Question

A decision-maker is a father who has two sons.They are going to purchase a new house. Suppose there are five possible choices ω1, ω2, ... , ω5. The rankings of houses from the most favorite to the least favorite are as following: • Father: ω2, ω5, ω3, ω4, ω1; • Son 1: ω5, ω4, ω1, ω3, ω2; • Son 2: ω1, ω2, ω3, ω5, ω4. Suppose the decision-maker cares and only cares the feeling of his sons, and thus he comes up the following method to choose the new house: • For houses that are available, he asks both of his sons to rank them according to their preferences and assign a number to them starting from 1. For example, if the available houses are ω1 and ω4 and Son 1 ranks them truthfully, then he will assign number 1 to ω4 and 2 to ω1. After gathering those numbers, the father will then choose the house such that the sum of the numbers are minimum. If there is a tie, then the decision-maker will choose the one that he prefers. Suppose the sons always rank truthfully, will the decision-maker’s choice function satisfy the independence condition? and why?

Answer 1

The House’s that are available choices are: w1 , w2 , w3 , w4 & w5

The Fathers choice of the houses from most favorite to least favorite is : w2,w5,w3,w4,w1

Son1 ‘s choice of the houses from most favorite to least favorite is : w5,w4,w1,w3,w2

Son2 ‘s choice of the houses from most favorite to least favorite is : w1,w2,w3,w5,w4

           Since the father has decided that he would respect the choice made by hi Sons and in the process of determining the best choice has asked his sons to number the best houses they prefer in the order of 1 and 2 and 3 etc. depending upon their choice. He will then add up the numbers and the one with the least total will be the choice. If there is tie, then the Father’s choice will decide the final outcome.

                           From the given choices of Son1 and Son2 , we see that below are the total number the choices will make :

Choice w1 : 3 + 1 = 4

Choice w2 : 5 + 2 = 7

Choice w3 : 4 + 3 = 7

Choice w4 : 2 + 5 = 7

Choice w5 : 1 + 4 = 5

Therefore, depending in the choice the two Sons have made, Choice of house w1 is the most preferred choice. Now, house w1 is the house with the least preferability for the father, but since he gives more importance to his Son’s wishes, therefore he will go with house w1, and since there is no tie for the first choice, therefore the decision makers decision function will satisfy the independence condition.