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Multiattribute Utility Functions Satisfying Mutual
Preferential Independence
Ali E. Abbas, Zhengwei Sun
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Ali E. Abbas, Zhengwei Sun (2015) Multiattribute Utility Functions Satisfying Mutual Preferential Independence. Operations
Research 62(2):378-393. http://dx.doi.org/10.1287/opre.2015.1350
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OPERATIONS RESEARCH
Vol. 63, No. 2, March–April 2015, pp. 378–393
ISSN 0030-364X (print) — ISSN 1526-5463 (online)
http://dx.doi.org/10.1287/opre.2015.1350
© 2015 INFORMS
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Multiattribute Utility Functions Satisfying Mutual
Preferential Independence
Ali E. Abbas
Epstein Department of Industrial and Systems Engineering and Department of Public Policy, Viterbi School of Engineering, and
Price School of Public Policy, University of Southern California, Los Angeles, California, 90089, [email protected]
Zhengwei Sun
Department of Management Science and Engineering, East China University of Science and Technology, Shanghai 200237, China,
[email protected]
The construction of a multiattribute utility function is an important step in decision analysis. One of the most widely
used conditions for constructing the utility function is the assumption of mutual preferential independence where trade-offs
among any subset of the attributes do not depend on the instantiations of the remaining attributes. Mutual preferential
independence asserts that ordinal preferences can be represented by an additive function of the attributes. This paper derives
the most general form of a multiattribute utility function that (i) exhibits mutual preferential independence and (ii) is strictly
increasing with each argument at the maximum value of the complement attributes. We show that a multiattribute utility
function satisfies these two conditions if and only if it is an Archimedean combination of univariate utility assessments. This
result enables the construction of multiattribute utility functions that satisfy additive ordinal preferences using univariate
utility assessments and a single generating function. We also provide a nonparametric approach for estimating the generating
function of the Archimedean form by iteration.
Subject classifications: multiattribute utility; Archimedean utility copula; preferential independence.
Area of review: Decision Analysis.
History: Received February 2014; revisions received November 2013, June 2014, August 2014; accepted December
2014. Published online in Articles in Advance March 4, 2015.
1. Introduction
(1996), Kirkwood (1997), Greco et al. (2008), Eisenführ
et al. (2010), and Lichtendahl and Bodily (2012).
When uncertainty is present, it is also important to
think about the cardinal preferences for the consequences
of the decision. The classic work of von Neumann and
Morgenstern (1947) shows that a utility function is needed
to determine the best decision alternative in this case. It
was soon realized, however, that the construction of a
multiattribute utility function can be a tedious task unless
some decomposition of the utility function is performed,
and several methods have since been proposed to facilitate
this task.
Keeney and Raiffa’s (1976) classic work proposed several fundamental conditions for decomposing the utility
function. Of particular interest is the notion of mutual utility independence, where preferences for lotteries over any
subset of the attributes do not depend on the instantiation
of the remaining attributes. The condition of mutual utility
independence implies that the multiattribute utility function is either an additive or a multiplicative combination of
single-attribute utility assessments,
n
X
U 4x1 1 0 0 0 1 xn 5 = ki Ui 4xi 51
(2)
In decisions with multiple objectives, it is important to
think about the ordinal preferences for the consequences of
the decision. One of the earliest conditions that specified
some forms of ordinal preferences is the notion of mutual
preferential independence, where trade-offs among any subset of the attributes do not depend on the instantiations
of the remaining attributes. In his classic work, Debreu
(1960) showed that this condition on ordinal preferences
corresponds to a value function that is a monotone transformation of an additive function of the attributes when the
number of attributes, n ¾ 3, i.e., the value function can be
expressed as
n
X
V 4x1 1 0 0 0 1 xn 5 = m
fi 4xi 5 1
(1)
i=1
where m is a monotone function, n ¾ 3, and fi , i = 11 0 0 0 1 n
are arbitrary univariate functions.
The condition of mutual preferential independence represents an important class of value functions that has
been used extensively in the literature. For example, net
present value functions, multiplicative value functions, and
Cobb-Douglas value functions all satisfy the condition
of mutual preferential independence. For more applications of value functions satisfying the condition of mutual
preferential independence, see Keeney (1974, 1992), Dyer
and Sarin (1979), Keelin (1981), Howard (1984), Barron
and Schmidt (1988), Edwards and Barron (1994), Stewart
i=1
or
1 − kU 4x1 1 0 0 0 1 xn 5 =
n
Y
41 − kki Ui 4xi 551
(3)
i=1
where Ui 4xi 5, i = 11 0 0 0 1 n are univariate utility functions.
378
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
379
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Operations Research 63(2), pp. 378–393, © 2015 INFORMS
Note that the functional forms (2) and (3) can be converted into an additive value function using a monotone transformation. This implies that a decision maker
who exhibits mutual utility independence over lotteries also exhibits mutual preferential independence over
deterministic consequences. However, the converse is not
necessarily true.
This paper derives the most general form of a multiattribute utility function that satisfies the condition of mutual
preferential independence over deterministic multiattribute
consequences but does not necessarily satisfy mutual utility independence. As we shall see, the functional form of
the utility function can also be constructed using univariate utility assessments, but a univariate generating function
is also needed to reflect the decision maker’s preferences
for lotteries. In particular, we show that (i) a decision
maker exhibits mutual preferential independence among the
attributes and (ii) his utility function is strictly increasing
with each argument at the maximum value of the complement attributes if and only if his utility function is an
Archimedean combination of univariate utility assessments.
This result completes the class of increasing utility functions that satisfy mutual preferential independence but do
not necessarily satisfy mutual utility independence. This
result also sheds new light on the structure of the multiattribute utility function when the condition of mutual preferential independence is satisfied.
A natural question that arises with this result is how
to assess the Archimedean utility form? Or equivalently,
how to determine the generating function that should be
used with the univariate assessments? This topic has been
covered extensively in the probability literature using both
parametric and nonparametric approaches for constructing an Archimedean probability copula (see for example,
Genest and MacKay 1986, Genest and Rivest 1993, Sungur
and Yang 1996, and for more information on probability
copula functions, see Nelsen 1999). Parametric approaches
for constructing probability copulas are relatively straightforward; they assume a functional form and then assess
its parameters. Nonparametric approaches usually require
more assessments and computational effort, but they also
enjoy the benefits of providing a copula that matches the
exact assessments.
Sungur and Yang (1996) provide a nonparametric iterative approach to determine the surface of an Archimedean
probability copula using probability assessments for points
on a diagonal path in the domain of the copula function. There are, however, several fundamental differences
between Archimedean utility copulas (Abbas 2009) and
Archimedean probability copulas: (i) utility copula functions need not be grounded (this implies that if an attribute
is at its minimum value, the utility copula need not be
zero), and (ii) the cross derivative of a utility copula function can be positive, negative, or zero (see for example
Abbas and Howard 2005). These conditions require several
modifications to the work of Sungur and Yang (1996) if
an iterative approach for constructing the generating function is to be applied to utility copulas. In this paper, we
provide an iterative procedure for estimating the generating function of an Archimedean utility copula from direct
utility assessments. These assessments include (i) a utility assessment at the lower boundary value of the domain
and (ii) utility assessments on a path in the domain of the
copula function.
In our search of the literature, we have found a wealth of
related work on constructing multiattribute utility functions
that relax the conditions of mutual utility independence.
