Consider a committee of Ann, Bob. and Carol that is choosing between three alter

Question

Consider a committee of Ann, Bob. and Carol that is choosing between three alternatives, {x.y.z}. Player i's preferences are represented by the ordering Suppose the three players preferences are: x A y Az Y B Bx z cx cy Each player has complete and transitive strict preference. Suppose the committee decides to operate on the basis of majority-rule: the committee strictly prefers alternative a to alternative 6 if a strict majority prefer alternative a to b. Let M be the committee's preference. Based on the preferences of the committee members. Ls complete and transitive? Can you represent committee preferences by a utility function?

Explanation / Answer

A) We have 3 indivdual preferences, so we count the number of individuals for whom  X i Y or Y i X. Since, X i Y 2 times and Y i X for 1 time only, we can say the group preference (m) satisfies X Y, when these two alternatives are considered in isolation. So we can say that m   is COMPLETE.

Using majority-rule to define m , we’d have X Y (since X is preferred to Y 2timesthan 1), Y Z (since Y is preferred to Z 2 times than 1), and Z X (since Z is preferred to X 2 times than 1). This violates transitivity, which would require X Z once we have X Y and Y Z.

B) There are 3 alternatives A3 = AX , Ay , Az. And the committee has N members (N=3) A,B,C. Ui represents utility of an individual.

Committee preferences by a utility function:

UG (AX , Ay , Az) = Ni=1 Ui (AX , Ay , Az)

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