Question:
Let us consider a simultaneous game involving two players \(X\) and \(Y\). \(X\) has two strategies, namely ‘Top’ and ‘Bottom’, and \(Y\) has two strategies, namely ‘Left’ and ‘Right’. The payoff matrix is as given below. In payoff \((i,j)\), \(i\) and \(j\) refer to payoffs of \(X\) and \(Y\), respectively. Then, which one of the following statements is CORRECT?
Let us consider a simultaneous game involving two players \(X\) and \(Y\). \(X\) has two strategies, namely ‘Top’ and ‘Bottom’, and \(Y\) has two strategies, namely ‘Left’ and ‘Right’. The payoff matrix is as given below. In payoff \((i,j)\), \(i\) and \(j\) refer to payoffs of \(X\) and \(Y\), respectively. Then, which one of the following statements is CORRECT?
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If no pure strategy Nash equilibrium exists in a \(2\times2\) game, check for a mixed strategy equilibrium by making each player indifferent between their two strategies.
Updated On: Jun 5, 2026
- Pure strategy Nash equilibria are hidden here.
- Nash equilibrium is reached when each of them purely randomises their behavior.
- The game is similar to a Matching Pennies game.
- The game is similar to a Battle of Sexes game.
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The Correct Option is B
Solution and Explanation
Step 1: Write the payoff matrix.
\[ \begin{array}{c|cc} & \text{Left} & \text{Right} \\ \hline \text{Top} & (1,2) & (4,1) \\ \text{Bottom} & (2,3) & (3,4) \end{array} \]
Step 2: Find player \(X\)'s best responses.
If \(Y\) chooses Left, then \(X\)'s payoffs are
\[ Top=1,\qquad Bottom=2 \] So, \(X\)'s best response is Bottom.
If \(Y\) chooses Right, then \(X\)'s payoffs are
\[ Top=4,\qquad Bottom=3 \] So, \(X\)'s best response is Top.
Step 3: Find player \(Y\)'s best responses.
If \(X\) chooses Top, then \(Y\)'s payoffs are
\[ Left=2,\qquad Right=1 \] So, \(Y\)'s best response is Left.
If \(X\) chooses Bottom, then \(Y\)'s payoffs are
\[ Left=3,\qquad Right=4 \] So, \(Y\)'s best response is Right.
Step 4: Check pure strategy Nash equilibrium.
No cell contains mutual best responses.
Therefore, there is no pure strategy Nash equilibrium.
Step 5: Find mixed strategy equilibrium.
Let \(X\) play Top with probability \(p\), and \(Y\) play Left with probability \(q\).
For \(X\) to randomise, payoff from Top must equal payoff from Bottom.
\[ 4-3q=3-q \] \[ q=\frac{1}{2} \]
Step 6: Find \(Y\)'s mixing probability.
For \(Y\) to randomise, payoff from Left must equal payoff from Right.
\[ 3-p=4-3p \] \[ p=\frac{1}{2} \]
Step 7: Final conclusion.
Thus, Nash equilibrium is obtained when both players randomise between their strategies.
\[ \boxed{\text{Nash equilibrium is reached when each of them randomises their behavior.}} \]
Hence, the correct option is (B).
\[ \begin{array}{c|cc} & \text{Left} & \text{Right} \\ \hline \text{Top} & (1,2) & (4,1) \\ \text{Bottom} & (2,3) & (3,4) \end{array} \]
Step 2: Find player \(X\)'s best responses.
If \(Y\) chooses Left, then \(X\)'s payoffs are
\[ Top=1,\qquad Bottom=2 \] So, \(X\)'s best response is Bottom.
If \(Y\) chooses Right, then \(X\)'s payoffs are
\[ Top=4,\qquad Bottom=3 \] So, \(X\)'s best response is Top.
Step 3: Find player \(Y\)'s best responses.
If \(X\) chooses Top, then \(Y\)'s payoffs are
\[ Left=2,\qquad Right=1 \] So, \(Y\)'s best response is Left.
If \(X\) chooses Bottom, then \(Y\)'s payoffs are
\[ Left=3,\qquad Right=4 \] So, \(Y\)'s best response is Right.
Step 4: Check pure strategy Nash equilibrium.
No cell contains mutual best responses.
Therefore, there is no pure strategy Nash equilibrium.
Step 5: Find mixed strategy equilibrium.
Let \(X\) play Top with probability \(p\), and \(Y\) play Left with probability \(q\).
For \(X\) to randomise, payoff from Top must equal payoff from Bottom.
\[ 4-3q=3-q \] \[ q=\frac{1}{2} \]
Step 6: Find \(Y\)'s mixing probability.
For \(Y\) to randomise, payoff from Left must equal payoff from Right.
\[ 3-p=4-3p \] \[ p=\frac{1}{2} \]
Step 7: Final conclusion.
Thus, Nash equilibrium is obtained when both players randomise between their strategies.
\[ \boxed{\text{Nash equilibrium is reached when each of them randomises their behavior.}} \]
Hence, the correct option is (B).
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