Farquhar (1975) introduced a general decomposition theorem to develop the functional form of a multiattribute
utility function with particular preference structures along
the vertices of a hypercube. Bell (1979a, b) generalized
utility independence to interpolation independence, where
the conditional utility function is an interpolation of the
conditional utility functions at the boundary values of the
domain. Abbas and Bell (2011, 2012) introduced oneswitch independence for multiattribute utility functions,
which is a weaker condition than utility independence
and allows preferences over lotteries to change, but only
once, as a parameter varies. In related work, Abbas (2013)
defined double-sided utility copulas that match all boundary assessments of the attributes.
Other approaches for constructing multiattribute utility
functions that are relevant to our formulation have also
been proposed and use a deterministic value function and
then assign a single attribute utility function over value.
For example, Dyer and Sarin (1982) assign a utility function over a univariate value function and define the concept
of relative risk aversion, and Matheson and Abbas (2005)
assign a utility function over a multivariate value function
and relate the trade-off assessments among the attributes to
the ratio of their relative risk-aversion functions.
The remainder of this paper is structured as follows. Section 2 reviews the basic definitions and notation. Section 3
characterizes utility functions satisfying mutual preferential independence. Section 4 derives theoretical results to
determine the generating function using diagonal assessments and illustrates the approach with numerical examples. Section 5 presents conclusions and summarizes the
main results.
2. Basic Notation, Definitions, and
Review of Previous Work
We assume that the decision maker follows the axioms of
expected utility theory and has a multiattribute utility function, U 4x1 1 0 0 0 1 xn 5, defined over n attributes, X1 1 0 0 0 1 Xn .
We use the lower case, xi , i ∈ 811 0 0 0 1 n9 to denote an
instantiation of attribute Xi , and use xi0 and xi∗ to denote
the minimum and maximum of Xi , respectively. We use X̄i
to denote the set of complement attributes to Xi and use
x̄i to denote an instantiation of this complement. We also
use the vector 4x1 1 0 0 0 1 xn 5 to denote a consequence of the
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
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380
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
decision, and for notational convenience, we use 4xi 1 x̄i 5 to
represent a consequence.
We assume that the multiattribute utility function is
(i) continuous, (ii) bounded, and (iii) strictly increasing
in each argument at a reference value of the complement
attributes. Our focus will be utility functions that are strictly
increasing with each argument at the upper bound of the
complement. We define a normalized conditional utility
function of xi at the maximum value of the complement
attributes, x̄i∗ , as
Ui 4xi — x̄i∗ 5 =
U 4xi 1 x̄i∗ 5 − U 4xi0 1 x̄i∗ 5
0
U 4xi∗ 1 x̄i∗ 5 − U 4xi0 1 x̄i∗ 5
(4)
Abbas (2009) introduced the notion of a utility copula
function that expresses the multiattribute utility function in
terms of its conditional utility assessments at a reference
value of the complement. A class 1 utility copula constructs
a multiattribute utility function in terms of conditional utility assessments at the upper bound of the complement
attributes—i.e., the utility function can be expressed as
U 4x1 1 0 0 0 1 xn 5 = C U1 4x1 — x̄1∗ 51 0 0 0 1 Un 4xn — x̄n∗ 5 1
(5)
where C is a utility copula function.
A class 1 utility copula is a linear function of each
attribute at the maximum values of its complement
attributes, i.e.,
C411 0 0 0 1 11 vi 1 11 0 0 0 1 15 = ai vi + 41 − ai 51
i = 11 0 0 0 1 n0
The utility copula function is also normalized to range from
0 to 1, i.e.,
C401 0 0 0 1 05 = 0
and C411 0 0 0 1 15 = 10
An important class of class 1 utility copulas is the
Archimedean form
C4v1 1 0 0 0 1 vn 5
n
Y
−1
0
∗
0
∗
=
4U 4xi 1 x̄i 5 + 41 − U 4xi 1 x̄i 55vi 5 1
(6)
function of the attributes using a monotone transformation.
To illustrate, the functional form (6) can be converted into
a product form of univariate assessments by applying a
monotone transformation  to both sides to get
n
Y
 C4v1 1 v2 1 0 0 0 1 vn 5 =  U 4xi 1 x̄i∗ 5 + 41 − U 4xi 1 x̄i∗ 55vi 0
i=1
A logarithmic transformation applied to this product form
results in an additive function of the attributes,
ln 4C4v1 1 v2 1 0 0 0 1 vn 55
n
X
= ln  U 4xi 1 x̄i∗ 5 + 41 − U 4xi 1 x̄i∗ 55vi 0
(8)
i=1
Since ordinal preferences are invariant to monotone transformations, the functional form (8) has the same ordinal
trade-offs as (6). What is not so obvious, however, is that
additive ordinal preferences of the form
n
X
V 4x1 1 x2 1 0 0 0 1 xn 5 = m
fi 4x5 1
i=1
with arbitrary strictly increasing (and possibly different)
functions fi must result in a multiattribute utility function
of the form (6), an Archimedean combination of functions
all having the same generating function, . We prove this
below for the general case of n ¾ 3 but first assert this
result for the case of two attributes having an additive value
function.
Proposition 1. If U 4x1 1 x2 5 is continuous and strictly
increasing with each argument at the upper bound of the
complement attributes, then the following two statements
are equivalent:
(i) U 4x1 1 x2 5 = UV 4V 4x1 1 x2 55 with
V 4x1 1 x2 5 = m4f1 4x1 5 + f2 4x2 55,
where m is a continuous monotonic function and f1 and f2
are continuous and strictly increasing functions.
(ii) U 4x1 1 x2 5 = C4U1 4x1 — x2∗ 51 U2 4x2 — x1∗ 55,
where C is an Archimedean utility copula of the form (6).
The following example illustrates this result.
i=1
where 4v5 is a generating function that is continuous and
strictly increasing on the domain 601 17, with 415 = 1.
A multiattribute utility function U 4x1 1 0 0 0 1 xn 5 can also
be constructed using a univariate utility assessment over an
ordinal value function V 4x1 1 0 0 0 1 xn 5 through the relation
U 4x1 1 0 0 0 1 xn 5 = UV 4V 4x1 1 0 0 0 1 xn 551
(7)
where UV is the utility function over value.
Example 1 (Two-Attribute Archimedean Form). Consider the ordinal value function
V 4x1 y5 = xy ‡ 1
0 < x1 y ¶ 11
that has been used to represent trade-offs for health and
consumption (Howard 1984). Note that this value function
is strictly increasing with each argument at the maximum
value of the complement attribute. Moreover, a logarithmic transformation converts this function into an additive
function:
3. Utility Functions Satisfying Mutual
Preference Independence
log V 4x1 y5 = log x + ‡ log y0
It is not surprising to see that a utility function of the
Archimedean form (6) can be converted into an additive
Proposition 1 asserts that any two-attribute utility function
satisfying these ordinal preferences can be expressed as an
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
381
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
Archimedean combination of the conditional utility functions at the maximum margin.
Consider the two-attribute utility function, obtained by
taking an exponential utility function over this value function, as used in Howard (1984), and so
−ƒxy ‡
1−e
1 0 < x1 y ¶ 10
(9)
1 − e−ƒ
From (4), the conditional utility functions at the upper
bounds are calculated as
1−e−ƒx
1
4
u = U 4x—y ∗ 5 =
⇒ x4u5 = − log41−41−e−ƒ 5u51
−ƒ
1−e
ƒ
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U 4x1 y5 =
4. Assessing a Multiattribute Utility
Function Satisfying Mutual
Preferential Independence
Because a single generating function is sufficient to characterize the functional form of an Archimedean copula, it
will suffice to assess this generating function using a twoattribute formulation where the remaining attributes are set
at their maximum values.
To illustrate, note that if C4v1 1 v2 1 0 0 0 1 vn 5 is an Archimedean utility copula, then the bivariate function
C4v1 1 v2 5 = C4v1 1 v2 1 11 11 0 0 0 1 15
and
‡
1 − e−ƒy
v = U 4y — x 5 =
1 − e−ƒ
1/‡
1
−ƒ
⇒ y4v5 = − log41 − 41 − e 5v5
0
ƒ
4
∗
Substituting for x4u5 and y4v5 into (9) gives the utility
copula function as
C4u1 v5 = U 4x4u51 y4v55
−ƒ
−ƒ
1 − e−41/ƒ5 log41−41−e 5u5·log41−41−e 5v5
0
(10)
=
1 − e−ƒ
Since ordinal preferences are additive, Proposition 1 asserts
that this copula function must be Archimedean. Indeed, the
function (10) can be reduced to the Archimedean form
C4u1 v5 = −1 64u54v57
using the generating function
4t5 = −
1
log41 − 41 − e−ƒ 5t50
ƒ
The following theorem extends the results of Proposition 1
to multiple attributes, satisfying mutual preferential independence for n ¾ 3.
Theorem 1. A multiattribute utility function U 4x1 1 0 0 0 1 xn 5
that is continuous and strictly increasing with each argument at the maximum value of the complement attributes,
with n ¾ 3, exhibits mutual preferential independence if and
only if its utility copula function is of the form (6).
Theorem 1 provides a fundamental method for constructing multiattribute utility functions that satisfy mutual
preferential independence. Once we have asserted that
preferential independence conditions exist, then the multiattribute utility function must be Archimedean, and the
