# MATHEMATICS

## 10. Consider the following two-person zero-sum game. Assume the two players have the same two strategy options. The payoff table shows the gains for Player A.

Player B | ||

Player A | Strategy b_{1} |
Strategy b_{2} |

Strategy a_{1} |
3 | 9 |

Strategy a_{2} |
6 | 2 |

Determine the optimal strategy for each player. What is the value of the game?

11.consider the following two-person zero-sum game. identify the pure strategy what is value game?

Player a | Player b | |||

B1 | B2 | B3 | ||

a1 | 8 | 5 | 7 | |

a2 | 2 | 4 | 10 |

2. Suppose that a decision maker faced with four decision alternatives and four states of nature develops the following profit payoff table:

State of Nature

Decision Alternative S1 S2 S3 S4

D1 14 9 10 5

D2 11 10 8 7

D3 9 10 10 11

D4 8 10 11 13

A. If the decision maker knows nothing about the probabilities of the four states of nature, what is recommended decision using the optimistic, conservative and minimax regret approaches?

B. Which approach do you prefer? Explain. Is establishing the most appropriate approach before analyzing the problem important for the decision maker? Explain.

C. Assume that the payoff table provides cost rather than profit payoffs. What is the recommended decision using the optimistic, conservative and minimax regret approaches?

9- Myrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full-price service using the company’s new fleet of jet aircraft and a discount service using smaller capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based upon two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars).

Demand for service

Service Strong Weak

Full price 960 -490

discount 670 320

a. What is the decision to be made, what is the chance event and what is the consequence for this problem? How many outcomes are there for the chance event?

b. If nothing is known about the probabilities of the chance outcomes, determine the recommended decision using the optimistic, conservative, and minimax regret approaches.

8- The following payoff table shows profit for a decision analysis problem with two decision alternatives and three states of nature:

States of Nature

Decision Alternative

S1

S2

D1

10

4

D2

1

3

a- Use the graphical sensitivity analysis to determine the range of values of the probability of state of nature s1 over which each of the decision alternatives has its largest expected value

b- Suppose the decision maker obtains the probabilities P(S1) = 0.2, P(S2) = 0.8, What is the expected value approach?