A commander faces a battlefield mobility situation. He has three feasible routes, R1, R2 and R3. The payoffs of his choice will be determined by weather conditions W1, W2 and W3. The best result is to gain 200 miles and the worst result is to lose 160 miles. The commander is indifferent between a 60 mile gain, and an 80% chance of gaining 200 miles with a 20% chance of losing 160 miles. He is also indifferent between a 100 mile loss and an 80% chance of losing 160 miles and a 20% chance of gaining 200 miles. The probabilities of weather events occur are the following: P(W1)=0.6; P(W2)=0.3; P(W3)=0.1

Question

A commander faces a battlefield mobility situation. He has three feasible routes, R1, R2 and R3. The payoffs of his choice will be determined by weather conditions W1, W2 and W3. The best result is to gain 200 miles and the worst result is to lose 160 miles. The commander is indifferent between a 60 mile gain, and an 80% chance of gaining 200 miles with a 20% chance of losing 160 miles. He is also indifferent between a 100 mile loss and an 80% chance of losing 160 miles and a 20% chance of gaining 200 miles. The probabilities of weather events occur are the following: P(W1)=0.6; P(W2)=0.3; P(W3)=0.1

Mileage payoff table
W1 W2 W3
R1 200 -160 60
R2 60 60 60
R3 -100 200 60

If the commander maximizes expected utility, which route will he choose?

Answer 1

To determine which route the commander will choose if they maximize expected utility, we need to calculate the expected payoff for each route.

The expected payoff of a route is calculated by multiplying the payoff for each weather condition by the probability of that weather condition occurring, and then summing these values for each weather condition.

Let's calculate the expected payoff for each route:

For Route R1:
Expected payoff = (P(W1) * 200) + (P(W2) * (-160)) + (P(W3) * 60)
= (0.6 * 200) + (0.3 * (-160)) + (0.1 * 60)
= 120 - 48 + 6
= 78

For Route R2:
Expected payoff = (P(W1) * 60) + (P(W2) * 60) + (P(W3) * 60)
= (0.6 * 60) + (0.3 * 60) + (0.1 * 60)
= 36 + 18 + 6
= 60

For Route R3:
Expected payoff = (P(W1) * (-100)) + (P(W2) * 200) + (P(W3) * 60)
= (0.6 * (-100)) + (0.3 * 200) + (0.1 * 60)
= -60 + 60 + 6
= 6

Now, let's compare the expected payoffs of each route:

Expected payoff for R1 = 78
Expected payoff for R2 = 60
Expected payoff for R3 = 6

The commander will choose the route that maximizes the expected utility, which means they will choose the route with the highest expected payoff. In this case, the commander will choose Route R1, as it has an expected payoff of 78, which is the highest among the three routes.