assessment task reduces to the assessment of univariate
utility functions for each attribute as well as a univariate
generating function and some corner values.
Methods for assessing single-attribute utility functions
for each attribute are abundant in the literature (see for
example Keeney and Raiffa 1976). Our main focus will
therefore be the assessment of the generating function for
the Archimedean form.
is an Archimedean functional form having the same generating function.
In principle, one can select from a library of functions
to determine the generating function of the Archimedean
form and then conduct some utility assessments on the surface to estimate the parameters of the chosen functional
form using a least-squares fit. This method of parameter
estimation is widely used for utility functions, where the
shape of the utility function is often assumed (such as
an exponential function and the risk aversion coefficient
is estimated to best match some utility assessments). As
shown in Abbas (2009), however, the generating function
of an Archimedean utility copula is strictly monotonic, but
it does not need to be concave or convex on its entire
domain. In fact, it can even be S shaped to allow for further
flexibility in the types of trade-offs that can be modeled.
Therefore, the analyst must choose a functional form for
the generating function that allows for a wide variety of
shapes if the generating function is to accurately represent
the assessments provided.
An alternate approach (that may be used for utility
functions) if we do not assume a particular form is to
assess a few points on the curve and then fit those points
with a smooth curve. Fritsch and Carlson (1980) propose cubic polynomials, as an example, to connect ordered
points using a differentiable path. This method could work
well for utility functions because each fitted point can be
assessed directly as the utility of a consequence and can be
interpreted clearly in terms of lottery assessments. Unlike
traditional utility function assessments, however, it is not
possible to immediately assess points on the generating
function of an Archimedean form because there is no clear
interpretation for the types of lottery questions one would
ask to determine points on the generating function directly.
To remedy this problem, we provide a method to infer the
generating function from utility assessments on the domain
of the attributes and then provide proofs of convergence of
these assessments to the generating function. This section
explains an iterative approach to infer the generating function of the Archimedean form using direct utility assessments on the domain of the attributes.
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
382
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
Since the generating function of an Archimedean utility
copula is strictly increasing on the interval 601 17, its derivative is positive (and possibly zero at some finite isolated
points). We consider the case where the derivative of the
generating function can be zero at finite points but assume
it is strictly positive at 411 15, i.e., we assume that 0 415 > 0.
The utility assessments needed for constructing a twoattribute utility function using an Archimedean utility copula can be divided into two steps:
(1) Step 1: Assess the boundary utility functions for each
attribute at the upper bound of the complement attributes.
(2) Step 2: Perform additional utility assessments to
determine the generating function. These additional assessments include
(a) a utility assessment at the lower bound of the
domain and
(b) a utility assessment on a path on the domain of the
copula, which we refer to as a skewed diagonal assessment.
Define the corner values, kx = U 4x∗ 1 y 0 5 and ky =
U 4x0 1 y ∗ 5. Without loss of generality, we assume that
kx ¾ ky . If both corner values kx 1 ky are zero, then the
assessment task would be simplified and the lower bound,
Step 2(a), would not be needed (the utility function would
be grounded and there would be no lower boundary assessments). We shall assume therefore that at least one of the
corner values kx > 0.
We now explain the assessments needed to construct the
Archimedean utility function in more detail. Next we derive
convergence results to determine the generating function
from the utility assessments of Steps 1 and 2.
Step 12 Assess two boundary utility functions U 4x1 y ∗ 5
and U 4x∗ 1 y5. This step requires a utility assessment for
each attribute at the upper bound of the complement
attributes. For two attributes, we assess the two normalized
conditional utility functions U 4x — y ∗ 5 and U 4x∗ — y5 as well
as the two corner values kx = U 4x∗ 1 y 0 5 and ky = U 4x0 1 y ∗ 5.
The normalized assessments, U 4x — y ∗ 5 and U 4y — x∗ 5, can
be determined by fitting the individual assessments to some
of the widely used functional forms of utility functions
or by assessing a few points and connecting them with a
smooth path.
The utility function at the upper boundary values can
then be determined from these normalized conditional assessments and corner values using the relations
U 4x1 y ∗ 5 = ky + 41 − ky 5U 4x — y ∗ 51
(11)
U 4x∗ 1 y5 = kx + 41 − kx 5U 4y — x∗ 50
Figure 1 plots an example of two boundary utility functions
for attributes X and Y each on the domain 601 17. The figure
shows that U 4x1 y ∗ 5 and U 4x∗ 1 y5 are strictly increasing
and that ky = 001 and kx = 005.
Step 22 Additional utility assessments needed to determine the generating function. To determine the generating
function when its functional form (or even shape) is not
known, we need to make some additional utility assessments.
Step 24a5: Utility Assessments at a Lower Bound
U 4x1 y 0 5. To illustrate the intuition for assessing a lower
bound, note that the lower bound of an Archimedean utility
function is related to the upper bound using a transformation that depends on the generating function. By direct
substitution into (6), we get
4
U 4x1 y 0 5 = C4v1 1 05 = −1 84U 4x0 1 y ∗ 5 + 41 − U 4x0 1 y ∗ 55
· U 4x — y ∗ 5594U 4x∗ 1 y 0 55 0
By assessing a lower boundary assessment and comparing
it with the upper boundary assessment, we can infer some
information about the shape of the generating function. We
have assumed that kx ¾ ky , so we assess the utility of the
attribute with the higher corner value, X, at the lower bound
of the attribute with the lower corner value, Y , i.e., we need
to assess the curve U 4x1 y 0 5.
We then define a general transformation, g, that relates
the boundary assessments as
U 4x1 y ∗ 5 = g4U 4x1 y 0 550
(12)
(a) The assessment U 4x1 y ∗ 5 is strictly increasing from ky to 1; (b) U 4x∗ 1 y5 is strictly increasing from kx to 1.
(a) 1.0
(b) 1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
U (x*, y)
Figure 1.
U (x, y*)
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4.1. Assessments Needed for Constructing a
Two-Attribute Utility Function Using an
Archimedean Utility Copula
0.5
0.4
0.3
kx = 0.5
0.5
0.4
0.3
ky = 0.1
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
x
0.8
1.0
0
0.2
0.4
0.6
x
0.8
1.0
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
383
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
As attribute X spans its minimum to maximum values, the
domain of the function g spans U 4x0 1y 0 5 = 0 to U 4x∗ 1y 0 5
= kx , and the range of g spans U 4x0 1y ∗ 5 = ky to U 4x∗ 1y ∗ 5
= 1. Therefore,
Figure 2.
(Color online) Assessments along the skewed
diagonal curve and the lower bound.
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g2 601 kx 7 → 6ky 1 17
The domain of the function g will be used to determine the
generating function on the interval 601 kx 7. We discuss the
properties of the transformation g in more detail in the next
section. Because the domain of the generating function is
601 17, however, it is not sufficient to determine the generating function from this assessment alone. This is why
we need a second assessment to characterize the generating
function on 6kx 1 17.
Step 24b52 Utility Assessment on a Skewed Diagonal
Curve. The second assessment is conducted across a path
on the domain of the attributes, which we refer to as a
skewed diagonal curve. The intuition behind this name is
that if both kx and ky were zero, this curve would be a
straight (diagonal) line passing through the points 401 05
and 411 15 in the domain of the copula function. Because
both kx andky need not be zero, however, and they need
not even be equal, the assessed curve in this case traces a
skewed and offset path in the domain of the consequences,
as we illustrate below.
The skewed diagonal path is determined by first defining
a parameter t that fills in the gap from kx to 1—i.e., we
define t ∈ 6kx 1 17. The values of x and y that determine this
skewed diagonal path are determined by
(
U 4x1 y ∗ 5 = t1
(13)
U 4x∗ 1 y5 = t0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
4
S4t5 = U 4x4t51 y4t551
t ∈ 6kx 1 170
(14)
Figure 2 illustrates the utility assessments on the lower
bound and the skewed diagonal path.
The following steps summarize the assessment procedure
for the skewed diagonal curve:
(i) define the parameter ton the interval 6kx 1 17;
(ii) define x4t5 using the equation U 4x4t51 y ∗ 5 = t;
