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### Multi-Attribute Utility Models with Interactions

Dr. Yan Liu

Department of Biomedical, Industrial & Human Factors Engineering

Wright State University

Introduction

- Attributes Can be Substitutes to One Another
- e.g. You have invested in a number of different stocks. The simultaneous successes of all stocks may not be very important (although desired) because profits may be adequate as long as some stocks perform well
- Attributes Can be Complements to One Another
- High achievement on all attributes is worth more than the sum of the values obtained from the success of individual attributes
- e.g. In a research-development project that involves multiple teams, the success of each team is valuable in its own right, but the success of all teams may lead to substantial synergic gains

x+, y+

1-p

A

x-, y-

B

x, y

Multi-Attribute Utility Function- Direct Assessment
- Find U(x,y), where x-≤ x ≤ x+and y-≤ y ≤ y+ using reference lottery

When you are indifferent between A and B, EU(A) = EU(B)

p•U(x+, y+ ) + (1-p)•U (x-, y- ) = U (x, y)

p=U (x, y)

After you find the utilities for a number of (x, y) pairs, you can plot the assessed points on a graph and sketch rough indifference curves

y+

x+, y+

0.45

0.66

0.30

0.70

0.35

0.42

Interaction between x and y?

0.30

0.10

0

x-, y-

x+

- The point values are assessed utility values for the corresponding (x, y) pair. Sketching indifferent curves.

, then U (x, y) is said to be separable

Multi-Attribute Utility Function (Cont.)- Mathematical Expression

Additive utility function:

Multilinear utility function (captures a limited form of interaction):

Decisions with Certainty/Under Risk

- Decision with Certainty
- Decision Maker knows for sure the consequences of all alternatives
- e.g. A decision regarding which automobile to purchase with consideration of the color and advertised price and life span
- Decision Under Risk
- Decision maker does not know the consequence of every alternative but can assign the probabilities of the various outcomes
- The consequences depend on the outcomes of uncertain events as well as the alternative chosen
- e.g. A decision regarding which investment plan to choose with the objective of maximizing the payoffs

Preferential Independence

- Attribute Y is said to be preferentially independent of attribute X if preferences for specific outcomes of Y do not depend on the level of attribute X
- If Y is preferentially independent of X and X is preferentially independent of Y, then X and Y are mutually preferentially independent

e.g. X = the cost of a project ($1,000 or $2,000)

Y = time-to-completion of the project (5 days or 10 days)

If you prefer the 5-day time-to-completion to the 10-day time-to-completion no matter whether the cost is $1,000 or $2,000, then Y is preferentially independent of X

If you prefer the lower cost of the project regardless of its time-to-completion, then X is preferentially independent Y.

X and Y are mutually preferentially independent

Preferential Independence (Cont.)

- For a decision with certainty, mutual preferential independence is the sufficient condition for the additive utility function to be appropriate
- If E1 E2, then E1 is the sufficient condition for E2
- For a decision under risk, mutual preferential independence is a necessary condition but not sufficient enough for obtaining a separable multi-attribute utility function
- If E3 + E4 E5, then E3 is a necessary (but not the sufficient) condition for E5

Utility Independence

- Slightly stronger than preferential independence
- Attribute X is considered utility independent of Y if certainty equivalent (CE) for risky choices involving different levels of X are independent of the value of Y
- If Y is utility independent of X and X is utility independent of Y, then X and Y are mutually utility independent

X = the cost of a project ($1,000 or $2,000)

Y = time-to-completion of a project (5 days or 10 days)

If your CE to an option that costs $1,000 with probability 50% and $2,000 with probability 50% does not depend on the time-to-completion of the project, then X is utility independent of Y

Multilinear Utility Function

- If attributes X and Y are mutually utility independent, then

where

UX (x) = utility function of X scaled so that UX (x-) =0 and UX (x+) =1

UY (y) = utility function of Y scaled so that UY (y-) =0 and UY (y+) =1

kX = U(x+, y-) NOT relative weight of UX

kY = U(x-, y+) NOT relative weight of UY

kX +kY ≠1

Additive Independence

- Attributes X and Y are additively independent if X and Y are mutually utility independent, and you are indifferent between lotteries A and B

Lottery A: (x-, y-) with probability 0.5, (x+, y+) with probability 0.5

Lottery B: (x-, y+) with probability 0.5, (x+, y-) with probability 0.5

(additive utility function)

excellent service, excellent reliability

(0.5)

A

poor service, poor reliability

B

(0.5)

poor service, excellent reliability

(0.5)

excellent service, poor reliability

Additive Independence (Cont.)- Additive independence is a reasonble assumption in decision under certainty
- Additive independence does not usually hold in decision under risk

e.g. You are considering buying a car, and reliability and quality of service are the two attributes you consider

Which assessment lottery for this car decision will you choose, A or B?

