies.
**Matrix Structure:**
- **Players:**
- Player A has two strategies: \(a_1\) and \(a_2\).
- Player B has two strategies: \(b_1\) and \(b_2\).
- **Payoff Combinations:**
- The payoff for each player is represented in the form \((x,y)\), where \(x\) is the payoff for Player A and \(y\) is the payoff for Player B.
**Payoff Matrix:**
\[
\begin{array}{c|c|c}
& b_1 & b_2 \\
\hline
a_1 & (2, 5) & (4, 2) \\
\hline
a_2 & (3, 4) & (1, 8) \\
\end{array}
\]
**Question:**
How many pure Nash equilibria does this game have?
- Options:
- 1
- 0
- 2
- 4
**Analysis:**
To determine the number of pure Nash equilibria, examine each strategy combination where neither player can unilaterally improve their payoff by changing their strategy.
- **Equilibrium Analysis:**
- \((a_1, b_1):\) Check if either player can benefit from switching strategies.
- \((a_1, b_2):\) Check value (4, 2).
- \((a_2, b_1):\) Check value (3, 4).
- \((a_2, b_2):\) Check value (1, 8).
After this evaluation, identify the Nash equilibria and count them accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e84a40-2ab9-4b1b-97f3-40eaaaf2763c%2F84199985-bf86-47bf-94d4-c5ec304cf3cc%2Fx340e4r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Payoff Matrix Analysis for a Game with Players A and B**
The image depicts a payoff matrix for a game involving two players, A and B. The matrix shows the payoffs for each combination of strategies.
**Matrix Structure:**
- **Players:**
- Player A has two strategies: \(a_1\) and \(a_2\).
- Player B has two strategies: \(b_1\) and \(b_2\).
- **Payoff Combinations:**
- The payoff for each player is represented in the form \((x,y)\), where \(x\) is the payoff for Player A and \(y\) is the payoff for Player B.
**Payoff Matrix:**
\[
\begin{array}{c|c|c}
& b_1 & b_2 \\
\hline
a_1 & (2, 5) & (4, 2) \\
\hline
a_2 & (3, 4) & (1, 8) \\
\end{array}
\]
**Question:**
How many pure Nash equilibria does this game have?
- Options:
- 1
- 0
- 2
- 4
**Analysis:**
To determine the number of pure Nash equilibria, examine each strategy combination where neither player can unilaterally improve their payoff by changing their strategy.
- **Equilibrium Analysis:**
- \((a_1, b_1):\) Check if either player can benefit from switching strategies.
- \((a_1, b_2):\) Check value (4, 2).
- \((a_2, b_1):\) Check value (3, 4).
- \((a_2, b_2):\) Check value (1, 8).
After this evaluation, identify the Nash equilibria and count them accordingly.
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