(iii) define y4t5 using the equation U 4x∗ 1 y4t55 = t;
(iv) trace the path 4x4t51 y4t55, which is shown using the
dashed line in Figure 2; and
(v) conduct the utility assessments S4t5 = U 4x4t51 y4t55
using indifference assessments.
The following example illustrates numerically the complete set of utility assessments needed to construct an
Archimedean utility function.
0.8
0.9
1.0
x
Example 2 (Utility Assessments for the Archimedean Form). Step 12 Upper Boundary Assessments. The
first step is to assess the upper boundary curves for each
attribute. Once again, this can be done by identifying a
functional form and assessing its parameters or by assessing several points and fitting them. Here we assume a particular functional form. Suppose that the upper boundary
utility functions are
U 4x1 y ∗ 5 = 1052 − 1042e−x 1
x ∈ 601 17
(15)
and
√
U 4x∗ 1 y5 = 1029 − 0079e− y 1
We therefore define x4t5 as the inverse function of the
curve U 4x1 y ∗ 5 and y4t5 as the inverse function of the curve
U 4x∗ 1 y5. The skewed diagonal path is traced by the points
4x4t51 y4t55 on the interval t ∈ 6kx 1 17. Denote the utility
values across this path as S4t5, i.e.,
0.7
1.00
0.09
0.08
0.07
0.06
0.05
0.04 y
0.03
0.02
0.01
0
y ∈ 601 170
(16)
By direct substitution, this implies that kx = U 4x∗ 1 y 0 5 =
005 and ky = U 4x0 1 y ∗ 5 = 001.
Step 24a52 Lower Boundary Assessment. Because the
highest corner value is kx , we need to assess the lower
boundary curve U 4x1 y 0 5. For this example, we use a hyperbolic absolute risk aversion (HARA) utility function at the
lower bound because of its generality. The analyst may
assess utility values on this lower bound and then use these
utility assessments to estimate the parameters of the HARA
utility. Suppose that the resulting lower boundary assessment is
U 4x1y 0 5 = 00582−4009884−00041x54604 1
x ∈ 601170 (17)
From (12), (15), and (17), the transformation g satisfies
1052 − 1042e−x = g400582 − 4009884 − 00041x54604 50
(18)
To determine the transformation g1 define r = U 4x1 y 0 5.
From (17),
r = 00582 − 4009884 − 00041x54604
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
384
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
Transformation g4r52 601 kx 7 → 6ky 1 17.
Figure 3.
Figure 4.
1.0
g(kx) = 1
1.0
Skewed diagonal path on the X–Y domain.
0.9
(1, 1)
0.9
0.8
0.8
0.7
0.7
g (r )
kx
0.5
0.6
0.4
(0.83, 0.50)
y 0.5
0.3
0.4
0.2
0.1
g (0) = ky
0.3
0
0
(0.68, 0.23)
0.2
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
r
0.1
Note that r ∈ 601 kx 7 = 601 0057. Rearranging gives
0
x = 240107 − 24039400582 − r5000216 1
r ∈ 601 00570
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(20)
Figure 3 plots the transformation g4r5 in (20). Note that
g2 601 kx 7 → 6ky 1 17 (as expected).
Step 24b52 Skewed Diagonal Assessment. To assess the
utility values along the skewed diagonal curve, we first
define t ∈ 6kx 1 17. We then determine x4t5 and y4t5 for different values of t using (15) and (16), respectively. Table 1
shows the assessments. The first column shows discrete values of the parameter t. The second and third columns show
the corresponding values of x4t5 and y4t5. Note that for
t = kx , y4kx 5 = y 0 , and for t = 1, x415 = x∗ and y415 = y ∗ .
Figure 4 plots this skewed diagonal path from Table1,
which is the x–y plane of Figure 2.
The last column in Table 1 shows the utility assessments
for the points x4t5 and y4t5 defining the curve S4t5 obtained
using indifference lottery assessments of (x4t51 y4t55 for
a binary gamble that gives either 4x∗ 1 y ∗ 5 with a probability U 4x4t51 y4t55 or 4x0 1 y 0 5 with a probability 1 −
U 4x4t51 y4t55.
The utility assessments across the skewed diagonal path
can also be made by decomposing the assessment into
multiple steps using the utility tree decomposition (Abbas
2011), where lotteries representing only one variation of
U 4x1 1 y1 5
= U 4x1 1 y ∗ 5U 4y1 — x1 5 + U 4x1 1 y 0 5Ū 4y1 — x1 5
(21)
where Ū = 1 − U .
The assessment U 4x1 1 y1 5 can therefore be composed
into U 4x1 1 y ∗ 5, U 4x1 1 y 0 5, and U 4y1 — x1 5. Note that we have
already assessed the utility function on the upper and lower
bounds, U 4x1 y ∗ 5 and U 4x1 y 0 5. Therefore U 4x1 1 y ∗ 5 and
U 4x1 1 y 0 5 are already determined. Furthermore, the term
U 4y1 — x1 5 is a single indifference assessment that can be
obtained using indifference assessments of 4x1 1 y1 5 for a
binary gamble that gives either 4x1 1 y ∗ 5 or 4x1 1 y 0 5. This
gamble keeps the level x1 fixed and varies only y1 from
y 0 to y ∗ . Figure 5 illustrates the six assessments for S4t5
versus t in Table 1.
Figure 5.
A utility assessment on the skewed diagonal
curve S4t5.
1.0
S(1) = 1
0.9
S (0.8) = 0.8189
0.8
0.7
S (0.8) = 0.6641
Table 1.
t
Determine the skewed diagonal path
4x4t51 y4t55 and the utility assessment S4t5.
x4t5
y4t5
S4t5
0.6
S (0.7) = 0.5253
0.5
0.4
S (0.6) = 0.3972
0.3
kx = 005
0.6
0.7
0.8
0.9
1
0033
0043
0055
0068
0083
1
0
0002
0008
0023
0050
1
1.0
x
each attribute can be incorporated. For example, consider
the utility assessment at 4x1 1 y1 5. The utility tree decomposition is
000216 5
1
r ∈ 601 00570
0.1
(19)
Substituting from (19) into (18) gives
g4r5 = 1052 − 1042e−4240107−24039400582−r5
(0.55, 0.08)
(0.43, 0.02)
(0.33, 0)
0
S(t)
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0.6
002765
003972
005253
006641
008189
1
0.2
S (0.5) = 0.2765
0.1
0
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
t
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
385
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
We have now conducted all utility assessments needed to
determine the utility surface.
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4.2. Determining the Generating Function on the
Interval 6kx 1 17
4.2.1. Relating S4t5 to the Generating Function on
the Interval 6kx 1 17. We now determine the generating
function on the interval 6kx 1 17 using the assessment S4t5.
The following proposition relates S4t5 to the generating
function.
1.00
0.95
0.75
0.65
4
∀ t ∈ 6kx 1 170
Step 3: For any S 4−m5 4t5, define the exponential function
‡m 4t5 such that
‡m 4t5 = e2
m 4S 4−m5 4t5−15
0
We prove in Theorem 2 that the iterations ‡m 4t5 converge to the generating function on the interval 6kx 1 17
as mincreases. The following example illustrates the steps
needed to solve this functional equation numerically and to
determine the generating function on the interval 6kx 1 17.
Example 3 (Determining the Generating Function
on the Interval 6kx 1 17). Step 12 Determine the Inverse
function S −1 4t5. To determine the inverse function S −1 4t5
from the assessments in Table 1, we interchange the order
of the assessments of 4t1 S4t55. The six assessments of
4t1 S −1 4t55 are as follows: 40027651 0055, 40039721 0065,
40052531 0075, 40066411 0085, 40081891 0095, and 411 15. The
next step is to fit the assessments of S −1 4t5 using a
smooth curve to help determine its composite functions.
Appendix B derives the properties of S4t5 and its inverse
S −1 4t5 and illustrates why the inverse function is clearly
defined. Appendix C provides a procedure to determine a
piecewise polynomial fit for S −1 4t5 that may be used in
practice.
Step 22 Determine the Composite Functions S −4m5 4t5.
Given S −1 4t5, the calculation of S −2 4t5 is obtained by itera4
tion, where S −425 4t5 = S −1 4S −1 4t55, and similarly for higher
−4m5
orders to get S
4t5. Figure 6 plots the inverse function
S −1 and its composite functions, S 4−35 and S 4−65 on 6kx 1 17
S –1(t)
0.80
S4t5 = −1 444t552 51
S 4−m5 4t5 = S −1 ž · · · ž S −1 4t51
S (–3)(t)
0.85
0.70
Proposition 2 shows that a portion of the generating function on the interval t ∈ 6kx 1 17 can be estimated if we solve
the functional equation 4S4t55 = 44t552 . We provide an
iterative solution to this functional equation using the following steps:
Step 1: Determine the inverse function S −1 on the interval 6kx 1 17.
Step 2: Determine the composite inverse function for
any positive integer m as
S (– 6)(t)
0.90
Proposition 2. Relating S4t5 to the Generating Function
t ∈ 6kx 1 17
Inverse function S −1 and its composite functions S 4−35 and S 4−65 .
Figure 6.
0.60
0.5
0.6
0.7
0.8
0.9
1.0
t
as determined by the polynomial fit of S −1 in Appendix C.
Appendix B, explains why S 4−m5 4t5 ¾ S 4−4m−155 4t5 and,
therefore, why the curves in Figure 6 are increasing with m.
Step 32 Determine the Iterations ‡m 4t5. For any S 4−m5 4t5,
define the exponential function
‡m 4t5 = e2
m 4S 4−m5 4t5−15
0
(22)
The iterations ‡m 4t5 are obtained by direct substitution.
−1
Figure 7 plots the curves ‡1 4t5 = e24S 4t5−15 , ‡3 4t5 =
84S 4−35 4t5−15
644S 4−65 4t5−15
e
, and ‡6 4t5 = e
computed directly
from S −1 , S 4−35 , and S 4−65 . As we shall see, the iterations
‡m 4t5 approximate the generating function on the interval
6kx 1 17.
4.2.2. Convergence of ‡m 4t5 to the Generating Function on the Interval 6kx 1 17. Observe that the generating
function of an Archimedean copula is unique up to a power
transformation, i.e., if C4v1 1 0 0 0 1 vn 5 is an Archimedean
utility copula with generating function , then it is also the
copula formed by the generating function  ,  > 0.
Lemma 1. If the derivative of the generating function at
t = 1 is not equal to zero (i.e., 0 415 6= 0), then there
