Answer :
To find the value of the game using a mixed strategy for a two-player game with the given payoff matrix:
[tex]\[ \left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] \][/tex]
we will follow the following steps:
### Step 1: Define the Expected Payoffs
Let's denote the probabilities that Player 1 chooses Row 1 by [tex]\( p \)[/tex] and Row 2 by [tex]\( 1 - p \)[/tex]. Similarly, let’s denote the probabilities that Player 2 chooses Column 1 by [tex]\( q \)[/tex] and Column 2 by [tex]\( 1 - q \)[/tex].
The expected payoff for Player 1 (E1) when playing mixed strategies is:
[tex]\[ E1 = p \cdot (a_{11}q + a_{12}(1 - q)) + (1 - p) \cdot (a_{21}q + a_{22}(1 - q)) \][/tex]
The expected payoff for Player 2 (E2) when playing mixed strategies is:
[tex]\[ E2 = q \cdot (a_{11}p + a_{21}(1 - p)) + (1 - q) \cdot (a_{12}p + a_{22}(1 - p)) \][/tex]
### Step 2: Set Up the System of Equations
To find the mixed strategy Nash equilibrium, both players should be indifferent to their choices. This means the expected payoffs for each player should be equal when switching strategies.
For Player 1 to be indifferent:
[tex]\[ a_{11}q + a_{12}(1 - q) = a_{21}q + a_{22}(1 - q) \][/tex]
For Player 2 to be indifferent:
[tex]\[ a_{11}p + a_{21}(1 - p) = a_{12}p + a_{22}(1 - p) \][/tex]
### Step 3: Solve the Equations for [tex]\( p \)[/tex] and [tex]\( q \)[/tex]
#### Finding [tex]\( q \)[/tex]:
[tex]\[ a_{11}q + a_{12}(1 - q) = a_{21}q + a_{22}(1 - q) \][/tex]
Simplify the equation:
[tex]\[ a_{11} q + a_{12} - a_{12} q = a_{21} q + a_{22} - a_{22} q \][/tex]
Combine like terms:
[tex]\[ (a_{11} - a_{12}) q + a_{12} = (a_{21} - a_{22}) q + a_{22} \][/tex]
Rearrange to solve for [tex]\( q \)[/tex]:
[tex]\[ (a_{11} - a_{12} - a_{21} + a_{22}) q = a_{22} - a_{12} \][/tex]
[tex]\[ q = \frac{a_{22} - a_{12}}{a_{11} - a_{12} - a_{21} + a_{22}} \][/tex]
#### Finding [tex]\( p \)[/tex]:
[tex]\[ a_{11}p + a_{21}(1 - p) = a_{12}p + a_{22}(1 - p) \][/tex]
Simplify the equation:
[tex]\[ a_{11} p + a_{21} - a_{21} p = a_{12} p + a_{22} - a_{22} p \][/tex]
Combine like terms:
[tex]\[ (a_{11} - a_{21}) p + a_{21} = (a_{12} - a_{22}) p + a_{22} \][/tex]
Rearrange to solve for [tex]\( p \)[/tex]:
[tex]\[ (a_{11} - a_{21} - a_{12} + a_{22}) p = a_{22} - a_{21} \][/tex]
[tex]\[ p = \frac{a_{22} - a_{21}}{a_{11} - a_{21} - a_{12} + a_{22}} \][/tex]
### Step 4: Determine the Value of the Game
The value of the game [tex]\( V \)[/tex] is the expected payoff for either player using the equilibrium strategies.
[tex]\[ V = p \cdot (a_{11} q + a_{12} (1 - q)) + (1 - p) \cdot (a_{21} q + a_{22} (1 - q)) \][/tex]
Substitute [tex]\( p \)[/tex] and [tex]\( q \)[/tex] into the expected payoff equations to calculate [tex]\( V \)[/tex].
### Step 5: Final Substitution (Example Calculation)
If numerical values are given, substitute [tex]\( p \)[/tex] and [tex]\( q \)[/tex] into the expected payoff equations and compute [tex]\( V \)[/tex]. However, without specific values for [tex]\( a_{11}, a_{12}, a_{21}, \)[/tex] and [tex]\( a_{22} \)[/tex], we leave the solution in terms of the expressions we derived above.
Thus, the mixed strategy Nash equilibrium for the game and the value of the game can be calculated using these results:
[tex]\[ p = \frac{a_{22} - a_{21}}{a_{11} - a_{21} - a_{12} + a_{22}} \][/tex]
[tex]\[ q = \frac{a_{22} - a_{12}}{a_{11} - a_{12} - a_{21} + a_{22}} \][/tex]
The value of the game [tex]\( V \)[/tex] is then computed by substituting [tex]\( p \)[/tex] and [tex]\( q \)[/tex] back into the expected payoff formula.
