Decide whether the game is strictly determined. If it is,...

Decide whether the game is strictly determined. If it is, give the players' optimal pure strategies and the value of the game. [HINT]: See Example $4 .$
$$
\left.\begin{array}{c|ccc}
P & 1 & -1 & -5 \\
Q & 4 & -4 & 2 \\
R & 3 & -3 & -10 \\
S & 5 & -5 & -4
\end{array}\right]
$$

Key Concepts

Saddle Point

A saddle point is an element in a payoff matrix that serves both as the minimum element in its row and the maximum element in its column. Its existence guarantees that a particular pair of pure strategies is optimal for both players, thus determining the game’s value.

Matrix Games

Matrix games are two-player zero-sum games represented by a matrix in which rows and columns correspond to the strategies available to the respective players, and the entries represent the payoffs. This format is used to analyze the interactions and optimal strategies in competitive situations.

Pure Strategies

Pure strategies involve choosing a single action with certainty rather than randomizing over multiple actions. In settings where pure strategy optimality exists, players can reliably select one strategy that maximizes their outcome regardless of the opponent's move.

Strictly Determined Game

A game is strictly determined if there is a saddle point in the payoff matrix, meaning each player has an optimal pure strategy that secures the game's value independent of the opponent’s actions. This directly implies that the outcome is predictable and stable.

Minimax Theorem

The minimax theorem states that in zero-sum games, the maximal value that a player can guarantee themselves by minimizing the opponent's maximum loss is equal to the minimal value the opponent can force upon them. This principle is fundamental in identifying equilibrium strategies and determining the value of strictly determined games.

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