Question

Multiple choice:

Despite the presence of many safety measures, fire hazard in whisky distilleries is very high. We consider three distilleries, Dalwhinnie, Talisker and Lagavulin. Each of these distilleries has 726 barrels of whisky to be left to age in their warehouses. We assume that a distillery catches fire with probability 0.4 and whether a distillery catches fire is independent of what happens to other distilleries. If a fire happens, the distillery loses all the barrels stored in its warehouse. We assume that each distillery's utility function is u(b) = b^_1/2 (the square-root of b), where b is the final amount of barrels the distillery is able to sell.

Three managers at Dalwhinnie one day start discussing about possible ways to increase their expected utility. These are the plans they suggest:

Manager 1: we should exchange 1/2 of our barrels with 1/2 of Talisker's barrels [Note: if this plan is implemented, both Dalwhinnie's and Talisker's warehouses will contain 1/2 of Dalwhinnie's barrels and 1/2 of Talkisker's barrels.]  

Manager 2: we should exchange 1/3 of our barrels with 1/3 of each of the other two distilleries' barrels. [Note: if this plan is implemented, all distilleries' warehouses will contain 1/3 of Dalwhinnie's barrels, 1/3 of Talisker's barrels and 1/3 of Lagavulin's barrels.]  

Manager 3: our best option is to keep all our barrels.

Let U(1) denote Dalwhinnie's expected utility if Manager 1's plan is implemented, U(2) its expected utility if Manager 2's plan is implemented and U(3) its expected utility if Manager 3's plan is implemented.

Which of the following statements is true? (Numbers are approximated to two decimal digits)

Select one:

a. U(1) = 14.27 , U(2)= 10.48, U(3)= 16.17

b. U(1) = 18.85 , U(2) = 28.00, U(3)= 16.17

c. U(1) = 14.27 , U(2)= 10.48, U(3)= 435.60

d. U(1) = 22.86, U(2) = 28.00, U(3)= 16.17

e. U(1) = 18.85 , U(2) = 19.80 and U(3)= 16.17

f. U(1) = 22.86, U(2) = 28.00, U(3)= 435.60

g. U(1) = 18.85 , U(2)= 19.80, U(3)= 435.60

Please show work, thank you!

Answer 1

Each of these distilleries has 726 barrels of whisky to be left to age in their warehouses.

Probability of a distillery catching a fire = 0.4 (which is independent of other distilleries)

Utility function of each distillery:

where b is final number of barrels the distillery is able to sell.

In case of fire in a distillery, all barrels lost => b = 0

Manager 1's plan is implemented:

Manager 2's plan is implemented:

Manager 3's plan is implemented:

Correct Ans - D