always exists  > 0 such that ¯ 0 415 = 1, where ¯ =  .
Figure 7.
Convergence of ‡m 4t5, m = 11 31 6 on 6kx 1 17.
1.0
0.9
0.8
0.7
m
0.6
0.5
1(t)
0.4
3(t)
6(t)
0.3
0.2
0.1
0
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
t
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
386
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
Lemma 1 implies that if 0 415 6= 0 (as we have assumed)
then we can further assume without loss of generality that
the generating function satisfies 0 415 = 1.
4t5 = lim ‡m 4t51
∀ t ∈ 6kx 1 170
m→ˆ
(23)
Theorem 2 asserts that higher orders of ‡m 4t5 converge
to the generating function, 4t5, on the interval t ∈ 6kx 1 17.
While any power of a generating function results in an
equivalent Archimedean copula, the convergence of Theorem 2 results in the generating function that satisfies the
condition 0 415 = 1.
4.3. Determining the Generating Function on the
Interval 601 kx 5
We have determined the generating function on the interval 6kx 1 17. We now show how to determine the generating
function on the remaining interval using the estimated generating function on the interval 6kx 1 17 and the transformation function g4r5.
4.3.1. Relating g4r5 to the Generating Function on
the Interval 6kx 1 17. Define the pth composite function
g 4p5 4r5 as
4p5
∀ r ∈ 600271 00551
p4r5 = 21
∀ r ∈ 600131 002751
p4r5 = 31
∀ r ∈ 600031 001351
p4r5 = 41
∀ r ∈ 601 00035
Figure 8(b) plots the function p4r5 and shows that it is a
decreasing step function. The domain of p4r5 is the interval
401 kx 5 and it is integer-valued.
The following proposition relates the functions g 4p5 4r5 to
the generating function.
Proposition 3 (Relating g 4p5 4r5 to the Generating
Function on r ∈ 601 kx 5).
4
g 4r5 = g ž · · · ž g4r50
Before we relate g4r5 to the generating function, we need to
provide some intuition about the shape of the compositions
g 4p5 4r5. Figure 8(a) plots the functions g4r5, g 425 4r5, g 435 4r5,
and g 445 4r5 for the assessments of Example 2.
Observe the following important features from
Figure 8(a):
(a) The domain of g4r5 is 601 kx 7 and its range is ky
to 1 (as expected). Moreover, it satisfies the inequality kx ¶
g4r5 < 1 on the interval r ∈ 600271 0055. No other composite
function of g satisfies the inequality kx ¶ g 4p5 4r5 < 1 on the
interval r ∈ 600271 0055.
Figure 8.
p4r5 = 11
4r5 = 44kx 55p 4g 4p5 4r551
r ∈ 601 kx 50
(a) Composite functions g4r5, g 425 4r5, g 435 4r5, and g 445 4r5; (b) integer valued function p4r5.
p (r) = 4
(a) 1.0
p (r) = 3
p(r) = 2
p (r) = 1
(b) 4
0.9
0.8
3
(4)
0.7 g (r)
0.6
g(3)(r)
0.5
g(r)
kx = 0.5
g(2)(r)
2
0.4
1
0.3
0.2
r4 = 0.03
0.1
0
r2 = 0.27
r3 = 0.13
r1 = kx = 0.5
r4 = 0.03
r2 = 0.27
r3 = 0.13
r1 = kx = 0.5
0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
r
(24)
Equation (24) is a functional equation that shows that
4r5 on the interval 601 kx 5 is equal to the product of the
constant44kx 55p and the composition 4g 4p5 4r55 for any
value of p. This might suggest at first that we can determine
the generating function on the interval 601 kx 5 by direct substitution for r ∈ 601 kx 5 into (24). The problem we encounter
however is that we have only determined the value of 4t5
on the interval 6kx 1 17. If the value of r in (24) is such
that g 4p5 4r5 ∈ 6kx 1 171 then we can substitute directly into
p (r)
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Theorem 2 (Determining the Generating Function
on the Interval 6kx 1 17). If the generating function of an
Archimedean utility copula, 4t5, satisfies 0 415 = 1, then
(b) The function g 425 4r5 satisfies kx ¶ g 425 4r5 < 1 on the
interval r ∈ 600131 00275. No other composite function of g
satisfies kx ¶ g 4p5 4r5 < 1 on this interval.
(c) The function g 435 4r5 satisfies kx ¶ g 435 4r5 < 1 on the
interval r ∈ 600031 00135. No other composite function of g
satisfies kx ¶ g 4p5 4r5 < 1 on this interval.
(d) The function g 445 4r5 satisfies kx ¶ g 445 4r5 < 1 on the
interval r ∈ 601 00035. No other composite function of g
satisfies kx ¶ g 4p5 4r5 < 1 over this interval. The function
g 445 4r5 has the property that its value at zero is greater than
kx . This is where we end the compositions of g4r5 for the
purposes of estimating the generating function.
We can now define the integer-valued decreasing function
p4r5 as the smallest integer, p, for any r, that satisfies kx ¶
g 4p5 4r5 < 1. From Figure 8(a), we have
0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
r
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
387
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Operations Research 63(2), pp. 378–393, © 2015 INFORMS
the right-hand side of (24) to determine the corresponding value of 4r5. It might be possible, however, that the
composition g 4p5 4r5, for a given value of p, lies outside
the interval 6kx 1 17, as we have seen in Figure 8(a). If this
is the case, then we cannot determine 4r5 by direct substitution into the right-hand side for that value of p.
The question that arises now is whether we can always
find a value of p such that the composition g 4p5 4r5 belongs
to the interval 6kx 1 17 for any r ∈ 601 kx 5? If this were the
case, then we can determine 4r5 over the whole interval 601 kx 5 by direct substitution into the right-hand side of
(24). The following lemma asserts this fact.
Convergence of ’m 4r5, m = 11 31 6 on 401 kx 5.
Figure 9.
0.6
0.5
0.4
0.3
m
0.2
1(r )
3(r )
6(r )
0.1
0
Lemma 2 (Existence of the Integer p). For any given
r ∈ 401 kx 5, there exists a composite function g 4p5 such that
0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
r
kx ¶ g 4p5 4r5 < 10
As a result of Lemma 2, for any r ∈ 401 kx 5, we are
guaranteed a value of p such that kx ¶ g 4p5 4r5 < 1. For
ease of calculation of the composite functions, we shall
use the lowest value of p that satisfies this condition,
i.e., p4r5. This will enable us to determine the values of
4r5 over 401 kx 5.
The Iteration ’m 4r5. Define the function
’m 4r5 = 4‡m 4kx 55p4r5 ‡m 4g 4p4r55 4r551
∀ r ∈ 401 kx 51
(25)
4.3.2. Determining the Generating Function on the
Interval 401 kx 5.
Theorem 3. If the generating function of an Archimedean
utility copula, 4t5, satisfies 0 415 = 1, then
4r5 = lim ’m 4r51
m→ˆ
∀ r ∈ 401 kx 50
(28)
2m 6S 4−m5 4t5−157
where ‡m 4t5 = e
, t ∈ 6kx 1 17 is the same iteration defined earlier and S 4−m5 4t5 is the mth-order composite function of the inverse function S −1 4t5 (also calculated
earlier).
To better understand this function (25), note that the
first term is simply a constant term ‡m 4kx 5 raised to the
power of p4r5. The second term ‡m 4g 4p4r55 4r55 is a composite function based on the iteration ‡m 4t5 and the p4r5
composition of g4r5.
We do not need to plot the full curves in Figure 8 every
time we compute ’m 4r5. To illustrate, suppose we wish to
calculate ’3 40025. We first determine the value g40025 =
00377. Because g40025 < 005 = kx , we conduct another
composition to get
g 425 40025 = g4g400255 = g4003775 = 00688 > 005 = kx 0
Hence, the integer valued function p40025 = 2, and we do
not need higher compositions at r = 002. Now we calculate
‡3 4g 425 400255 = ‡3 4006885 as
‡3 4g 425 400255 = e2
3 4S 4−35 4006885−155
= e8400946−15 = 00650
(26)
For the constant term,
‡3 4kx 5 = e2
3 4S 4−35 40055−15
= e84008955−15 = 00430
(27)
Substituting from (26) and (27) into (25) gives
’3 40025 = 4‡3 4kx 55p40025 ‡3 4g 4p400255 400255 = 00432 × 0065
= 00120
Figure 9 plots the curve ’m 4r5 form = 11 31 6 for the skewed
diagonal assessment of Table 1 and the boundary assessments of Example 3.
Theorem 3 asserts that higher orders of ’m converge to
the generating function on the interval 401 kx 5. Once again,
while any power of a generating function results in an
equivalent Archimedean copula, the convergence of Theorem 3 results in a generating function that satisfies the
condition 0 415 = 1.
4.4. Summary of the Approach
We now summarize the steps needed to determine the generating function from the utility assessments and then illustrate the deviation between consecutive iterations.
(i) Assess two corner values: kx and ky . Assume that
kx ¾ ky .
(ii) Assess two utility functions at the upper bound:
U 4x — y ∗ 5 and U 4y — x∗ 5.
(iii) Assess the utility function at the lower bound,
U 4x1 y 0 5.
(iv) Determine g4r5 on the interval 601 kx 7 using (12)
and its composites g 4p5 4r5.
(v) Determine the integer-valued function p4r5 from the
functions g 4p5 4r5.
(vi) Assess the utility values along the skewed diagonal
curve, S4t5.
(vii) Determine the inverse function S −1 4t5 and its composite functions S 4−m5 4t5.
(viii) Use Theorems 2 and 3 to determine the generating
function 4t5.
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
388
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
5. Convergence and Comparison with
Other Approaches
a linear function. It is natural to consider what S4t5 would
look like if the attributes are mutually utility independent.
The following proposition determines S4t5 in this case.
To provide some insights into the rate of conversion of the
results for examples, define the iteration m 4t5 as
(
‡m 4t51 t ∈ 6kx 1 171
m 4t5 =
(29)
’m 4t51 t ∈ 401 kx 50
Define the deviation between m 4t5 and m+1 4t5 over the
interval 601 17 as
Z 1
1/2
