Naval Operations Analysis, Additional Problems for Chapter 2

ANSWERS

 

1. For the following payoff matrix, which is the optimal strategy (S1, S2 or S3) for the indicated criterion?  Show all work.
 

	N1	N2	N3	N4
S1	5	3	4	1
S2	2	6	5	3
S3	0	2	3	5

(a) Using the Wald (Maximin or "best guaranteed payoff") criterion.
(b) Using the minimum (or least) regret criterion.
(c) Using the Laplace (rationality) criterion where each state is given equal weight.

2. Suppose an executive must make a decision that has four possible outcomes.  Given in order of preference (best to worst) they are identified as A, B, C, and D.
(a) Assume that the executive has decided that he is indifferent to a situation where B is the certain outcome and a situation where A is the outcome 75% of the time, B and C are never the outcome, and D is the outcome 25% of the time.  He is also indifferent to a situation where C is the certain outcome and a situation where A and C are never the outcome, B is the outcome 2/3 of the time and D is the outcome 1/3 of the time. Find a utility function that you could use to help this executive in his decision.
(b) The assumptions made in (a) do not apply to (b).  Suppose the executive has retained an OR consultant who has helped him find the utility function given in the following table.

outcome	utility
A	4.0
B	3.2
C	1.8
D	0.0

If the executive used this utility function, which of the following decisions would he prefer?  The decisions have the following probabilities of outcomes:

	A	B	C	D
Decision 1	0.2	0.3	0.1	0.4
Decision 2	0.1	0.4	0.2	0.3

 

3.  Suppose the following is a payoff matrix for a game between Blue and Red.

	Red 1	Red 2
Blue 1	1	4
Blue 2	5	2

(a) Find the optimal pure strategies for both Blue and Red.
(b) Find the optimal mixed strategy for Blue.
(c) What is the value of the game?

 

ANSWERS

1. (a) S2, (b) S2, (c) S2.
2. Decision 2.
3. (a) B2, R2, (b) Pr{B1}=.5, Pr{B2}=.5, (c) 3.