Multi-Attribute Utility Models with Interactions

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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) pU(x, y ) (1-p)U (x-, y- ) U (x,
y) ?pU (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
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1
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.

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Multi-Attribute Utility Function (Cont.)

• Mathematical Expression

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

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
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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)
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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
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
If (1 kX kY ) gt 0,
so X and Y complement each other
If (1 kX kY ) lt 0,
so X and Y substitute each other
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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 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 makers
utility over these consequences.
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• Assessment of Utility Function

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 Y0,
2, 5, 8, and 10.
If CEX does not change for different values of Y,
then X is utility independent of Y
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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
X0, 2, 5, 8, and 10.
Y
(0.5)
0
If CEY does not change for different values of X,
then Y is utility independent of X
(0.5)
A
10
CEY
B
Suppose the nurses assessments suggest mutual
independence between X and Y, then the utility
function is of the multilinear form
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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.
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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 (X4.75, Y0) and (X0, Y10)
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Suppose the nurse is also indifferent between the
consequence (X6, Y6) and a 50-50 lottery
between (X0, Y0) and (X10, Y10)
(Equation 2)
Solving Equations 1 and 2 simultaneously for KX
and kY, we find KX0.72 and kY0.13
Therefore, the two-attribute utility function can
be written as
Implications?
Because 1- kX - kY gt0, X and Y are complements
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Utilities for Shortage and Outdating Rates in
the Blood Bank
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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) 0.31
0.7 kX 0.30.70.72 0.804
EU(Low) 0.3 U(10,0) 0.7 U(0,0) 0.3 kY
0.7 1 0.30.130.7 0.739
Because EU(High)gtEU(Low), the high inventory
level is preferred