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 $4 .$
$$
\begin{array}{rr}
p & q \\
a\left[\begin{array}{rr}
-1 & 2 \\
10 & -1
\end{array}\right]
\end{array}
$$

Key Concepts

Minimax Principle

The minimax principle is a fundamental concept in game theory used to determine the optimal strategy for a player by minimizing the maximum possible loss. In strictly determined games, this principle leads directly to the identification of a saddle point, confirming that the players’ optimal strategies secure the game’s value.

Optimal Pure Strategies

Optimal pure strategies refer to the set of strategies that yield the best guaranteed outcome for each player without requiring randomization. When a game is strictly determined, each player’s best move is a pure strategy, and these strategies remain optimal regardless of the opponent’s decision.

Value of the Game

The value of the game is the payoff that results when both players implement their optimal strategies. This value represents the equilibrium outcome of the game, ensuring that the row player cannot improve by switching strategies and the column player cannot decrease the loss by deviating from their chosen strategy.

Strictly Determined Game

A strictly determined game is one in which both players have optimal pure strategies that yield a game value regardless of the opponent's actions. In these games, the best response of each player is unambiguous and typically identified through the existence of a saddle point in the payoff matrix.

Saddle Point

A saddle point in a payoff matrix is an entry which is simultaneously the minimum of its row and the maximum of its column. This point represents the outcome where neither player can improve their payoff by unilaterally changing their strategy, confirming that the game is strictly determined.

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