Hello, all! I need help solving these equations using Multi-Attribute Utility Theory (MAUT). If you can show your work, that would be great. Thank you! 1.1. A decision maker (DM) wants to buy a...

Hello, all! I need help solving these equations using Multi-Attribute Utility Theory (MAUT). If you can show your work, that would be great. Thank you!

1.1. A decision maker (DM) wants to buy a new car from one of the dealers in the city. S/he thinks the most important attributes are customer service and reliability. Each attribute is assigned a score from 0 to 100 (0 is the worst and 100 is the best).
Suppose that US(x) is the utility function on customer service and UR(y) is the utility function on reliability. The DM uses the following multiplicative form of the utility function: US,R(x, y) = kSUS(x) + kRUR(y) + (1 − kS − kR)US(x)UR(y).
Suppose that the DM is indifferent between the following two scenarios A and B:
A:
50% chance with customer service score 100 and reliability score 0
50% chance with customer service score 0 and reliability score 100
B:
50% chance with customer service score 0 and reliability score 0
50% chance with customer service score 100 and reliability score 100

What can you say about 1 − kS − kR?

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mathsworkmusic | (Level 2) Educator

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The Utility Function depicts the relative usefulness of different scenarios in a process where a decision is to be made.

Here we have two variables and so are considering the properties of the bivariate utility function US,R(x,y) that is taken over the two variables S and R where these relate respectively to customer service score and reliability score.

First consider scenario A and what this tells us about US,R(x,y). In this scenario, the two pairs of scores (S,R) = (100,0) and (S,R) = (0,100) occur with equal probability. No other outcomes are possible, or at least we have no other outcomes to consider. Each of these pairs will have a bivariate utility US,R(x,y), but we don't know the value or the relative heights at the two points. Similarly, for scenario B the two outcomes (S,R) = (0,0) and (S,R) = (100,100) occur with equal probability and are the only possible outcomes. Again we don't know the height of the bivariate utility function at these points, nor the relative height of this utility at the two points. However, we do know that the sum of the utility at the two points described in each of the scenarios A and B is equal, as we are told the customer is indifferent between the two scenarios A and B, meaning that the usefulness or utility of the outcome portfolio for the customer is the same in both scenarios.

But what can we say about the coefficient (1-kS-kR) in the function that relates the bivariate utility US,R(x,y) to the two univariate utilities US(x) and UR(y)? Unfortunately, not knowing the relative heights of US,R(x,y) in each of the scenarios A and B of the two defined points on the (x,y) plane, means that we cannot know the relative heights of US(x) at x=0 or x=100 (the only two possible scores in either scenario) or of UR(y) at y=0 and y=100. The sum of the univariate utility in each case may depend on the value of the other variable. That is, US(x=0) + US(x=100) may differ depending on whether y=0 or y=100. This is the key to understanding what we can say about (1-kS-kR), as the bivariate utility US,R(x,y) isn't necessarily comprise of the two univariate utilities as independent entities, but may also contain a non-zero interaction term involving the product of the two utilities. If that interaction term exists, that is is non-zero, then the coefficient (1-kS-kR) would be non-zero. Without the knowledge that the two univariate utility functions are independent of each other when the other variable is taken into account, we cannot say for certain that (1-kS-kR) is zero.

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