91Ó°ÊÓ

In a zero-sum game, a key approach for a player is to employ a maximin strategy. This is a defensive strategy used by a player to minimize their maximum possible loss, which can be quite useful when facing uncertainty about an opponent's moves. In essence, player 1, the row player, looks at the worst-case scenario payoff in each of their strategic options, which corresponds to the smallest number in each row of the payoff matrix.

To determine their maximin strategy, player 1 finds the minimum value for each of their possible moves (rows) and then chooses the move that has the highest of these minimal values. For example, in our exercise:

• Row 1's minimum value is \( -1 \)
• Row 2's minimum value is \( 2 \)

Then, player 1 chooses the higher value between the two, which in this case is \( 2 \) from Row 2. This approach ensures that, regardless of what the opponent does, the player secures the best of the worst-case outcomes.

Conversely, the minimax strategy is adopted by the opposing player (player 2 or the column player) in the game, which acts as an offensive strategy mirroring the concept of the maximin. Player 2 seeks to maximize the minimum gain that can be secured, indicative of the best worst-case outcome from their viewpoint.

To apply this strategy, player 2 looks at the maximum values they could force the opponent to receive within each of their strategic moves (columns) and then select the move that gives them the least benefit. In the given exercise:

• Column 1's maximum value is \( 2 \)
• Column 2's maximum value is \( 5 \)

Out of these, player 2 will prefer to give player 1 the lower payment, hence they'll choose the \( 2 \) from Column 1. It guarantees the least advantage to the opponent, which, in turn, is the best outcome for player 2 under the assumption that player 1 is using their maximin strategy.

A saddle point in the context of zero-sum games is a position on the payoff matrix where the strategies of both players intersect at the same value, reflecting the optimal decision for both. It presents a stable solution as it represents a scenario where neither player can benefit by unilaterally changing their strategy.

Finding a saddle point involves checking if the maximin value for player 1 matches the minimax value for player 2. If such a point exists, the payoff represented by that point is the value of the game. In other words, both players have optimized their strategies such that any deviation would only lead to equal or less favorable outcomes.

In our exercise, the value \( 2 \) serves as a saddle point found at the intersection of Row 2 and Column 1. This indicates that the optimal strategy for player 1 is to play Row 2 and for player 2 is to play Column 1, with the game's value being \( 2 \). This equality of maximin and minimax values ratifies the presence of a saddle point and, consequently, the most strategic moves both players can make.