Find (a) the optimal mixed row strategy, (b) the optimal...

Find (a) the optimal mixed row strategy,
(b) the optimal mixed column strategy, and (c) the expected value of the game. [HINT: See Example 3.]
$$
P=\left[\begin{array}{ll}
-2 & -1 \\
-1 & -3
\end{array}\right]
$$

Key Concepts

Zero-Sum Game

A zero?sum game is one in which one player's gain is exactly balanced by the losses of the other player. In these games, the total benefit to all players in every outcome always adds up to zero, which is why each player's strategy must account for the opponent's moves—maximizing their own minimum gain (or minimizing their maximum loss) in competitive settings.

Payoff Matrix

The payoff matrix is a tabular representation of the outcomes of a game, listing the payoffs for each combination of strategies chosen by the players. It serves as a foundational tool in game theory as it allows the analysis of players’ possible decisions and the resulting gains or losses.

Mixed Strategy

A mixed strategy involves randomizing over available pure strategies by assigning a probability to each. Instead of choosing a single strategy, players distribute their choices in a way that makes them unpredictable, which can help prevent opponents from gaining an advantage by exploiting predictable behavior.

Optimal Strategy

An optimal strategy is a strategy that maximizes a player's expected payoff given the strategies chosen by the opponent. In the context of zero-sum games, this usually means selecting a strategy that guarantees the highest possible minimum payoff regardless of the opposition’s action.

Expected Value of the Game

The expected value of the game is the average outcome a player can anticipate when both parties use their optimal strategies. It represents the value or payoff that, under rational play, each player can secure regardless of the moves of their opponent due to the balance inherent in zero-sum situations.

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