A decision-maker is a father w
A decision-maker is a father who has two sons.They are going topurchase a new house. Suppose there are five possible choices ω1,ω2, … , ω5. The rankings of houses from the most favorite to theleast 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 thedecision-maker cares and only cares the feeling of his sons, andthus he comes up the following method to choose the new house: •For houses that are available, he asks both of his sons to rankthem according to their preferences and assign a number to themstarting from 1. For example, if the available houses are ω1 and ω4and Son 1 ranks them truthfully, then he will assign number 1 to ω4and 2 to ω1. After gathering those numbers, the father will thenchoose the house such that the sum of the numbers are minimum. Ifthere is a tie, then the decision-maker will choose the one that heprefers. Suppose the sons always rank truthfully, will thedecision-maker’s choice function satisfy the independencecondition? and why?
Answer:
The House’s that are available choices are: w1 , w2 , w3 , w4& w5
The Fathers choice of the houses from most favorite to leastfavorite is : w2,w5,w3,w4,w1
Son1 ‘s choice of the houses from most favorite to leastfavorite is : w5,w4,w1,w3,w2
Son2 ‘s choice of the houses from most favorite to leastfavorite is : w1,w2,w3,w5,w4
Since the father has decided that he would respect the choice madeby hi Sons and in the process of determining the best choice hasasked his sons to number the best houses they prefer in the orderof 1 and 2 and 3 etc. depending upon their choice. He will then addup 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 finaloutcome.
From the given choices of Son1 and Son2 , we see that below are thetotal 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 isthe house with the least preferability for the father, but since hegives more importance to his Son’s wishes, therefore he will gowith house w1, and since there is no tie for the first choice,therefore the decision makers decision function will satisfy theindependence condition.
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