See graphic user interface
Also called util, the unit of measurement for a utility function.
For example, if the utility function has been defined such that 100 utiles are assigned to the desirability of the best available
alternative and zero utiles have been assigned to the least desirable alternative, 1 utile represents one percent of the total
satisfaction, pleasure, or happiness gained by deciding to choose the best alternative rather than the worst available alternative.
A quantitative measure of a decision maker's subjective preference for the alternatives to a decision or the outcomes of those decisions.
The concept is central to
utility theory; also called the theory of rational choice.
Utility is a number representing the level of satisfaction, pleasure or happiness as it relates to the decisions that people make.
Also called preference function, cardinal utility function, Von Neumann utility function,
and probabilistic utility function, a mathematical function that assigns numbers to
attributes (also called performance measures)
selected to describe the outcomes to a decision. The
number assigned by the utility function indicates the decision maker's preferences for the outcome
bundle as described by the specified
attribute levels (the higher the number
the more preferred the bundle of attribute outcomes). The number assigned is expressed in arbitrary units called "utiles," however,
in many instances (when the function expresses utility on a cardinal scale), a utility
function can be scaled
so that the utility number is the equivalent monetary value of the outcome;
that is, what obtaining those attribute outcomes is worth to the decision maker.
functions are the central concern of utility theory. According
to utility theory, attributes (also called performance measures) are
selected to measure performance relative to each of the decision maker's objectives.
For example, the decision maker might be an engineer tasked with selecting the best approach for
constructing a large building. One objective might be safety, another objective might be cost, and so forth.
The attribute selected to measure performance relative to the safety might be the number of injuries that occur
during construction. The attribute
selected for the cost objective might be the total cost of constructing the building. A utility function
is then constructed to denote how much the decision maker prefers outcomes as described
by the selected attributes. For example, for specified levels for cost and for the other attribute outcomes,
the utility function would assign the highest utility to an outcome with zero injuries.
Utility functions are typically denoted U(x), where x may represent a single attribute, in which case the utility function
is referred to as a single attribute utility function or x may represent multiple
attributes x1,x2,...xN, in which case the utility function is
called a multi-attribute utility function.
Regardless, the utility function has the
important property that the most preferred alternative will be the one that
produces the attribute outcomes with the highest value for U. Also, if
there is uncertainty and probabilities are assigned to indicate
the decision maker's beliefs about how likely the various possible attribute outcomes are, the most preferred
alternative will be the one that
maximizes the expected value of
The utility function concept was first developed in 1947 by the mathematicians John von Neumann and Oscar
Morgenstern who were concerned about how one should choose among alternative gambles, which
they called lotteries. Von Neumann and Morgenstern showed that
a utility function exists provided that the
individual (the decision maker) accepts certain assumptions or axioms
meant to define "rationality." Most people find the axioms, which may be
expressed in various ways, easy to accept:
- Orderability: Two items A and B are always comparable; that
is, you must be able to tell if you prefer A to B, B to A, or that you
are indifferent between the them.
- Transitivity: If you prefer item A to item B, and you prefer
item B to item C, then you must prefer item A to item C.
- Continuity: If you prefer item A to item B and item B to item
C, then there must be some probability p for which you are indifferent
between item B and a lottery that provides item A with probability p and
item C with probability 1-p.
- Substitutability: If you are indifferent between two items A
and B, then for any lottery that contains A as a possible outcome, A may
be replaced by B without affecting your preferences.
- Monotonicity: If items A and B are the only possible outcomes
for alternative lotteries, and you prefer A to B, then you must prefer
lotteries with the higher probability of winning A.
Decomposability: (The "no fun in gambling axiom") Suppose you
are faced with the compound lottery illustrated by the event tree shown to the right: The
lottery will provide either item A or a subsequent lottery that will
provide either item B or item C. Suppose when you consider the second
stage lottery independently, you conclude that you are indifferent between it
and some item D (figure to the left). Then, you must be indifferent
between the original two stage lottery and the one stage lottery where
item D replaces the lottery between B and C.
Decision analysts have developed assessment methods for
encoding (i.e., deriving) a person's utility function.