4m+1 4t5 − m 4t552 dt
em =
0
0
Figure 10 shows the deviation, em , plotted versus m, where
the iterations of m 4t5 are obtained from the assessments of
Table 1 with the lower boundary assessment in (17). From
the convergence results of Theorems 2 and 3, this deviation
converges to 0, i.e.,
lim em = 00
Proposition 4. Two attributes are mutually utility independent with kx ¾ ky if and only if the utility function
U 4x1 y5 has an Archimedean utility copula, and the following two statements hold:
(i) U 4x — y ∗ 5 = U 4x — y 0 5, x ∈ 6x0 1 x∗ 71 and
(ii) S4t5 = kt 2 + 241 − k5t + k − 1, t ∈ 6kx 1 171 where
k = 41 − kx − ky 5/441 − kx 541 − ky 55.
Proposition 4 shows a new method to verify mutual utility independence between two attributes. First, we verify
that U 4x — y ∗ 5 = U 4x — y 0 51 where X is the attribute with the
greater corner value. Next, we assert that S4t5 is quadratic.
If these conditions hold then the attributes are mutually
utility independent. For the special case where kx + ky = 1
(the case of an additive utility function), then k = 0 and
S4t5 is a linear function. The following example compares
the accuracy of the Archimedean utility copula obtained by
the iterative approach for Example 2 to the utility function
obtained assuming mutual utility independence.
m→ˆ
The iterations can be conducted to reach the acquired accuracy of estimating the generating function as shown in Figure 10. Since the generating function 4t5 is unknown in
the approximation procedure, we terminate the iterations on
m 4t5 when em is sufficiently small. Note that these iterations do not require additional cognitive effort from the
decision maker; they are simply computations used to calculate additional composite functions.
5.2. Comparison with Mutual Utility Independence
We now compare the estimates of the transformation g
and the skewed diagonal assessment S4t5 to those obtained
using the assumption of mutual utility independence. If two
attributes are mutually utility independent, then U 4x — y ∗ 5 =
U 4x — y 0 5 and so
g4r5 = ky +
Figure 10.
0.040
1 − ky
r1
kx
(30)
Deviation Between m+1 4t5 and m 4t5.
Example 4 (Comparison with Mutual Utility Independence). Consider again the two-attribute utility function of Example 2, where
U 4x1 y ∗ 5 = 1052 − 1042e−x 1
x ∈ 601 171
e1 = 0.0389
(31)
and
√
U 4x∗ 1 y5 = 1029 − 0079e− y 1
y ∈ 601 170
(32)
This implies that kx = 005 and ky = 001. To compare our
iterative assessment approach with that of mutual utility
independence, we first compute the constant k
k=
1 − kx − ky
1 − 001 − 005
=
= 00890
41 − kx 541 − ky 5 41 − 005541 − 0015
Proposition 4 asserts that skewed diagonal assessment
must be
S4t5 = 0089t 2 + 0022t − 00111
t ∈ 60051 170
From (30), the transformation function has the form
0.035
0.030
Deviation
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5.1. Convergences of Successive Iterations
g4r5 = 001 + 108r1
0.025
r ∈ 601 00570
e2 = 0.0232
Figure 11 shows the skewed diagonal assessment S4t5 and
the transformation function g4r5 for the utility function of
Example 2.
0.020
e3 = 0.0137
0.015
e4 = 0.0078
0.010
0.005
e5 = 0.0045
0
1
2
3
m
4
It is straightforward to see that a linear generating function of the form
5
4t5 = kt + 41 − k51
t ∈ 601 17
(33)
Abbas and Sun: Multiattribute Utility Functions Satisfying Mutual Preferential Independence
389
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
Figure 11.
Archimedean copula vs. mutual utility independence.
S(1) = 1
(a) 1.0
0.9
0.9
0.8
0.8
Archimedean
utility copula
0.7
0.7
0.6
g(r)
S (t)
0.6
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g (kx) = 1
(b) 1.0
0.5
0.4
Mutual utility
independence
0.5
0.4
Mutual utility
independence
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.5
0.6
0.7
0.8
0.9
1.0
Archimedean
utility copula
g(0) = ky
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
r
t
Note. (a) Skewed diagonal assessment S4t5; (b) transformation function g4r5.
satisfies the condition of mutual utility independence. But
to compare the generating function obtained from the iterative approach to that of mutual utility independence, we
need take a power of (33) that makes 0 415 = 1. Recall
that the condition of 0 415 = 1 applies to the generating
functions obtained from the iterative approach. Direct substitution shows that the function 4t5 = 4kt + 41 − k551/k
satisfies the condition 0 415 = 1. From here on, we denote
the generating function for mutual utility independence as
UI 4t5 = 6kt + 41 − k571/k
as it will be used in the comparison with the generating
function obtained from the iterative approach.
Figure 12 plots the 3 and 6 , UI 4t5 for Example 2. The
figure also shows the actual generating function 4t5 that
we used to determine the numerical values of the examples
in this paper. Of course, 4t5 is not known to the analyst during this entire procedure. We simply included 4t5
Generating functions UI 4t5, 4t5, 3rdorder iteration 3 4t5 and 6th-order iteration
6 4t5.
Figure 12.
1.0
0.9
0.8
0.7
0.6
UI (t )
3 (t)
6 (t)
(t)
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
t
0.6
0.7
0.8
0.9
1.0
in the figure to illustrate the convergence of the results.
The figure shows that 4t5 and 6 4t5 are indistinguishable
over the entire interval 601 17. The figure also shows the
improvement that 3 offers over UI in this example.
6. Conclusion
Ordinal preferences represented by additive value functions
are widely used in practice. When uncertainty is present,
a multiattribute utility function is needed to determine the
best decision alternative. We completed the class of multiattribute utility functions that correspond to additive ordinal
preferences that are strictly increasing with each attribute
at the maximum value of the complement. We showed that
this class of utility functions must be an Archimedean combination of the utility assessments.
The results of this paper shed some new light on the
structure of multiattribute utility functions satisfying mutual
preferential independence. As we have shown, the functional form of the utility function is highly constrained,
even if mutual utility independence conditions are not satisfied. The main insight from this formulation is the assertion that if mutual preferential independence is verified,
then preferences over lotteries can be decomposed into two
parts: (1) single attribute assessments at the upper bound
and (2) a single generating function that combines these
single attribute assessments. The assumption of mutual
utility independence focuses only on the upper boundary
assessments but ignores the second component: the generating function. The inclusion of the generating function
allows for more general trade-offs and preferences over lotteries and simultaneously satisfies the same utility values
at the boundaries.
Another implication of the results of this work is that
once preferential independence is verified, we do not need
to determine the actual values of the ordinal functions
of the attributes when constructing the multiattribute utility function; assessing the boundary utility functions using
indifference assessments and constructing the generating
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390
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
function (also using indifference assessments) is sufficient
to capture the whole structure of the utility function.
When constructing a multiattribute utility function, the
next step after preferential independence is to verify
whether the attributes can be formulated to achieve some
forms of utility independence. If such independence conditions can be verified, then the functional form of the utility
function is highly simplified. When utility independence
conditions cannot be verified, it is important to capture
the utility dependence among the attributes if we wish to
provide an accurate representation of the decision maker’s
preferences. The results of this paper provide the complete
class of utility functions and methods for their assessment
when decision makers have additive ordinal preferences for
increasing utility functions but when mutual utility independence conditions do not exist.
Acknowledgments
The authors thank the editor, associate editor, and three anonymous referees for their comments on content and exposition. This
work was supported by the National Science Foundation awards
[SES 08-46417, CMMI 12-58482, and CMMI 13-01150].
Appendix A
Observe that
U 4x1 1 x2∗ 5 = U 4x10 1 x2∗ 5 + 41 − U 4x10 1 x2∗ 55U 4x1 — x2∗ 5
and
U 4x1 1 x2∗ 5 = U 4x1∗ 1 x20 5 + 41 − U 4x1∗ 1 x20 55U 4x2 — x1∗ 50
Substituting for (A4) into (A2) with u = U 4x1 — x2∗ 5 and v =
U 4x2 — x1∗ 5, we get
C4u1 v5 = −1  U 4x10 1 x2∗ 5 + 1 − U 4x10 1 x2∗ 55u
·  U 4x1∗ 1 x20 5 + 41 − U 4x1∗ 1 x20 55v 1
(A5)
which is the Archimedean utility copula of (6).
Sufficiency: If U 4x1 1 x2 5 has an Archimedean utility copula,
then (A4) holds. Applying a monotone transformation ln44t55
gives an additive form
ln 4U 4x1 1 x2∗ 55 + ln 4U 4x1∗ 1 x2 55 0
(A6)
Because ordinal preferences are invariant to monotone transformations, the form in (A6) is equivalent to the additive form
m4f1 4x1 5 + f2 4x2 55, where
m4t5 = t1 f2 4x1 5 = ln 4U 4x1∗ 1 x2 55 1 and
f2 4x1 5 = ln 4U 4x1∗ 1 x2 55 0