If you are indifferent between A and B, then additive independence holds for attributes reliability ad quality of service; otherwise, additive independence does not hold

Substitutes and Complements

If (1– kX –kY ) > 0,

so X and Y complement each other

If (1– kX –kY ) < 0,

so X and Y substitute each other

Blood Bank

In a hospital bank it is important to have a policy for deciding how much of each type of blood should be kept on hand. For any particular year, there is a shortage rate, the percentage of units of blood demanded but not filled from stock because of shortages. Whenever there is a shortage, a special order must be placed to locate the required blood elsewhere or to locate donors. An operation may be postponed, but only rarely will a blood shortage result in a death. Naturally, keeping a lot of blood stocked means that a shortage is less likely. But there is also a rate at which it must be discarded. Although having a lot of blood on hand means a low shortage rate, it probably also would mean a high outdating rate. Of course, the eventual outcome is unknown because it is impossible to predict exactly how much blood will be demanded. Should the hospital try to keep as much blood on hand as possible so as to avoid shortages? Or should the hospital try to keep a fairly low inventory in order to minimize the amount of outdated blood discarded? How should the hospital blood bank balance these two objectives?

Demand

Y

X

High

Low

Inventory Decision

The final consequence at the blood bank depends on not only the inventory level chosen (high or low) but also the uncertain blood demand over the year. Therefore, this problem is a decision under risk.

Attributes?

Shortage rate (X) and outdating rate (Y)

Shortage rate: annual percentage of units demanded but not in stock

Outdating rate: annual percentage of units that are discarded due to aging

To choose an appropriate inventory level, we need to assess probability distributions of shortage rate and outdating rate consequences for each possible inventory level and the decision maker’s utility over these consequences.

(0.5)

0%

(0.5)

A

10%

CEX

B

Who is the decision maker ?

- Assessment of Utility Function

The nurse who is in charge of ordering blood is responsible for maintaining an appropriate inventory level, so the utility function will reflect his/her personal preferences

Ranges of attributes ?

The nurse judges that 0%(best case) ≤X ≤10%(worst case) and 0%(best case)≤ Y ≤10%(worst case)

Mutual Independence between X and Y ?

The nurse is asked to assess the certainty equivalent for uncertain shortage rate (X), given different fixed outdating rates (Y), say Y=0%, 2%, 5%, 8%, and 10%.

If CEX does not change for different values of Y, then X is utility independent of Y

Mutual Independence between X and Y ? (Cont.)

The nurse is asked to assess the certainty equivalent for uncertain outdating rate (Y), given different fixed shortage rates (X), say X=0%, 2%, 5%, 8%, and 10%.

Y

(0.5)

If CEY does not change for different values of X, then Y is utility independent of X

0%

(0.5)

A

10%

CEY

B

Suppose the nurse’s assessments suggest mutual independence between X and Y, then the utility function is of the multilinear form:

Suppose UX (x) can be modeled using an exponential function:

UX(x) and UY(y)?

kX and kY?

The trick is to use as much information as possible to set up equations based on indifferent outcomes and lotteries, and then to solve the equations for the weight.

(Equation 1)

There are two unknown weights in this problem, so we need to set up two equations in the two unknowns, which requires two utility assessments.

Suppose the nurse is indifferent between two consequences (X=4.75%, Y=0) and (X=0, Y=10%)

Suppose the nurse is also indifferent between the consequence (X=6%, Y=6%) and a 50-50 lottery between (X=0, Y=0) and (X=10%, Y=10%)

(Equation 2)

Solving Equations 1 and 2 simultaneously for KX and kY, we find KX=0.72 and kY=0.13

Therefore, the two-attribute utility function can be written as

Implications?

Because 1- kX - kY= >0, X and Y are complements

Y

(0.3)

0

0

High

(0.7)

0

10

(0.3)

10

0

Low

(0.7)

0

0

Use the derived utility function to choose between the inventory level in the following decision tree.

EU(High) = 0.3* U(0,0) +0.7* U(0,10) = 0.3*1 + 0.7* kX = 0.3+0.7*0.72 = 0.804

EU(Low) = 0.3* U(10,0) +0.7* U(0,0) = 0.3* kY + 0.7* 1 = 0.3*0.13+0.7 = 0.739

Because EU(High)>EU(Low), the high inventory level is preferred

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