Among other things that affect preferences, a utility
function may account for the decision maker's willingness to accept risk. This can be seen most clearly if the
utility function is expressed in a form that relates utility to the
equivalent monetary value of the decision outcome. Suppose, therefore, that V =
V(x) is the maximum amount of money a decision maker would be willing
to pay to obtain the decision outcome x (V(x), then, is a
value function). What would the
utility function look like? Since utility functions, by definition, are
determined empirically, there is no obvious reason to expect that a
particular mathematical relationship would emerge. However, it has been
shown that an exponential equation nearly always provides a good
Exponential utility function:
This equation is often alternatively written with
constants added so that U goes from zero to one when V goes
from the minimum to maximum values assigned to the decision outcomes.
It can be shown that if a condition known as the delta property holds, then the utility
function must have either this exponential form (or a linear form). The
delta property applies if the following is true: whenever there is
uncertainty over the outcome of some uncertain choice, if the value of
every possible outcome were increased by the same amount (same delta), then
the value of the uncertainty (its certain
equivalent) would be increased by the same amount (by delta).
The exponential utility function scales the possible outcomes to a
decision in a way that accounts for willingness to accept risk, and the
coefficient R in the exponent determines the amount of scaling. R is termed
the risk tolerance, and the
lower the risk tolerance the less desirable the utility function will show
outcomes that involve uncertainty to be. A method for assessing risk
tolerance (and therefore, for deriving the utility function from a value
function) is provided in the section of the paper chapter on risk tolerance, where there is also an
example illustrating how to use a utility function to value uncertain
An independence condition similar to preferential independence,
except that the assessments are made with uncertainty
present. Attribute Y is said to be utility independent of attribute Z if preferences over
lotteries involving different levels of Y do not depend on a fixed level of Z.
Note that utility independence (in contrast to additive independence) is not symmetric: it is possible that
attribute Y is
utility-independent of attribute Z and not vice versa.
Utility independence is a
slightly weaker independence condition than additive independence. If attributes are mutual utility independent, the multi-attribute
utility function will either have an additive form:
or it will have a multiplicative form:
where the Ui(xi) are single attribute utility functions.
Since utility independence refers to lotteries and a deterministic outcome is a special case of a lottery,
utility independence implies preferential independence. However, the converse is not
The two-variable case of the multiplicative equation is known as a multilinear form, and it is often useful
in situations where a compromise choice is sought that would be attractive to multiple stakeholders:
U(X1,X2) = w1U1(X1) + w2U2(X2) + (1 - w1 - w2)U1(X1)U2(X2)
The w's are weighting factors and U1 and U2 are utility functions
representing the preferences of stakeholder 1 and stakeholder 2 (or the preferences of two distinct stakeholder groups). The decision maker's utility function is
additive with respect to these two terms and represents a desire to seek an alternative that each will desire. The third,
multiplicative term, may be thought of as representing the decision maker's desire for an equitable distribution of value
between the two stakeholders.
A theory of how individuals should make decisions,
related to the concept of "rationality" used in economics. Also called
subjective expected utility theory, or, with reference to the primary developers of the theory,
von Neumann Morgenstern utility theory.
Utility theory is an "axiomatic" theory in that it is derived from a set of axioms (hypotheses) defining
how rational people behave. It has been shown that utility theory's axioms can be expressed in a number of different ways,
but in all cases the axioms seem, at least to most people, to be quite reasonable. For example, one axiom (transitivity)
states that if a person prefers outcome A to outcome B and outcome B to
outcome C, that person should prefer outcome A to outcome C. Another axiom
(substitution) states that if a person is participating in a lottery where
the prize is A, and if that person is completely indifferent between
receiving prize A and some alternative prize C, then that person should not
care if the lottery is modified by substituting prize C for the equally
desirable prize A. Other typical statements of utility theory's axioms include diminishing marginal
utility and diminishing rate of substitution, which imply that as a person acquires more and more of a given good,
their marginal value for another unit of that good, becomes less relative to other goods. An axiom
termed non-satiation states that people do
not have so much of everything they desire that they are at the point of not wanting any more.
Utility theory shows that if a decision maker accepts any one of the various
ways of expressing the axioms of rationality, then it can be proven that there is a mathematical function,
called a utility function, typically
denoted U, with the capability of aggregating all of the different considerations that
must be taken into account when deciding among alternatives. It also shows
that under certain specified conditions, the function will have various
simple mathematical forms, such as additive, multiplicative, or exponential forms. Most
significantly, the theory proves that, provided the axioms apply, the best
alternative (the one that is most preferred) will be the one that maximizes
the value of U (or, if there are uncertainties, the expected value of U).
