Midterm Test, Oct. 24, 2018 Page 2/4 is expecting his 3 nephews to dinner togeth

Question

Midterm Test, Oct. 24, 2018 Page 2/4 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 are, respectively, 1/8, 2/8, 2/8 and 3/8. He must decide how many meals to cook. His preferences may be represented by the utility function u(x,y,z)=2x-2y-z? where x number of nephews who are fed; y 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 a. b. How many meals should he cook? Do you think if Mr. Smith's decision will be the same under the new utility function u(x, y, z) = x-2y-3x2 (without detailed calculation of utilities).

Explanation / Answer

a.

Below table shows the utility for each possible outcome.

Expected utility when number meals cooked is 0 = (1/8) * 0 + (2/8) * (-2) + (2/8) * (-4) + (3/8) * (-6) = -30/8

Expected utility when number meals cooked is 1 = (1/8) * (-1) + (2/8) * (2) + (2/8) * (0) + (3/8) * (-2) = -3/8

Expected utility when number meals cooked is 2 = (1/8) * (-4) + (2/8) * (1) + (2/8) * (4) + (3/8) * (2) = +12/8

Expected utility when number meals cooked is 3 = (1/8) * (-9) + (2/8) * (-2) + (2/8) * (3) + (3/8) * (6) = +11/8

As, the expected utility is maximum when number meals cooked is 2, number of meals cooked should be 2.

b.

The new utiliy function u(x,y,z) = x - 2y -3z2 gave more negative weights to number of meals wasted, so the decision will be more inclined towards less number of meals prepared (either 0 or 1) so that the number of meals wasted is minimum. So, Mr. Smith's decision will change under the new utility function.

Meal Cooked Number of Nephews arrived 0 1 2 3 0 0 -2 -4 -6 1 -1 2 0 -2 2 -4 1 4 2 3 -9 -2 3 6

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