We are asked to solve a 4×3 zero-sum game with the payoff matrix: $$ A = \begin{bmatrix} 8 & 5 & 8 \\ 8 & 6 & 5 \\ 7 & 4 & 5 \\ 6 & 5 & 6 \end{bmatrix} $$ Step 1: Check for a saddle point * Find row minima: * Row 1: min(8,5,8) = 5 * Row 2: min(8,6,5) = 5 * Row 3: min(7,4,5) = 4 * Row 4: min(6,5,6) = 5 * Find column maxima: * Col 1: max(8,8,7,6) = 8 * Col 2: max(5,6,4,5) = 6 * Col 3: max(8,5,5,6) = 8 * Compare max of row minima (maximin) and min of column maxima (minimax): * Maximin = max(5,5,4,5) = 5 * Minimax = min(8,6,8) = 6 Since maximin ≠ minimax, there is no saddle point. So we must use mixed strategies. --- Step 2: Reduce the matrix (if possible) * Check for dominated strategies: Rows: * Compare Row 4 (6,5,6) with Row 1 (8,5,8): * Row 1 ≥ Row 4 in all entries? 8≥6, 5≥5, 8≥6 → Yes → Row 4 is dominated by Row 1, so remove Row 4. Rows after removal: $$ \begin{bmatrix} 8 & 5 & 8 \\ 8 & 6 & 5 \\ 7 & 4 & 5 \end{bmatrix} $$ Columns: * Compare Column 1 (8,8,7) and Column 3 (8,5,5): * Column 3 ≤ Column 1 in all entries? 8≤8,5≤8,5≤7 → Yes → Column 3 is dominated by Column 1, so remove Column 3. Reduced matrix: $$ \begin{bmatrix} 8 & 5 \\ 8 & 6 \\ 7 & 4 \end{bmatrix} $$ --- Step 3: Use mixed strategies Let Player A (row player) use strategies with probabilities $x_1, x_2, x_3$ for the 3 rows (sum = 1), and Player B (column player) use $y_1, y_2$ for 2 columns (sum = 1). Let value of the game = $V$. Expected payoff for column 1: $$ 8x_1 + 8x_2 + 7x_3 = V $$ Expected payoff for column 2: $$ 5x_1 + 6x_2 + 4x_3 = V $$ Also, $x_1 + x_2 + x_3 = 1$. --- Step 4: Solve the linear system Subtract the second equation from the first: $$ (8x_1+8x_2+7x_3) - (5x_1+6x_2+4x_3) = V - V \implies 3x_1 + 2x_2 + 3x_3 = 0 $$ Also, $x_1+x_2+x_3=1$. * Solve $3x_1 + 2x_2 + 3x_3 = 0$ → $3x_1 + 2x_2 + 3x_3 = 0$ * Try to express one