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.
$$
\begin{gathered}
\text { B } \\
p & q \\
\text { A } \left.\begin{array}{rr}
a \\
1 & 1 \\
2 & -4
\end{array}\right]
\end{gathered}
$$

Key Concepts

Payoff Matrix

A payoff matrix is a tabular representation of the possible outcomes in a game, where each cell represents the result of a combination of strategies chosen by the players. It is the foundation used to analyze strategies and determine optimal plays, depicting the gains or losses for each player under various strategic choices.

Saddle Point

A saddle point in a game is an outcome in the payoff matrix where the row minimum (from the perspective of the player choosing rows) equals the column maximum (from the perspective of the player choosing columns). This intersection represents an equilibrium where one player's best-response strategy coincides with the other’s, ensuring that neither player can unilaterally improve their payoff.

Strict Determinacy

A game is strictly determined if it possesses a saddle point, meaning that the optimal strategies for both players are pure (non-randomized) strategies. In such a game, the outcome is predetermined and is referred to as the value of the game, indicating that both players can guarantee a specific payoff regardless of the opponent's actions.

Optimal Pure Strategies

Optimal pure strategies are the specific choices made by players, without randomness, that maximize a player's minimum payoff (for a maximizer) or minimize the opponent's maximum payoff (for a minimizer). When a saddle point exists, these are the strategies that both players employ to secure the value of the game.

Value of the Game

The value of the game is the payoff that results when both players use their optimal strategies. In strictly determined games, this value reflects a stable outcome since neither player can deviate unilaterally to achieve a better result, ensuring equilibrium in the strategic interaction.

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