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Question 1173267: Shown below is the solution to the linear program for finding Player A's optimal mixed strategy in a two-person, zero-sum game.
OBJECTIVE FUNCTION VALUE = 3.500
VARIABLE
VALUE
REDUCED COSTS
PA1
0.050
0.000
PA2
0.600
0.000
PA3
0.350
0.000
GAINA
3.500
0.000
CONSTRAINT
SLACK/SURPLUS
DUAL PRICES
1
0.000
−0.500
2
0.000
−0.500
3
0.000
0.000
4
0.000
3.500
a.
What is Player A's optimal mixed strategy?
b.
What is Player B's optimal mixed strategy?
c.
What is Player A's expected gain?
d.
What is Player B's expected loss?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down the information from the linear program output to answer each question:
**a. Player A's Optimal Mixed Strategy**
* **PA1 = 0.050:** Player A should play strategy 1 with a probability of 0.05 (or 5%).
* **PA2 = 0.600:** Player A should play strategy 2 with a probability of 0.60 (or 60%).
* **PA3 = 0.350:** Player A should play strategy 3 with a probability of 0.35 (or 35%).
Therefore, Player A's optimal mixed strategy is (0.05, 0.60, 0.35).
**b. Player B's Optimal Mixed Strategy**
To find Player B's optimal mixed strategy, we look at the **DUAL PRICES** of the constraints. These represent the optimal probabilities for Player B's strategies.
* Constraint 1's Dual Price = -0.500
* Constraint 2's Dual Price = -0.500
* Constraint 3's Dual Price = 0.000
However, the Dual Prices must be converted into positive values, and then normalized, as they represent the negative of the player B's strategy.
Also, the dual prices represent the negative of the player B's strategy, and the 4th constraint represents the gain.
Therefore, the dual prices of constraints 1,2, and 3 are the negative of the probabilities of player B's strategies.
To normalize the dual prices, we must take the absolute value of the dual prices. Then add them together.
0.5 + 0.5 + 0 = 1
Then we can divide each dual price by the sum.
* Player B strategy 1: 0.5/1 = 0.5
* Player B strategy 2: 0.5/1 = 0.5
* Player B strategy 3: 0/1 = 0
Therefore, Player B's optimal mixed strategy is (0.5, 0.5, 0).
**c. Player A's Expected Gain**
* **OBJECTIVE FUNCTION VALUE = 3.500**
* **GAINA = 3.500**
Player A's expected gain is 3.5.
**d. Player B's Expected Loss**
In a zero-sum game, Player B's expected loss is equal to Player A's expected gain.
Therefore, Player B's expected loss is also 3.5.
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