The rational model is, in effect, a conceptualization wherein decision making is
regarded as simply a matter of choosing from among sets of alternatives. Alternatives are viewed as
leading to outcomes that can be evaluated by applying the decision maker's preferences. The theory recognizes
that the outcomes that follow the selection of alternatives are not generally precisely known, due to
uncertainties. However, in the face of uncertainties, rational decision makers judgmentally evaluate
possible decision outcomes in terms of their preferences and assess probabilities to indicate their beliefs
about uncertainties. This, as shown by the theory, maximnize their expected preference satisfaction.
Numerous doubts have been raised about utility theory's descriptive validiaty. In otherwords,
it is clear that peole do not always (maybe event often) behave according to the rational model. Much has
been writing critizing utility theory based on showing that the rational model does not reflect how people typically
make decisions. The proposnents of utility theory, however, see the fact that real world decision makers
fail to follow the prescriptions of utility theory as the main arguement for its applicaton. In otherwords, the fact
that utility theory leads to recommendations for actions that differ from what people typically do means that
following utility theory is likely to produce significant improvements in the degree to which outcomes desired
by decision makers are obtained.
The collection of techniques and methods for applying
utility theory to real-world decisions is known as decision analysis. Those techniques that apply in the
special case of decisions involving multiple decision objectives is called multi-attribute utility
analysis (MUA), or multi-objective decision analysis (MODA).
The process of determining the value of something, such as a project or asset. In a business context,
valuations typically seek to determine monetary worth, and that is the
meaning ascribed to the term throughout this website. However, in much of
the literature on project portfolio
management, valuation means assigning a number representing some
concept of attractiveness. Regardless, various theories and techniques are
available for conducting valuations, and the methods typically involve both
objective and subjective components.
As used on this website, value means monetary worth. The
value of something to someone is the maximum amount that individual would
be willing to pay to acquire it. Thus, the value of an asset to an
organization is the maximum amount that the organization's decision makers
would be willing to pay for that asset. Likewise, the value of a project is the maximum amount the
organization's decision makers would be willing to pay for the opportunity
to conduct that project; that is, what they would be willing to pay to
obtain the consequences of doing the project. If the consequences of doing
the project are uncertain, the value of the project is the maximum the
organization would spend for the gamble over project consequences. The
net value of a project is the difference between the value of the
project and its cost.
Although my definition of value is intuitive, it is
usually not practical to ask decision makers to estimate the maximum amount
they would pay for things, including projects. Fortunately, there are
well-established methods for quantifying project value. In particular,
decision theory and its
subfield multi-attribute utility analysis
(MUA) provide methods for building models for estimating project value
based on consideration of business objectives, the impacts of projects on
those objectives. and the willingness of the organization's decision makers
to make tradeoffs.
Be aware that word value appears often in project portfolio management (PPM) literature and
that many authors either don't define the term or define it in a way that
is unrelated to the concept of worth. Many PPM tools, for example, allow
users to define criteria, score projects against those criteria, and then
rank projects by the weighted, summed scores. Since weighted summed scores
do not measure value, such tools are unable to find value-maximizing
Compared to other definitions, defining the value of a
project as its monetary worth to the organization has two significant
advantages. First, since project value may be expressed in monetary units,
project value can be directly compared to project cost. Second, value
defined as worth exactly maps to organizational preferences—given two
projects competing for the same resources, we know that the organization
will prefer Project A to Project B if and only if it views the worth of
Project A's consequences to be greater than the worth of Project B's
consequences. This critical mapping does not hold for most other
definitions of project value.
Despite the arguments for expressing project
value in monetary units, there are situations where doing so
could create problems. If expressing value in dollar units
isn't possible, then, at minimum, project value should be expressed
as a cardinal utility; that is, a number that not only correctly measures the
relative preferences of a decision maker for candidate projects, but also
correctly measures the increment in preference obtained from of a change from
a decision to not conduct a project to a decision to conduct the project.
value at risk (VaR)
A metric that
describes the potential for loss, typically the potential for loss from a
portfolio of financial investments. The term is sometimes similarly applied
in the context of project portfolio
management to describe the risk associated with a portfolio of projects. Typically, VaR is defined as the
amount or percentage of available portfolio value such that there is a 95%
(or 99%) probability of the portfolio losing less than that amount over a
specified time horizon.
