COU 2: Utility Representation Basics

Part 1: Representing Preference Orderings with Utility

Below are four different preference orderings over three alternatives, A, B and C. For each preference ordering, write a utility representation of that ordering. More specifically, for each ordering, specify utility levels for each of the three alternatives such that if in the preference ordering an alternative x is preferred to an alternative y, then in your utility representation, the utility level at x is higher than the utility level at y.

For instance, consider the following preference ordering over three alternatives X, Y and Z:

Y is preferred to X
Z is preferred to X
Y is preferred to Z

One utility representation of this preference ordering is:

Utility level at X is 2
Utility level at Y is 500
Utility level at Z is 2.4

This is a utility representation of the preference ordering above because:

Y is preferred to X and 500 is larger than 2
Z is preferred to X and 2.4 is larger than 2
Y is preferred to Z and 500 is larger than 2.4.

Note that any assigned utility levels represent a given preference ordering if those levels are ordered relative to one another in the right way. So there’s good news and bad news. The good news is that there are an infinite number of correct answers to each of the following four prompts. The bad news is that there are an infinite number of incorrect answers to each of the following four prompts!

Remember the academic honesty rules for this course: You may not look at answers written by another student and you may not show or dictate answers you have written to another student. Thus given the infinity of correct answers to the following four prompts, it is exceedingly unlikely that you will submit the same answer to any one of them as another student without having violated the academic honesty rules for this course!

Preference Ordering 1

A is preferred to B
A is preferred to C
B is preferred to C

Preference Ordering 2

B is preferred to A
A is preferred to C
B is preferred to C

Preference Ordering 3

B is preferred to A
C is preferred to A
B is preferred to C

Preference Ordering 4

B is preferred to A
C is preferred to A
C is preferred to B

Part 2: Deriving Preference Orderings from Utility

Below are four different utility representations of a person’s preferences over three alternatives, A, B and C. For each utility representation, write the preference ordering that the representation depicts.

For instance, consider the following utility representation of a person’s preferences over three alternatives X, Y and Z:

Utility level at X is 42
Utility level at Y is 1{,}641
Utility level at Z is 89

The preference ordering depicted by that utility representation is:

Y is preferred to X
Z is preferred to X
Y is preferred to Z

That preference ordering is the one depicted by the above utility representation because

1{,}641 is larger than 42, so Y is preferred to X.
89 is larger than 42, so Z is preferred to X.
1{,}641 is larger than 89, so Y is preferred to Z.

Unlike the problems in part 1, there is only one correct answer to each of these!

Utility Representation 1

Utility level at A is -6.5
Utility level at B is -0.65
Utility level at C is -0.065

Utility Representation 2

Utility level at A is -6.5
Utility level at B is 42
Utility level at C is -42

Utility Representation 3

Utility level at A is 829
Utility level at B is 1{,}000{,}013
Utility level at C is 830

Utility Representation 4

Utility level at A is 829
Utility level at B is 828
Utility level at C is 0

Rubric

Each of the eight prompts in this COU is worth 1 point. On each, you get 1 point if you submit a correct answer and 0 points otherwise.