Find the optimal mixed row strategy, the optimal mixed column strategy, and the expected value of the game in the event that each player uses his or her optimal mixed strategy.
$P = \left[ \begin{array} { l l }
- 7 & - 2 \
- 2 & - 3
\end{array} \right]$
A) $R = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 5 } { 6 }$
B) $R = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
C) $R = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
D) $R = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
E) $R = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right]$ , $C = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right] ^ { T }$ , $e = - \frac { 5 } { 6 }$

Question

Find the optimal mixed row strategy, the optimal mixed column strategy, and the expected value of the game in the event that each player uses his or her optimal mixed strategy.   
$P = \left[ \begin{array} { l l } 
- 7 & - 2 \
- 2 & - 3
\end{array} \right]$  
A) $R = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 5 } { 6 }$
B)   $R = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
C)   $R = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , - \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
D)   $R = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right]$ , $C = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right] ^ { T }$ , $e = - \frac { 17 } { 6 }$
E)   $R = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right]$ , $C = \left[ \frac { 5 } { 6 } , - \frac { 1 } { 6 } \right] ^ { T }$ , $e = - \frac { 5 } { 6 }$

Quizplus · Accepted Answer

To find the optimal mixed strategy and the expected value of the game, we use the payoff matrix $P$. The mixed strategy for both players can be found by solving a system of linear equations derived from the condition that in equilibrium, each player is indifferent to the actions of the opponent. The expected value (e) of the game is the average payoff when both players play their optimal strategies.Given $P = \left[ \begin{array} { l l } -7 & -2 \ -2 & -3 \end{array} \right]$, let $R = [x, 1-x]$ for the row player and $C = [y, 1-y]$ for the column player, where $x$ and $y$ represent the probabilities of choosing the first strategy for the row and column players, respectively.The expected payoffs for the row player choosing the first and second row are:- For the first row: $E_1 = -7y - 2(1-y)$- For the second row: $E_2 = -2y - 3(1-y)$Setting $E_1 = E_2$ to find the row player's mixed strategy gives us the value of $y$.Similarly, for the column player, the expected payoffs for choosing the first and second column are equated to find $x$.The expected value of the game (e) is calculated using the mixed strategies found.Without solving these equations here (as it requires algebraic manipulation), the correct mixed strategies and expected value must satisfy the game's equilibrium conditions, where neither player can benefit by unilaterally changing their strategy.Option D provides positive probabilities that sum to 1 for both players ($R = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right]$, $C = \left[ \frac { 1 } { 6 } , \frac { 5 } { 6 } \right] ^ { T }$) and an expected value that is consistent with the payoff matrix, indicating it is the correct choice. The negative probabilities in other options are not feasible in the context of mixed strategies, as probabilities cannot be negative.

Find the Optimal Mixed Row Strategy, the Optimal Mixed Column P=[−7−2−2−3]P = \left[ \begin{array} { l l } - 7 & - 2 \\- 2 & - 3\end{array} \right]P=[−7−2​−2−3​]

Find the Optimal Mixed Row Strategy, the Optimal Mixed Column $P = \left[ \begin{array} { l l } - 7 & - 2 \\- 2 & - 3\end{array} \right]$