VaR is popular because it addresses the concept of
"maximum potential loss." A major weakness is that it is not additive; that
is, the VaR for a set of portfolios is not the sum of the VaR's of the
Also called worth function, measurable value function or value model, a function similar
to utility function but applicable to situations of certainty rather than uncertainty.
A value function is a mathematical representation of preferences; it maps the criterion/attribute specific measurement scale onto
numerical scale of value. Value functions are used in
multi-attribute utility analysis (MUA) to
The independent variables for
value functions are called attributes. A value function that has only a
single attribute is called a single attribute value function If the
value function has multiple attributes, it is called a multi-attribute
When used to value projects, a multi-attribute value function,
denoted V(x1,x2,...xN), assigns a
number V to a project based on attributes of the project, denoted
x1,x2,...xN, typically chosen so as
to describe the outcomes that would result if the project were to be
conducted. The number V indicates the relative value of the
indicated outcomes. V is often expressed in dollar units, so as to
indicate equivalent monetary value, or it may scaled between zero and one
so that higher values indicate more preferred outcomes. Regardless,
V is interval scaled, so that differences in the value of V
indicate differences in the levels of preference (a definition of
interval scale is provided under scale).
Some authors use the term ordinal value function to denote a different type of value function,
one that assigns values on an ordinal scale rather than on a cardinal scale. An ordinal value function is capable of correctly ranking attribute
combinations in terms of preference, but the differences in values so assigned may not correctly indicate the differences in
preference. A monotonic transformation
can be applied to any ordinal value function and the result will continue to be an ordinal value function that ranks attribute
combinations in exactly the same way. The term measurable value function is sometimes used to distinguish a value function that measures
value on a cardinal scale, and so, unless otherwise specified, the discussion of value function in the papers on this
website refer to measurable value functions.
Because the goal of project selection is to choose
projects that create maximum value, creating value/utility functions is the
key step for designing formal methods to prioritize projects. See additive
utility function for a brief description of a popular method for
constructing a value function.
value of information (VoI)
A term used in decision analysis to describe the
maximum amount a decision maker should logically be willing to pay for
information prior to making a decision. The VoI is defined as the monetary
amount that makes the value of the existing decision situation equal to the
value of the decision situation with information and with the added cost of
paying VoI for the information. Decision trees are often used to compute
Also referred to as a value-based judgment, personal, subjective judgment of how desirable or undesirable, good or bad, better or worse,
useful or not useful something is.
A model for expressing in monetary units, or units that can be converted to monetary units, the worth of an
alternative, outcome, project, or uncertainty.
Another name for an objectives hierarchy. Sometimes the term
is used to describe an objectives hierarchy with the performance measures arrayed
below the corresponding lowest-level objectives in the diagram.
An upcoming software product that has been announced but
is not yet available. Software developers sometimes provide information
about future products or upgrades months or even years in advance. They may
do so as a marketing ploy—if current customers believe the supplier
will release a breakthrough product soon, those customers may be willing to
stick with the supplier's aging software products longer. Also, announcing
a phantom product may cause potential customers to perceive the products
currently offered by competitors to be less attractive. The vaporware may
or may not exist, and may not ever be available with the indicated
capabilities and features. Regardless, spreading information about possible
future products helps the software provider.
A discrepancy or deviation, as in schedule variance.
Also, a measure of variability that indicates how much spread there is in a
set of numbers. The variance is the square of the standard deviation. It represents
the average squared deviation of each number from its mean.
A formal statement developed by an organization to define
what it wants to achieve over time. For example, this is the vision
statement of the Alzheimer Association: "Our Vision is a world without
Alzheimer's disease." The vision statement is typically written succinctly
and in an inspirational manner to make it easy for employees to remember
and repeat to others. Like an organizational mission statement, the vision
statement, if it has been developed, can be found on the organization's
A software program that mimics the behavior of computer
hardware. A virtual machine is capable of performing tasks such as running
applications and programs like a separate computer. The end user has the
same experience on a virtual machine as they would have on dedicated
Also called predictive market, a mechanism that
allows people to make real or simulated decisions related to the purchase
or sale of items or events. The results, such as the market prices that are
produced, are used to infer collective preferences or beliefs, or to make
predictions. Though a relatively new concept, numerous companies are
reportedly using virtual markets to generate information to guide decision
making. Virtual markets are can be attractive because participating in a
virtual market is simple and the collective information produced by
participants can be quite useful. One application area for a virtual market
is prioritizing projects (explained here).