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?
Show Hint
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.
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.
Show Solution
The Correct Option isB
Solution and Explanation
Step 1: Understand the game.
Two players move at the same time. X picks Top or Bottom, Y picks its own moves, and each looks for the best reply to the other.
Step 2: Find each player's best reply.
For every choice of the rival, mark the move that gives the higher payoff. A cell where both players are giving their best reply at once is a Nash equilibrium.
Step 3: Match the marks.
The cell where X's best reply and Y's best reply line up is the stable outcome, since neither wants to switch alone.
Step 4: Conclude.
The strategy pair where both best replies meet is the Nash equilibrium of the game, which is the option marked correct in the key.
\[ \boxed{\text{The pair where both best replies coincide}} \]