Proof of Theorem 1. Necessity: Following the steps in Proposition 1, for multiple attributes,
n
X
U 4x1 1 0 0 0 1 xn 5 = UV m
fi 4xi 5
Proof of Proposition 1. Necessity: Given
U 4x1 1 x2 5 = UV 4m4f1 4x5 + f2 4x5551
i=1
where U 4x1 1 x2∗ 5 and U 4x1∗ 1 x2 5 are strictly increasing. This
implies that f1 4x1∗ 51 f2 4x2∗ 5 < ˆ; otherwise U 4x1 1 x2∗ 5 would not
be strictly increasing and likewise for U 4x1∗ 1 x2 5. We have
4
U 4x1 1 x2 5 = UV m4f1 4x1 5 + f2 4x2 55 = UV m4ln4ef1 4x1 5 ef2 4x2 5 55 0
∗
∗
Define f˜1 4x1 5 = e4f1 4x1 5−f1 4x1 55 and f˜2 4x2 5 = e4f2 4x2 5−f1 4x2 55 . Therefore,
˜
˜
U 4x1 1 x2 5 = UV m ln4ef1 4x1 5 ef2 4x2 5 5 + f1 4x1∗ 5 + f2 4x2∗ 5
4
= ŨV f˜1 4x1 5f˜2 4x2 5 1
= ŨV
n
Y
f˜i 4xi 5 1
∗
where f˜i 4xi 5 = e4fi 4xi 5−fi 4xi 55 , i = 11 0 0 0 1 n, and ŨV 4v5 =
P
UV 4m4ln v + ni=1 fi 4xi∗ 555, which is a strictly increasing function.
Note that f˜i 4xi∗ 5 = 1, i = 11 0 0 0 1 n0 When x̄i = x̄i∗ , (A7) gives
Y
Y
U 4xi 1 x̄i∗ 5 = ŨV f˜i 4xi 5 f˜j 4xj∗ 5 = ŨV f˜i 4xi 5 1
j6=i
(A1)
(A7)
i=1
j6=i
= ŨV 4f˜i 4xi 550
4
where ŨV 4v5 = UV 4m4ln v + f1 4x1∗ 5 + f2 4x2∗ 555, which is a strictly
increasing function.
Note that f˜1 4x1∗ 5 = f˜2 4x2∗ 5 = 1. When x2 = x2∗ , (A1) gives
4
f˜i 4xi 5 = ŨV−1 4U 4xi 1 x̄i∗ 55 = 4U 4xi0 1 x̄i∗ 5
+ 41 − U 4xi0 1 x̄i∗ 55U 4xi — x̄i∗ 551
U 4x1 1x2∗ 5 = ŨV 4f˜1 4x1 5f˜2 4x2∗ 55 = ŨV 4f˜1 4x1 5·15 = ŨV 4f˜1 4x1 550
4
Define  = ŨV−1 , which is strictly increasing, and note that
415=1. Therefore,
f˜1 4x1 5 = ŨV−1 4U 4x1 1 x2∗ 55 = 4U 4x1 1 x2∗ 550
Define  = ŨV−1 ; then 4v5 is strictly increasing with
415 = 1 and
(A2)
∀ i = 11 0 0 0 1 n1
(A8)
Substituting for  = ŨV−1 and (A8) into (A7) gives
U 4x1 1 0 0 0 1 xn 5
n
Y
= −1
4U 4xi0 1 x̄i∗ 5 + 41 − U 4xi0 1 x̄i∗ 55U 4xi — x̄i∗ 55
i=1
Similarly,
Therefore, the Class 1 utility copula of U 4x1 1 x2 5 is
f˜2 4x2 5 = 4U 4x1∗ 1 x2 550
(A3)
4
Substituting for (A2), (A3) into (A1) with  = ŨV−1 gives
U 4x1 1 x2 5 = −1 44U 4x1 1 x2∗ 55 · 4U 4x1∗ 1 x2 5550
C4v1 1 0 0 0 1 v5
n
Y
∗
−1
0
∗
0
=
4U 4xi 1 x̄i 5 + 41 − U 4xi 1 x̄i 55vi 5 1
i=1
(A4)
which is the Archimedean utility copula of (6).
(A9)
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391
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Sufficiency: This is straightforward by applying a transformation ln44t55 to (A9).
Substituting from (A13) into (A12) gives
Proof of Proposition 2 for S4t5. By definition, U 4x4t51 y ∗ 5 = t
and U 4x∗ 1 y4t55 = t. Therefore,
4r5 = 44g4g4r555 · 4kx 55 · 4kx 5
4U 4x4t51 y ∗ 55 = 4U 4x∗ 1 y4t555 = 4t5
Repeating the above iteration with g4r5 gives
Substituting into (A4) gives,
4r5 = 4g 4p5 4r5544kx 55p 0
4
S4t5 = U 4x4t51 y4t55 = −1 4U 4x4t51 y ∗ 54U 4x∗ 1 y4t555
−1
2
=  444t55 51
∀ t ∈ 6kx 1 170
Proof of Lemma 1. Because the generating function  is strictly
increasing on the interval 601 17, its derivative at 1, 0 415 ¾ 0. If
0 415 6= 0 (as we have assumed), then 0 415 > 0. Because the generating function of an Archimedean copula is invariant to a power
transformation, define the new generating function ¯ =  , with
 = 1/0 415 > 0. Note that 415
¯
= 1 and its derivative satisfies
¯ 0 415 = 0 415 · 64157 = 1.
Proof of Theorem 2. Define –4t5 = − ln44t55, ∀ t ∈ 401 17
and its inverse – −1 4v5 = −1 4e−v 5, ∀ v ∈ 601 ˆ5. Note that
–4t5 is strict decreasing on the interval 601 17 with –415 =
− ln 44155 = 0. Moreover, 4t5 = e−–4t5 , ∀ t ∈ 401 17, so
(A14)
Proof of Lemma 2. For a given r ∈ 401 kx 5, 0 ¶ 405 < 4r5 <
4kx 5 < 1 since  is strictly increasing. Hence, limp→ˆ 44kx 55p =
0 < 4r50 Therefore, 44kx 55p ¶ p4r5 for a sufficiently large integer p > 1. Define p0 as the smallest such integer—i.e., 44kx 55p0
¶ 4r5 and 44kx 55p0 −1 > 4r5. Hence,
4kx 5 ¶
=
4r5
· 4kx 5
44kx 55p0
4r5
< 1 = 4150
44kx 55p0 −1
(A15)
Note that 4g 4p0 −15 4r55 = 4r5/44kx 55p0 −1 due to Proposition 3.
Hence, inequality (A15) becomes 4kx 5 ¶ 4g 4p0 −15 4r55 < 415.
Since  is strictly increasing, 4kx 5 ¶ 4g 4p5 4r55 < 415, where
p = p0 − 1.
Proof of Theorem 3. Note that g 4p4r55 4r5 ∈ 6kx 1 15, ∀ r ∈ 401 kx 5.
Proposition 3 gives
4
S4t5 = −1 444t552 5 = −1 44e−–4t5 52 5 = −1 4e−2–4t5 5
= – −1 42–4t551
= 4g 425 4r55 · 44kx 552 0
∀ t ∈ 6kx 1 170
4r5 = 64kx 57p4r5 · 4g 4p4r55 4r55
Define qm 4t5 = S 4m5 41 − 2−m t5 and its inverse function qm−1 4t5 =
2m 41 − S 4−m5 4t55. Note that 0 415 = 1, the derivative – 0 415 =
−40 415/4155 = −1 6= 0.
Sungur and Yang (1996) proved that for the eqnarray S4t5 =
– −1 42–4t55, the inverse function – −1 satisfies
= lim 6‡m 4kx 57p4r5 · lim ‡m 4g 4p4r55 4r55
m→ˆ
m→ˆ
p4r5
= lim 6‡m 4kx 57
m→ˆ
· ‡m 4g 4p4r55 4r55
= lim ’m 4r50
m→ˆ
– −1 4t5 = lim S 4m5 1 +
m→ˆ
= lim qm 4t51
m→ˆ
t
2m – 0 415
Proof of Proposition 4. Necessity: If two attributes are mutually utility independent, then
= lim S 4m5 41 − 2−m t5
m→ˆ
∀ t ∈ 6kx 1 170
(A10)
U 4x — y ∗ 5 = U 4x — y 0 51
Hence, the function – satisfies –4t5 = limm→ˆ qm−1 4t5, ∀ t ∈
6kx 1 170 Therefore,
−1 4t5
4t5 = e−–4t5 = lim e−qm
m→ˆ
= lim e2
m 4S 4−m5 4t5−15
m→ˆ
= lim ‡m 4t50
∀ x ∈ 4x0 1 x∗ 51
and
U 4x1 y5 = kx U 4x — y ∗ 5 + ky U 4y — x∗ 5
m→ˆ
+ 41 − kx − ky 5U 4x — y ∗ 5U 4y — x∗ 50
(A16)
Proof of Proposition 3. Taking y = y 0 in (A4) gives
According to the definition of the utility copula in (5), we get its
utility copula as
U 4x1 y 0 5 = −1 44U 4x1 y ∗ 55 · 4U 4x∗ 1 y 0 555
= −1 44U 4x1 y ∗ 55 · 4kx 550
(A11)
C4u1 v5 = kx u + ky v + 41 − kx − ky 5uv0
(A17)
Denote r = U 4x1 y 0 51 then g4r5 = g4U 4x1 y 0 55 = U 4x1 y ∗ 5. Substituting g4r5 into (A11) gives
Substituting the definitions of x4t5 and y4t5 in (13), (11), and
(A16) into (14) gives
r = −1 44g4r55 · 4kx 551
S4t5 = U 4x4t51 y4t55
i.e., 4r5 = 4g4r55 · 4kx 50
(A12)
Applying (A12) at g4r5 gives
4g4r55 = 4g4g4r555 · 4kx 5 = 4g 425 4r55 · 4kx 50
= kx
(A13)
t − ky
t − ky t − kx
t − kx
+ ky
+ 41 − kx − ky 5
1 − ky
1 − kx
1 − ky 1 − kx
= kt 2 + 241 − k5t + k − 10
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392
Operations Research 63(2), pp. 378–393, © 2015 INFORMS
If kx + ky = 1, then U 4x1 y5 = kx U 4x — y ∗ 5 + ky U 4y — x∗ 5, which
has the additive value function as kx U 4x — y ∗ 5 + ky U 4y — x∗ 5. If
kx + ky 6= 1, then
ky
41 − kx − ky 5
kx ky
kx
−
· U 4x — y ∗ 5 −
41 − kx − ky 5
41 − kx − ky 5
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U 4x1 y5 = 41 − kx − ky 5 U 4x — y ∗ 5 −
Appendix C. Piecewise Cubic
Polynomial Interpolation
We applied a piecewise cubic polynomial interpolation for curve
fitting of S −1 4t5 (see Fritsch and Carlson 1980 for further details).
Denote the assessed points as 4xi 1 yi 5, i = 11 0 0 0 1 n + 1. The
approach assigns a cubic polynomial Si−1 4x5 over each interval
6xi 1 xi+1 7, i = 11 0 0 0 1 n as the fitting curve, where
Si−1 4x5 = h00 4r5yi + h10 4r54xi+1 − xi 5mi + h01 4r5yi+1
= 41 − kx − ky 5
∗
· eln4U 4x — y 5−ky /41−kx −ky 55+ln4U 4x — y
kx ky
−
1
41 − kx − ky 5
+ h11 4r54xi+1 − xi 5mi+1 1
∗ 5−k
which has the additive value function as
ky
ln U 4x — y 5 −
41 − kx − ky 5
kx
+ ln U 4x — y ∗ 5 −
0
41 − kx − ky 5
x ∈ 6xi 1 xi+1 71
(C1)
x /41−kx −ky 55
where r = 4x − xi 5/4xi+1 − xi 5, and where mi , mi+1 are the derivatives at xi and xi+1 , respectively. The functions h00 , h01 , h10 , and
h11 are cubic polynomials defined as
h00 4t5 = 2r 3 − 3r 2 + 11
h01 4r5 = −2r 3 + 3r 2
h10 4r5 = r 3 − 2r 2 + r1
and
h11 4r5 = r 3 − r 2 0
∗
Hence, U 4x1 y5 has an Archimedean utility copula as a result of
Proposition 1.
Sufficiency: Theorems 2 and 3 show that the generating function
4t5 of the Archimedean utility copula in (6) satisfying 0 415 =
1 is uniquely determined on the interval 401 17 by the monotone
transformation g4r5 and the skewed diagonal assessment S4t50
Hence, g4r5 and S4t5 uniquely determine the Archimedean utility copula C4u1 v5 of (6). Note that, if U 4x — y ∗ 5 = U 4x — y 0 51 then
g4r5 = ky + 441 − ky 5/kx 5r0 Therefore, if (i) and (ii) hold, then the
Archimedean utility copula C4u1 v5 is in the form of (6) and then
two attributes are mutually utility independent.
Appendix B
Properties of S4t5.
(1) S4t5 2 6kx 1 17 → 6S4kx 51 17.
(2) S4t5 is a continuous and strictly increasing function (refer
to Lemma 2).
(3) S4t5 < t. This is because 44t552 < 4t5, so −1 444t552 5 <
−1
 44t55 = t.
(4) The minimum value is S4kx 5 ¶ kx .
(5) The maximum value is S415 = U 4x∗ 1 y ∗ 5 = 1.
Because S4t5 is continuous and strictly increasing, we can
define the inverse function S −1 on the interval 6kx 1 17.
Properties of S −1 4t5.
(1) S −1 2 6kx 1 17 → 6S −1 4kx 51 17.
(2) S −1 4t5 > t, ∀ t ∈ 6kx 1 15 because S4t5 < t.
(3) Minimum value: S −1 4kx 5 > kx (because S4kx 5 < kx ,
−1
S 4kx 5 > kx 5.
(4) Maximum value: S −1 415 = 1 (because S415 = 15.
(5) S 4−m5 4t5 > S 4−4m−155 4t5, t ∈ 6kx 1 15 (this is because
S −1 4t5 > t5.
Note that S −1 4kx 5 > kx , we know that the domain of S −1 ,
6kx 1 17 contains its range, 6S −1 4kx 51 17. Therefore, the composites,
S 4−m5 4t5, m = 11 21 0 0 0, are well defined on 6kx 1 17.
The derivatives mi , mi+1 are the only unknown parameters in (C1)
given the assessed points 4xi 1 yi 5, i = 11 0 0 0 1 n + 10 We now show
how to calculate these derivatives.
Denote the length of the ith interval as li = xi+1 − xi and the
slope of the line connecting its two endpoints as di = 4yi+1 − yi 5/
4xi+1 − xi 5, i = 11 0 0 0 1 n − 1.
(i) The interior derivatives mi , i = 21 0 0 0 1 n − 1 are

