Multi-Attribute Utility Models with Interactions

Dr. Yan Liu

Department of Biomedical, Industrial & Human Factors Engineering

Wright State University

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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

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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

1-px+, y+

x-, y-

x, y

A

B

p

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

Interaction between x and y?

x-, y-

x+, y+y+

x+

0.10

0.30

0.30

0.35

0.45

0.42

0.66

0.70

1

0

5

Multi-Attribute Utility Function (Cont.)

Mathematical Expression

)()()1()()(),( yUxUkkyUkxUkyxU YXYXYYXX

)()(),( yUkxUkyxU YYXX Additive utility function:

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

)](),([),( yUxUfyxU YXIf

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

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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

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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

X and Y are mutually preferentially independent

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

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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

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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

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Multilinear Utility Function

If attributes X and Y are mutually utility independent, then

)()()1()()(),( yUxUkkyUkxUkyxU YXYXYYXX 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

XYXYX

YXYXYYXX

kkkkkyUxUkkyUkxUkyxU

)0)(1)(1()0()1()()()1()()(),(

YYXYX

YXYXYYXX

kkkkkyUxUkkyUkxUkyxU

=)0)(1)(--1(+)1(+)0(=)()()--1(+)(+)(=),( +-+-+-

kX +kY ≠1

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)()()()()1()()(),(

yUkxUkyUxUkkyUkxUkyxU

YYXX

YXYXYYXX

Additive Independence

5.0),(5.0),(5.0)A(EU yxUyxU

1)(5.05.0)B(EU)A(EU YXYX kkkk

(additive utility function)

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

)(5.05.05.0),(5.0),(5.0)B(EU YXXY kkkkyxUyxU

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

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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

(0.5) (0.5) (0.5) (0.5)

excellent service, excellent reliability poor service, poor reliability poor service, excellent reliability excellent service, poor reliability

A

B

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

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Substitutes and Complements

)()()1()()(),( yUxUkkyUkxUkyxU YXYXYYXX

If (1– kX –kY ) > 0, )()()1()(,)( yUxUkkyUxU YXYXYX

so X and Y complement each other

If (1– kX –kY ) < 0,

so X and Y substitute each other

)()()1()(,)( yUxUkkyUxU YXYXYX

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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?

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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 stockOutdating 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.

X Consequences Demand Y

High

Low Inventory Decision

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Who is the decision maker ?

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%.

(0.5) (0.5)

0%

10%

AB C

EX

X

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

Assessment of Utility Function

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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%.

(0.5) (0.5)

0%

10%

AB C

EY

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

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

)()()1()()(),( yUxUkkyUkxUkyxU YXYXYYXX

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UX(x) and UY(y) ?

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

)1(375.01)( 692.7/xX exU

)1(033.21)( 25/yY eyU

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.

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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%)

Known: XY kYXUkYXU %)10,0(,)0%,10(

)]1(375.01)[1(%)75.4()1()1%)(75.4()1()1(%)75.4(

)0(%)75.4()1()0(%)75.4()0%,75.4(

692.7/75.4ekkUkkUkkkUk

UUkkUkUkU

YYXYY

XYXYXX

YXYXYYXX

XkYXU =%)10=,0=(

68.032.0)]1(375.01)[1( 692.7/75.4

YX

XYY

kkkekk

(Equation 1)

%)10=,0=(=)0%,75.4( YXUU

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5.0=)0(5.0+)1(5.0=%)10%,10(5.0+)0,0(5.0 UU

248.0198.0308.05.0)45.0)(56.0)(1()45.0()56.0(

YX

YXYX

kkkkkk

%)10%,10(5.0+)0,0(5.0=%)6%,6( UUU

)45.0)(56.0)(1()45.0()56.0()]1(033.21)][1(375.01)[1(

)]1(033.21[)]1(375.01[%)6(%)6()1(%)6(%)6(%)6%,6(

25/6692.7/6

25/6692.7/6

YXYX

YX

YX

YXYXYYXX

kkkkeekk

ekekUUkkUkUkU

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%)

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

)()(15.0)(13.0)(72.0),( yUxUyUxUYXU YXYX Implications? 1– kX – kY

(Equation 2)

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

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XY

0 2% 4% 6% 8% 10%

0 1 0.95 0.9 0.85 0.79 0.72

2% 0.9 0.86 0.81 0.76 0.7 0.64

4% 0.78 0.74 0.69 0.64 0.59 0.54

6% 0.62 0.58 0.54 0.5 0.45 0.4

8% 0.4 0.37 0.34 0.31 0.27 0.23

10% 0.13 0.11 0.08 0.06 0.03 0

Utilities for Shortage and Outdating Rates in the Blood Bank

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X

0

Y

High

Low

0

0 10

0 0

10 0

(0.3)

(0.7)

(0.3)

(0.7)

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