Find the optimal row or column strategy of the matrix game.
-Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r } 4 & - 1 \ - 4 & 5 \end{array} \right]$.
Find the optimal column strategy.
A) $\hat { y } = \left[ \begin{array} { l } \frac { 3 } { 7 } \ \frac { 4 } { 7 } \end{array} \right]$

B) $\hat { y } = \left[ \begin{array} { c } \frac { 5 } { 14 } \ \frac { 9 } { 14 } \end{array} \right]$

C) $\hat { \mathrm { y } } = \left[ \begin{array} { c } \frac { 9 } { 14 } \ \frac { 5 } { 14 } \end{array} \right]$

D) $\hat { \mathrm { y } } = \left[ \begin{array} { l } \frac { 4 } { 7 } \ \frac { 3 } { 7 } \end{array} \right]$

Question

Find the optimal row or column strategy of the matrix game.
-Let $M$ be the matrix game having payoff matrix $\left[ \begin{array} { r r } 4 & - 1 \ - 4 & 5 \end{array} \right]$.
Find the optimal column strategy.
A) $\hat { y } = \left[ \begin{array} { l } \frac { 3 } { 7 } \ \frac { 4 } { 7 } \end{array} \right]$

B) $\hat { y } = \left[ \begin{array} { c } \frac { 5 } { 14 } \ \frac { 9 } { 14 } \end{array} \right]$

C) $\hat { \mathrm { y } } = \left[ \begin{array} { c } \frac { 9 } { 14 } \ \frac { 5 } { 14 } \end{array} \right]$

D) $\hat { \mathrm { y } } = \left[ \begin{array} { l } \frac { 4 } { 7 } \ \frac { 3 } { 7 } \end{array} \right]$

Quizplus · Accepted Answer

To find the optimal column strategy for a 2x2 matrix game with matrix $\left[ \begin{array} { r r } 4 & - 1 \ - 4 & 5 \end{array} \right]$, we use the formula for mixed strategies in zero-sum games. The probability $p$ for the first strategy (row or column) is given by $p = \frac{b-d}{a-b-c+d}$, where $a, b, c, d$ are the elements of the matrix starting from the top left and going clockwise. Here, $a=4, b=-1, c=-4, d=5$. Thus, for the column player, the probability $q$ for choosing the first column is $q = \frac{5-(-4)}{4-(-1)-(-4)+5} = \frac{9}{9} = 1$, which means the formula was incorrectly applied. Correctly, for the column player, we calculate the probabilities considering the rows' perspective, leading to $q = \frac{4-(-4)}{4-(-1)-(-4)+5} = \frac{8}{9}$, which also indicates a mistake in calculation. The correct approach involves solving the system of linear equations or applying the correct formula for the column player, which involves considering the payoffs from the perspective of minimizing loss or maximizing gain against the row player's strategies. The correct calculation involves finding the equilibrium where neither player benefits from unilaterally changing their strategy. Given the mistake in the initial explanation, the correct probability distribution for the column player's optimal strategy must be recalculated properly, considering the game's nature and the strategies' expected payoffs. The correct probabilities for the column player, considering the minimax principle, should balance out the expected payoffs for both strategies, leading to the solution provided in option A, which correctly calculates the mixed strategy equilibrium for the column player.

Find the Optimal Row or Column Strategy of the Matrix MMM

Find the Optimal Row or Column Strategy of the Matrix $M$