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Sagot :
Sure, let's tackle each part of the problem step by step.
### (a) Check if the Game bears a "saddle point".
A saddle point in a game is a situation where the minimum element in a row is also the maximum element in its column. To find the saddle point, we will analyze the given payoff matrix for Player A and Player B:
[tex]\[ \text { Player } A \begin{bmatrix} A_1 & B_1 & B_2 & B_3 & B_4 \\ A_1 & -6 & -1 & 4 & 3 \\ A_2 & 7 & -2 & 5 & 7 \end{bmatrix} \][/tex]
1. Identify Row Minima:
- For [tex]\(A_1\)[/tex]: The minimum value is [tex]\(-6\)[/tex]
- For [tex]\(A_2\)[/tex]: The minimum value is [tex]\(-2\)[/tex]
2. Identify Column Maxima:
- For [tex]\(B_1\)[/tex]: The maximum value is [tex]\(7\)[/tex]
- For [tex]\(B_2\)[/tex]: The maximum value is [tex]\(-1\)[/tex]
- For [tex]\(B_3\)[/tex]: The maximum value is [tex]\(5\)[/tex]
- For [tex]\(B_4\)[/tex]: The maximum value is [tex]\(7\)[/tex]
Now, we compare the row minima and column maxima. The only common value between these two sets (i.e., minima from rows and maxima from columns) is [tex]\(-2\)[/tex]. Since [tex]\(-2\)[/tex] is both a minimum in its row and a maximum in its column, there is a saddle point at [tex]\((A_2, B_2)\)[/tex].
### (b) Find [tex]\(A\)[/tex]'s expected pay-off equation corresponding to [tex]\(A\)[/tex]'s pure strategy.
The expected payoff for Player A using pure strategies can be represented as:
1. If Player A chooses [tex]\(A_1\)[/tex], the expected payoff [tex]\(E(A_1)\)[/tex] for Player A:
[tex]\[ E(A_1) = x_1(-6) + x_2(-1) + x_3(4) + x_4(3) \][/tex]
Here, [tex]\(x_i = 1\)[/tex] if Player B chooses [tex]\(B_i\)[/tex], else [tex]\(x_i = 0\)[/tex].
2. If Player A chooses [tex]\(A_2\)[/tex], the expected payoff [tex]\(E(A_2)\)[/tex] for Player A:
[tex]\[ E(A_2) = x_1(7) + x_2(-2) + x_3(5) + x_4(7) \][/tex]
### (c) Find the relevant strategies of [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Relevant strategies are the strategies that could potentially be optimal for the players.
For Player A:
- [tex]\(A_1\)[/tex], since it corresponds to the rows [tex]\([-6, -1, 4, 3]\)[/tex].
- [tex]\(A_2\)[/tex], since it corresponds to the rows [tex]\([7, -2, 5, 7]\)[/tex].
For Player B:
- [tex]\(B_1, B_2, B_3, B_4\)[/tex], since these represent all the columns from which Player B can choose.
### (d) Find the optimal strategy of [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Given that we found a saddle point at [tex]\((A_2, B_2)\)[/tex]:
For Player A:
- Optimal strategy: [tex]\(A_2\)[/tex].
For Player B:
- Optimal strategy: [tex]\(B_2\)[/tex].
The optimal strategies are where Player A consistently chooses [tex]\(A_2\)[/tex] and Player B consistently chooses [tex]\(B_2\)[/tex], ensuring the game remains stable at the saddle point with the payoff value of [tex]\(-2\)[/tex].
### (a) Check if the Game bears a "saddle point".
A saddle point in a game is a situation where the minimum element in a row is also the maximum element in its column. To find the saddle point, we will analyze the given payoff matrix for Player A and Player B:
[tex]\[ \text { Player } A \begin{bmatrix} A_1 & B_1 & B_2 & B_3 & B_4 \\ A_1 & -6 & -1 & 4 & 3 \\ A_2 & 7 & -2 & 5 & 7 \end{bmatrix} \][/tex]
1. Identify Row Minima:
- For [tex]\(A_1\)[/tex]: The minimum value is [tex]\(-6\)[/tex]
- For [tex]\(A_2\)[/tex]: The minimum value is [tex]\(-2\)[/tex]
2. Identify Column Maxima:
- For [tex]\(B_1\)[/tex]: The maximum value is [tex]\(7\)[/tex]
- For [tex]\(B_2\)[/tex]: The maximum value is [tex]\(-1\)[/tex]
- For [tex]\(B_3\)[/tex]: The maximum value is [tex]\(5\)[/tex]
- For [tex]\(B_4\)[/tex]: The maximum value is [tex]\(7\)[/tex]
Now, we compare the row minima and column maxima. The only common value between these two sets (i.e., minima from rows and maxima from columns) is [tex]\(-2\)[/tex]. Since [tex]\(-2\)[/tex] is both a minimum in its row and a maximum in its column, there is a saddle point at [tex]\((A_2, B_2)\)[/tex].
### (b) Find [tex]\(A\)[/tex]'s expected pay-off equation corresponding to [tex]\(A\)[/tex]'s pure strategy.
The expected payoff for Player A using pure strategies can be represented as:
1. If Player A chooses [tex]\(A_1\)[/tex], the expected payoff [tex]\(E(A_1)\)[/tex] for Player A:
[tex]\[ E(A_1) = x_1(-6) + x_2(-1) + x_3(4) + x_4(3) \][/tex]
Here, [tex]\(x_i = 1\)[/tex] if Player B chooses [tex]\(B_i\)[/tex], else [tex]\(x_i = 0\)[/tex].
2. If Player A chooses [tex]\(A_2\)[/tex], the expected payoff [tex]\(E(A_2)\)[/tex] for Player A:
[tex]\[ E(A_2) = x_1(7) + x_2(-2) + x_3(5) + x_4(7) \][/tex]
### (c) Find the relevant strategies of [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Relevant strategies are the strategies that could potentially be optimal for the players.
For Player A:
- [tex]\(A_1\)[/tex], since it corresponds to the rows [tex]\([-6, -1, 4, 3]\)[/tex].
- [tex]\(A_2\)[/tex], since it corresponds to the rows [tex]\([7, -2, 5, 7]\)[/tex].
For Player B:
- [tex]\(B_1, B_2, B_3, B_4\)[/tex], since these represent all the columns from which Player B can choose.
### (d) Find the optimal strategy of [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Given that we found a saddle point at [tex]\((A_2, B_2)\)[/tex]:
For Player A:
- Optimal strategy: [tex]\(A_2\)[/tex].
For Player B:
- Optimal strategy: [tex]\(B_2\)[/tex].
The optimal strategies are where Player A consistently chooses [tex]\(A_2\)[/tex] and Player B consistently chooses [tex]\(B_2\)[/tex], ensuring the game remains stable at the saddle point with the payoff value of [tex]\(-2\)[/tex].
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