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.
$$
\begin{array}{rr}
& \mathbf{B} \\
p & q & r \\
\mathbf{A} & a & {\left[\begin{array}{rrr}
-2 & 1 & -3 \\
-2 & 3 & -2
\end{array}\right]}
\end{array}
$$

Key Concepts

Optimal Pure Strategies

Optimal pure strategies refer to choices for each player that guarantee the best outcome under the worst-case scenario when the saddle point exists. In a strictly determined game, these strategies are identified by the row corresponding to the maximin and the column corresponding to the minimax, ensuring that neither player can unilaterally improve their payoff.

Strictly Determined Game

A strictly determined game is a two?player zero-sum game in which there exists a pair of pure strategies (one for each player) such that neither player can benefit by unilaterally changing their choice. In other words, the game has a saddle point, meaning the maximum of the row minima equals the minimum of the column maxima.

Saddle Point

A saddle point in a payoff matrix is an entry that is simultaneously the smallest entry in its row and the largest in its column (or vice versa, depending on player roles). Its existence guarantees that both players have optimal pure strategies, leading directly to the value of the game without needing to resort to mixed strategies.

Maximin and Minimax

The maximin and minimax principles are fundamental in zero-sum game theory. The row player (maximizer) evaluates the worst-case (minimum) payoffs on each row and then selects the row with the highest minimum payoff (maximin). Conversely, the column player (minimizer) evaluates the worst-case (maximum) payoffs on each column and then selects the column with the smallest maximum payoff (minimax). Equality of these two values indicates a saddle point.

Value of the Game

The value of the game is the payoff that results from both players playing their optimal pure strategies. When a saddle point exists, this value is precisely the entry in the payoff matrix at the intersection of the optimal row and column. It represents the expected outcome if both players act rationally in a zero-sum context.

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