Multiattribute Utility Theory Two - Human Decision Making - Lecture Slides, Slides of Human-Computer Interaction Design

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Multi-Attribute Utility

Models with

Interactions

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

Interaction between x and y?

x - , y -

y + x +, y +

x +

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Multi-Attribute Utility Function (Cont’d)

 Mathematical Expression

U ( x , y ) kX UX ( x ) kYUY ( y )( 1  kX  kY ) UX ( x ) UY ( y )

Additive utility function: U ( x , y ) kX UX ( x ) kYUY ( y )

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

If U ( x , y ) f [ UX ( x ), UY ( y )], then U ( x , y ) is said to be separable

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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

U ( x , y )  kXU X ( x ) kYUY ( y )( 1  kX  kY ) UX ( x ) UY ( y )

where UX ( x ) = utility function of X scaled so that UX ( x- ) =0 and UX ( x +) = UY ( y ) = utility function of Y scaled so that UY ( y- ) =0 and UY ( y +) = kX = U ( x+ , y - ) NOT relative weight of UX kY = U ( x- , y +) NOT relative weight of UY

X Y X Y X

X X Y Y X Y X Y

k k k k k

U x y k U x k U y k k U x U y

X Y X Y Y

X X Y Y X Y X Y

k k k k k

U x y k U x k U y k k U x U y

kX + kY ≠

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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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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.

X

Consequences Demand

Y

High

Inventory^ Low Decision

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Mutual Utility Independence between X and Y? (Cont’d)

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%.

A 10%

B CE Y

Y

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

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

U ( x , y ) kX UX ( x ) kYUY ( y )( 1  kX  kY ) UX ( x ) UY ( y )

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UX ( x ) and UY ( y )?

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

U X ( x ) 1  0. 375 ( 1  ex /^7.^692 )

U Y ( y ) 1  2. 033 ( 1  ey /^25 )

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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0. 5 U ( 0 , 0 )+ 0. 5 U ( 10 %, 10 %)= 0. 5 ( 1 )+ 0. 5 ( 0 )= 0. 5

X Y

X Y X Y

k k

k k k k

U ( 6 %, 6 %)= 0. 5 U ( 0 , 0 )+ 0. 5 U ( 10 %, 10 %)

( 1 )[ 1 0. 375 ( 1 )][ 1 2. 033 ( 1 )]

[ 1 0. 375 ( 1 )] [ 1 2. 033 ( 1 )]

6 / 7. 692 6 / 25

6 / 7. 692 6 / 25

X Y X Y

X Y

X Y

X X Y Y X Y X Y

k k k k

k k e e

k e k e

U k U k U k k U U

Suppose the nurse is also indifferent between the consequence ( X =6%, Y =6%) and a 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. Therefore, the two-attribute utility function can be written as U ( X , Y ) 0. 72 UX ( x ) 0. 13 UY ( y ) 0. 15 UX ( x ) UY ( y ) Implications? 1 – kX – kY

(Equation 2)

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

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X

Y 0 2% 4% 6% 8% 10%

0 1 0.95^ 0.9^ 0.85^ 0.79^ 0.

2% 0.9^ 0.86^ 0.81^ 0.76^ 0.7^ 0.

4% 0.78^ 0.74^ 0.69^ 0.64^ 0.59^ 0.

6% 0.62 0.58 0.54 0.5 0.45 0.

8% 0.4 0.37 0.34 0.31 0.27 0.

10% 0.13^ 0.11^ 0.08^ 0.06^ 0.03^0

Utilities for Shortage and Outdating Rates in the Blood Bank