Answer :
Certainly! To solve this game graphically, we need to analyze the payoff matrix for player [tex]\( A \)[/tex].
### Payoff Matrix:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & a1 & a2 & a3 & a4 \\ \hline b1 & -7 & 7 & -4 & 8 \\ \hline b2 & 6 & -4 & -2 & -6 \\ \hline \end{array} \][/tex]
### Step-by-step Solution:
1. Identify the Strategies:
- Player [tex]\( A \)[/tex] has strategies: [tex]\( a1, a2, a3, a4 \)[/tex]
- Player [tex]\( B \)[/tex] has strategies: [tex]\( b1, b2 \)[/tex]
2. Graphical Approach:
We will plot the payoffs for each combination of strategies graphically to find the optimal strategies for both players.
3. Analyzing Player [tex]\( A \)[/tex]'s Payoffs:
- For [tex]\( a1 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(-7\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(6\)[/tex]
- For [tex]\( a2 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(7\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(-4\)[/tex]
- For [tex]\( a3 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(-4\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(-2\)[/tex]
- For [tex]\( a4 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(8\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(-6\)[/tex]
4. Plotting Payoffs for Each Strategy of Player [tex]\( A \)[/tex]:
We plot the payoffs for each strategy of Player [tex]\( A \)[/tex] against Player [tex]\( B \)[/tex]'s strategies.
- X-axis: Probability [tex]\( p \)[/tex] of Player [tex]\( B \)[/tex] playing [tex]\( b1 \)[/tex]
- Y-axis: Expected payoff of Player [tex]\( A \)[/tex]
Let's denote the probability [tex]\( p \)[/tex] that Player [tex]\( B \)[/tex] selects [tex]\( b1 \)[/tex], and [tex]\( 1 - p \)[/tex] the probability of selecting [tex]\( b2 \)[/tex].
5. Expected Payoff Equations:
- For [tex]\( a1 \)[/tex]: Expected payoff = [tex]\( -7p + 6(1-p) = -7p + 6 - 6p = -13p + 6 \)[/tex]
- For [tex]\( a2 \)[/tex]: Expected payoff = [tex]\( 7p + (-4)(1-p) = 7p - 4 + 4p = 11p - 4 \)[/tex]
- For [tex]\( a3 \)[/tex]: Expected payoff = [tex]\( -4p + (-2)(1-p) = -4p - 2 + 2p = -2p - 2 \)[/tex]
- For [tex]\( a4 \)[/tex]: Expected payoff = [tex]\( 8p - 6(1-p) = 8p - 6 + 6p = 14p - 6 \)[/tex]
6. Graphically Analysis:
To find the optimal mixed strategy for Player [tex]\( A \)[/tex], plot these lines on a graph and observe the intersection points.
- Line for [tex]\( a1 \)[/tex]: [tex]\( y = -13p + 6 \)[/tex]
- Line for [tex]\( a2 \)[/tex]: [tex]\( y = 11p - 4 \)[/tex]
- Line for [tex]\( a3 \)[/tex]: [tex]\( y = -2p - 2 \)[/tex]
- Line for [tex]\( a4 \)[/tex]: [tex]\( y = 14p - 6 \)[/tex]
Each line represents Player [tex]\( A \)[/tex]'s expected payoff for a given strategy based on the probability [tex]\( p \)[/tex] that Player [tex]\( B \)[/tex] plays [tex]\( b1 \)[/tex].
### Finding Optimal Strategy:
To solve graphically is quite intensive, so typically we would look for intersecting points of these lines. The optimal solution would involve the highest minimum values of these intersections. Alternatively, linear programming could be used to solve the game more precisely.
### Conclusion:
Graphical analysis helps visualize the payoffs and determine optimal strategies. The solution is typically found by identifying the intersection which gives the highest minimum expected payoff for Player [tex]\( A \)[/tex], but a detailed graph and further calculations are needed for exact numbers.