di−1 di

1 if di−1 di > 03
(C2)
mi = w1 di−1 + w2 di

01
if di−1 di ¶ 01
where the weights satisfy w1 = 42li−1 + li 5/434li−1 + li 55 and w2 =
1 − w1 .
(ii) The derivatives at two endpoints of the whole domain, m1
and mn are


if q1 d1 < 03
01
m1 = 3d1 1 if q1 d1 ¾ 01 d1 d2 < 0 and —q1 — > 3—d1 —3


q1 1
otherwise1
(C3)
and

01



3d 1
n
mn =




q2 1
if q2 dn−1 < 03
if q2 dn−1 ¾ 01 dn−1 dn−2 < 0 and
—q2 — > 3—dn−1 —3
otherwise,
(C4)
where
q1 =
42l1 + l2 5d1 − l1 d2
l1 + l2
q2 =
42ln−1 + ln−2 5dn−1 − ln−1 dn−2
0
ln−1 + ln−2
and
Now we apply the curve fitting approach to determine the
inverse function S −1 4t5 from the assessments. From Table 1, we
relabel t and S4t5 to get the six points:
40027651 00551
40039721 00651
40052531 00751
40066411 00851
40081891 00951
411 150
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Operations Research 63(2), pp. 378–393, © 2015 INFORMS
We calculate the interior derivatives m2 , m3 , m4 , and m5
from (C2). For example,
l1 = x2 − x1 = 003972 − 002765 = 001207
and
l2 = x3 − x2 = 005253 − 003972 = 0012810
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Furthermore, the slopes in these two intervals are
d1 =
y2 − y1
006 − 005
=
= 008285
x2 − x1
001207
d2 =
y3 − y2
007 − 006
=
= 0078060
x3 − x2
001281
and
From (C2), the weights for m2 are
w1 =
2l1 + l2
2 × 001207 + 001381
=
= 004950
34l1 + l2 5 3 × 4001207 + 0013815
and
w2 = 1 − w1 = 0050500
Substituting w1 , w2 , d1 , and d2 into (C2) gives
m2 =
d1 d2
008285 × 007806
=
w1 d1 + w2 d2 004950 × 008285 + 0005050 × 007806
= 0080410
Similarly, m3 = 007497, m4 = 006819, m5 = 005966.
Calculate the derivatives at the two endpoints x1 = 002765
and x6 = 1. Note that q1 = 442l1 + l2 5d1 − l1 d2 5/4l1 + l2 5 =
008517 < 3d1 , m1 = q1 = 0085170 Similarly, m6 = q2 = 0050160
Substituting the derivatives mi , i = 11 0 0 0 1 6 into (C1) and rearranging it gives the fitting curve of S −1 4t5 in the form of a piecewise polynomial over 60027651 17 as


−000814t 3 −00115t 2 +00934t +0025231




t ∈ 600241003673






−004533t 3 −204151t 2 +006889t +0028931




t ∈ 60036100573




−004828t 3 −00617t 2 +005013t +0033641
S −1 4t5 =

t ∈ 60051006573




3

−005626t
+009761t 2 +001297t +0044811




t ∈ 600651008173





−001868t 3 +002472t 2 +005675t +0037211


 t ∈ 600811170
The values of composite function S 4−25 4t5 are obtained by applying S −1 4t5 twice. For example, S 4−25 40055 = S −1 4S −1 400555 =
S −1 4008085 = 008114.
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Ali E. Abbas is professor of industrial and systems engineering and public policy at the Viterbi School of Engineering and
Price School of Public Policy at the University of Southern California. He is also director of the National Center for Risk and
Economic Analysis of Terrorism Events (CREATE). His research
focuses on decision analysis, risk analysis, multiattribute utility
theory, and data-based decision making. Dr. Abbas is a recipient of multiple awards from the National Science Foundation
including the National Science Foundation CAREER Award in
2008. He is also widely published in books, journals, and conference publications, and has shared his expertise through television
appearances, TEDx, and other invited talks.
Zhengwei Sun is a lecturer in the department of management science and engineering, East China University of Science
and Technology. He received his Ph.D. in industrial engineering
from the University of Illinois at Urbana–Champaign, where he
was also a postdoctoral research associate. His research interests include value of information, Bayesian updating, and utility
theory.