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.
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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