[tex]\[ \left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] \][/tex]
we will follow the following steps:
### Step 1: Define the Expected Payoffs
Let's denote the probabilities that Player 1 chooses Row 1 by [tex]\( p \)[/tex] and Row 2 by [tex]\( 1 - p \)[/tex]. Similarly, let’s denote the probabilities that Player 2 chooses Column 1 by [tex]\( q \)[/tex] and Column 2 by [tex]\( 1 - q \)[/tex].
The expected payoff for Player 1 (E1) when playing mixed strategies is:
[tex]\[ E1 = p \cdot (a_{11}q + a_{12}(1 - q)) + (1 - p) \cdot (a_{21}q + a_{22}(1 - q)) \][/tex]
The expected payoff for Player 2 (E2) when playing mixed strategies is:
[tex]\[ E2 = q \cdot (a_{11}p + a_{21}(1 - p)) + (1 - q) \cdot (a_{12}p + a_{22}(1 - p)) \][/tex]
### Step 2: Set Up the System of Equations
To find the mixed strategy Nash equilibrium, both players should be indifferent to their choices. This means the expected payoffs for each player should be equal when switching strategies.
For Player 1 to be indifferent:
[tex]\[ a_{11}q + a_{12}(1 - q) = a_{21}q + a_{22}(1 - q) \][/tex]
For Player 2 to be indifferent:
[tex]\[ a_{11}p + a_{21}(1 - p) = a_{12}p + a_{22}(1 - p) \][/tex]
### Step 3: Solve the Equations for [tex]\( p \)[/tex] and [tex]\( q \)[/tex]
#### Finding [tex]\( q \)[/tex]:
[tex]\[ a_{11}q + a_{12}(1 - q) = a_{21}q + a_{22}(1 - q) \][/tex]
Simplify the equation:
[tex]\[ a_{11} q + a_{12} - a_{12} q = a_{21} q + a_{22} - a_{22} q \][/tex]
Combine like terms:
[tex]\[ (a_{11} - a_{12}) q + a_{12} = (a_{21} - a_{22}) q + a_{22} \][/tex]
Rearrange to solve for [tex]\( q \)[/tex]:
[tex]\[ (a_{11} - a_{12} - a_{21} + a_{22}) q = a_{22} - a_{12} \][/tex]
[tex]\[ q = \frac{a_{22} - a_{12}}{a_{11} - a_{12} - a_{21} + a_{22}} \][/tex]
#### Finding [tex]\( p \)[/tex]:
[tex]\[ a_{11}p + a_{21}(1 - p) = a_{12}p + a_{22}(1 - p) \][/tex]
Simplify the equation:
[tex]\[ a_{11} p + a_{21} - a_{21} p = a_{12} p + a_{22} - a_{22} p \][/tex]
Combine like terms:
[tex]\[ (a_{11} - a_{21}) p + a_{21} = (a_{12} - a_{22}) p + a_{22} \][/tex]
Rearrange to solve for [tex]\( p \)[/tex]:
[tex]\[ (a_{11} - a_{21} - a_{12} + a_{22}) p = a_{22} - a_{21} \][/tex]
[tex]\[ p = \frac{a_{22} - a_{21}}{a_{11} - a_{21} - a_{12} + a_{22}} \][/tex]
### Step 4: Determine the Value of the Game
The value of the game [tex]\( V \)[/tex] is the expected payoff for either player using the equilibrium strategies.
[tex]\[ V = p \cdot (a_{11} q + a_{12} (1 - q)) + (1 - p) \cdot (a_{21} q + a_{22} (1 - q)) \][/tex]
Substitute [tex]\( p \)[/tex] and [tex]\( q \)[/tex] into the expected payoff equations to calculate [tex]\( V \)[/tex].
### Step 5: Final Substitution (Example Calculation)
If numerical values are given, substitute [tex]\( p \)[/tex] and [tex]\( q \)[/tex] into the expected payoff equations and compute [tex]\( V \)[/tex]. However, without specific values for [tex]\( a_{11}, a_{12}, a_{21}, \)[/tex] and [tex]\( a_{22} \)[/tex], we leave the solution in terms of the expressions we derived above.
Thus, the mixed strategy Nash equilibrium for the game and the value of the game can be calculated using these results:
[tex]\[ p = \frac{a_{22} - a_{21}}{a_{11} - a_{21} - a_{12} + a_{22}} \][/tex]
[tex]\[ q = \frac{a_{22} - a_{12}}{a_{11} - a_{12} - a_{21} + a_{22}} \][/tex]
The value of the game [tex]\( V \)[/tex] is then computed by substituting [tex]\( p \)[/tex] and [tex]\( q \)[/tex] back into the expected payoff formula.