### Payoff Matrix:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & a1 & a2 & a3 & a4 \\ \hline b1 & -7 & 7 & -4 & 8 \\ \hline b2 & 6 & -4 & -2 & -6 \\ \hline \end{array} \][/tex]
### Step-by-step Solution:
1. Identify the Strategies:
- Player [tex]\( A \)[/tex] has strategies: [tex]\( a1, a2, a3, a4 \)[/tex]
- Player [tex]\( B \)[/tex] has strategies: [tex]\( b1, b2 \)[/tex]
2. Graphical Approach:
We will plot the payoffs for each combination of strategies graphically to find the optimal strategies for both players.
3. Analyzing Player [tex]\( A \)[/tex]'s Payoffs:
- For [tex]\( a1 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(-7\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(6\)[/tex]
- For [tex]\( a2 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(7\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(-4\)[/tex]
- For [tex]\( a3 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(-4\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(-2\)[/tex]
- For [tex]\( a4 \)[/tex]:
- Against [tex]\( b1 \)[/tex]: [tex]\(8\)[/tex]
- Against [tex]\( b2 \)[/tex]: [tex]\(-6\)[/tex]
4. Plotting Payoffs for Each Strategy of Player [tex]\( A \)[/tex]:
We plot the payoffs for each strategy of Player [tex]\( A \)[/tex] against Player [tex]\( B \)[/tex]'s strategies.
- X-axis: Probability [tex]\( p \)[/tex] of Player [tex]\( B \)[/tex] playing [tex]\( b1 \)[/tex]
- Y-axis: Expected payoff of Player [tex]\( A \)[/tex]
Let's denote the probability [tex]\( p \)[/tex] that Player [tex]\( B \)[/tex] selects [tex]\( b1 \)[/tex], and [tex]\( 1 - p \)[/tex] the probability of selecting [tex]\( b2 \)[/tex].
5. Expected Payoff Equations:
- For [tex]\( a1 \)[/tex]: Expected payoff = [tex]\( -7p + 6(1-p) = -7p + 6 - 6p = -13p + 6 \)[/tex]
- For [tex]\( a2 \)[/tex]: Expected payoff = [tex]\( 7p + (-4)(1-p) = 7p - 4 + 4p = 11p - 4 \)[/tex]
- For [tex]\( a3 \)[/tex]: Expected payoff = [tex]\( -4p + (-2)(1-p) = -4p - 2 + 2p = -2p - 2 \)[/tex]
- For [tex]\( a4 \)[/tex]: Expected payoff = [tex]\( 8p - 6(1-p) = 8p - 6 + 6p = 14p - 6 \)[/tex]
6. Graphically Analysis:
To find the optimal mixed strategy for Player [tex]\( A \)[/tex], plot these lines on a graph and observe the intersection points.
- Line for [tex]\( a1 \)[/tex]: [tex]\( y = -13p + 6 \)[/tex]
- Line for [tex]\( a2 \)[/tex]: [tex]\( y = 11p - 4 \)[/tex]
- Line for [tex]\( a3 \)[/tex]: [tex]\( y = -2p - 2 \)[/tex]
- Line for [tex]\( a4 \)[/tex]: [tex]\( y = 14p - 6 \)[/tex]
Each line represents Player [tex]\( A \)[/tex]'s expected payoff for a given strategy based on the probability [tex]\( p \)[/tex] that Player [tex]\( B \)[/tex] plays [tex]\( b1 \)[/tex].
### Finding Optimal Strategy:
To solve graphically is quite intensive, so typically we would look for intersecting points of these lines. The optimal solution would involve the highest minimum values of these intersections. Alternatively, linear programming could be used to solve the game more precisely.
### Conclusion:
Graphical analysis helps visualize the payoffs and determine optimal strategies. The solution is typically found by identifying the intersection which gives the highest minimum expected payoff for Player [tex]\( A \)[/tex], but a detailed graph and further calculations are needed for exact numbers.
