Q7 7. Mr. Smith is expecting his 3 nephews to dinner together, but he is aware t

Question

Q7

7. Mr. Smith is expecting his 3 nephews to dinner together, but he is aware that not all of them might come. He thinks that the probabilities of 0, 1, 2, 3 nephews arriving 3/8. are, respectively, 1/8, 2/8, 2/8 and He must decide how many meals to cook. His preferences may be represented by the utility function u(x,y, z) = x-2y-22 where x number of nephews who are fed; y s number of nephews who are unfed because he did not cook enough food; z = number of meals wasted because he cooked too many. No meal can be shared. Each nephew can eat only one meal. How many meals should he cook? (Hint: Find the expected utility of cooking 0, 1, 2, 3 meals)

Explanation / Answer

When he cooks 0 meals:

E[u] = Pr[0 nephews arrive] * u[0] + Pr[1 nephews arrive] * u[1] + Pr[2 nephews arrive] * u[2] + Pr[3 nephews arrive] * u[3] = (1/8*0 + 2/8*(-2) + 2/8*(-4) + 3/8*(-6)) = (-4 - 8 - 18)/8 = -3.75

When he cooks 1 meals:

E[u] = Pr[0 nephews arrive] * u[0] + Pr[1 nephews arrive] * u[1] + Pr[2 nephews arrive] * u[2] + Pr[3 nephews arrive] * u[3] = (1/8*(-12) + 2/8*(1) + 2/8*(-1) + 3/8*(-3)) = (-1 + 2 - 2 - 9)/8 = -1.25

When he cooks 2 meals:

E[u] = Pr[0 nephews arrive] * u[0] + Pr[1 nephews arrive] * u[1] + Pr[2 nephews arrive] * u[2] + Pr[3 nephews arrive] * u[3] = (1/8*(-22) + 2/8*(1-12) + 2/8*(2) + 3/8*(2-2)) = (-4 + 0 + 4 - 0)/8 = 0

When he cooks 3 meals:

E[u] = Pr[0 nephews arrive] * u[0] + Pr[1 nephews arrive] * u[1] + Pr[2 nephews arrive] * u[2] + Pr[3 nephews arrive] * u[3] = (1/8*(-32) + 2/8*(1-22) + 2/8*(2-12) + 3/8*(3)) = (-9 - 6 + 2 + 9)/8 = -0.5

The maximum value of expected utility (= 0) occurs when he cooks 2 meals. So, he should cook 2 meals.

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