Question 5.p.6: Suppose that the values in the payoff table represent costs ...
Suppose that the values in the payoff table represent costs instead of profits.
a. Determine the choice you would make under each of these strategies: maximin, minimin, and Laplace.*
b. Develop the regret table, and identify the alternative chosen using minimax regret. Then, find the EVPI if P(new bridge) = .60.
c. Using sensitivity analysis, determine the range of P(no new bridge) for which each alternative would be optimal.
d. If P(new bridge) = .60 and P(no new bridge) = .40, find the alternative chosen to minimize expected cost.
a.
| New Bridge |
No New Bridge |
Maximin (worst) |
Minimin (best) |
Laplace (average) |
|
| A | 1 | 14 | 14 | 1 [best] | 15 ÷ 2 = 7.5 |
| B | 2 | 10 | 10 | 2 | 12 ÷ 2 = 6 |
| C | 4 | 6 | 6 [best] | 4 | 10 ÷ 2 = 5[best] |
b. Develop the regret table by subtracting the lowest cost in each column from each of the values in the column. (Note that none of the values is negative.)
| New Bridge |
No New Bridge |
Worst | |
| A | 0 | 8 | 8 |
| B | 1 | 4 | 4 |
| C | 3 | 0 | 3 |
| EVPI = .60(3) + .40(0) = 1.80 | |||
c. The graph is identical to that shown in Solved Problem 2. However, the lines now represent expected costs, so the best alternative for a given value of P(no new bridge) is the lowest line. Hence, for very low values of P(no new bridge), A is best; for intermediate values, B is best; and for high values, C is best. You can set the equations of A and B, and B and C, equal to each other in order to determine the values of P(no new bridge) at their intersections.
Thus,
A = B: 1 + 13P = 2 + 8P; solving, P = .20
B = C: 2 + 8P = 4 + 2P; solving, P = .33
Hence, the ranges are
A best: 0 ≤ P < .20
B best: .20 < P < .33
C best: .33 < P ≤ 1.00 (5.p.6.c)
d. Expected-value computations are the same whether the values represent costs or profits.
Hence, the expected payoffs for costs are the same as the expected payoffs for profits that were computed in Solved Problem 4. However, now you want the alternative that has the lowest expected payoff rather than the one with the highest payoff. Consequently, alternative C is the best because its expected payoff is the lowest of the three. Alternatively, the best choice (C) could be determined using the sensitivity ranges found in part c because .40 lies in the range .33 to 1.0.
