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Decision Analysis

VIEWS: 19 PAGES: 39

									DECISION ANALYSIS

                                    e
Hans Wolfgang Brachinger & Paul-Andr´ Monney, Department of Quantitative Economics, Uni-
versity of Fribourg, Switzerland

Keywords: decision making under uncertainty, multiple criteria decision making, dominance, effi-
ciency, decision rule, expected utility paradigm, rationality axioms, Allais paradox, behavioral deci-
sion theories, risk-value approach, decision tree, influence diagram.

Contents
1. Introduction
2. Examples
   2.1. Example 1: Decision Problem Under Uncertainty
   2.2. Example 2: Multiple Criteria Decision Problem
3. General Concepts
   3.1. Decision Matrix
   3.2. Generation of Alternatives and States of Nature
   3.3. Dominance and Efficiency
   3.4. Valuation Function
4. Decision Making Under Uncertainty
   4.1. Uncertainty, Risk, and Partial Probability Information
   4.2. Decision Rules Under Uncertainty
   4.3. Decision Rules Under Risk
   4.4. Decision Rules Under Partial Probability Information
5. The Expected Utility Paradigm
   5.1. The St. Petersburg Paradox
   5.2. Certainty Equivalent
   5.3. Utility Function
   5.4. Expected Utility Principle
   5.5. Expected Utility Theory
   5.6. Rationality Axioms
   5.7. Empirical Results
   5.8. Extensions of Expected Utility
6. The Risk-Value Approach
   6.1. General Characterization
   6.2. Risk-value Dominance
   6.3. Compensatory and Lexicographic Approaches
   6.4. Alternative Risk-Value Models
7. Graphical Representation of Decision Problems
   7.1. Decision Trees
   7.2. Influence Diagrams

Glossary

decision maker (DM): The individual, group of individuals or organization having the necessity and
the opportunity to choose between different options.
decision problem: A situation in which a decision maker has to make a decision.
structural stage: The step in the decision making process in which the decision maker collects and
organizes the information relevant for the decision problem.
decisional stage: The step in the decision making process in which the decision maker selects and

                                                  1
uses a particular method reach the best decision.
alternative: A possible decision to be made by the decision maker.
state of nature: One situation among a list of possible situations that might happen.
objective probabilities: Probabilities that can be approximated by relative frequencies.
subjective probabilities: Probabilities resulting from a subjective assessment by the decision maker.
outcome: The result of choosing any particular alternative under any particular state of nature.
decision matrix: A matrix containing, for each alternative, its outcomes for the various states of
nature.
criterion: For each state of nature, a mapping from the set of alternatives to the real numbers ex-
pressing a particular objective pursued by the decision maker.
payoff: The monetary outcome of any alternative under any state of nature in the case of a decision
problem under uncertainty with one criterion.
dominated alternative: An alternative whose outcome, under every state of nature or for every cri-
terion considered, is worse than the outcome of another alternative.
valuation function: A real-valued function giving the overall value of the various alternatives.
decision rule: A procedure applied by the decision maker to find the optimal decision.
expected monetary value: The mean dollar value of an alternative.
certainty equivalent: The cash payment that makes the decision maker indifferent between playing
a lottery and receiving that payment.
utility function: A function that associates a subjective value to every possible outcome.

Summary

This article presents the most fundamental concepts, principles and methods of the scientific disci-
pline called decision analysis. After a short introduction to the topic, first, some general concepts
of decision analysis are presented. Then well-known decision rules for decision making under un-
certainty are described by means of the general concept of a valuation function. Thereby, different
degrees of uncertainty are taken into account. In the main section of this article, the most important
normative approach to decision making under uncertainty, the so-called Expected Utility Paradigm
is presented in detail. Important concepts like the certainty equivalent and the utility function are
introduced, the expected utility principle and the general theory as well as the rationality axioms
behind are discussed, and, finally, essential empirical results and behaviorial extensions of expected
utility are pointed out. In another section, the so-called risk-value approach to decision making under
uncertainty is presented at length, including both compensatory and lexicographic methods, as well
as classic and recent alternative risk-value models. Finally, graphical approches to decision mak-
ing under uncertainty like decision trees and influence diagrams are pointed out. All concepts and
techniques presented in this article are motivated and illustrated by simple examples.

1.   Introduction

Decision analysis is a scientific discipline comprising a collection of principles and methods aiming to
help individuals, groups of individuals, or organizations in the performance of difficult decisions. In
1968, Howard announced the existence of this applied discipline to integrate two different streams of
research which now are the two pillars upon which most of modern decision analysis rests: normative
decision theory and psychological (descriptive) decision theory. The former develops theories of
coherent or rational behavior of decision making. Based on an axiomatic footing, certain principles
of rationality are developed to which a rational decision maker has to adhere if he or she wants to reach
the “best” decision. The latter, psychological decision theory, empirically investigates how (na¨ve)  ı
decision-makers really make their decisions and, based on empirical findings, develops descriptive
theories about real decision behavior. However, advancements to decision analysis have been made

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in as different disciplines as mathematics, statistics, probability theory, and artificial intelligence, as
well as economics, psychology, and operations research.

Decision problems are characterized by the fact that an individual, a group of individuals, or an orga-
nization, the decision maker (DM), has the necessity and the opportunity to choose between different
alternatives. As the name of the discipline suggests, decision analysis decomposes complex decision
problems into smaller elements or ingredients of different kinds. Some of these elements are proba-
bilistic in nature, others preferential or value-oriented. Thereby, the presumption is that for decision
makers it is easier to make specific statements and judgments on well-identified elements of their
decision problems than to make global ad-hoc statements about the quality of the different options
between which a choice has to be made. One major task of decision analysis is, at the structural
stage of the decision making process, to help decision makers to get aware of all the ingredients that
have necessarily to be identified in a particular decision problem and to guide them in defining and
structuring it. A second important task is, at the decisional stage of the decision making process, to
develop methods and technologies to reassemble these ingredients so that a choice can be made.

Decision problems play a pervasive role in many economic, political, social, and technological issues
but also in personal life. There are many different kinds of such decision problems that can be dis-
cerned. Economic decision problems include, e.g., more theoretical problems like the problem of the
optimal consumption plan of a household or the optimal production plan of a firm as well as more
practical problems like the choice of a house or a car. The manager of a firm has to decide on the op-
timal location of a new production plant, politicians on the optimal location of a nuclear power plant.
An investor has to make a choice on how to invest in different investment options, and engineers on
which of different technological alternatives to realize. Given the richness of decision problems, the
decision analytic approaches and methods recommended differ from one situation to another. Some
general decision analytic concepts can, however, always be identified. Today, decision analysis has
evolved into a general thinking framework containing theories, methods, and principles all aiming at
a better understanding of any decision-making problem for a better solution.

There are two main problems dealt with in decision analysis: uncertainty and multiple conflicting
objectives. Uncertainty arises when the quality of the different alternatives of a decision problem
depends on states of nature which cannot be influenced by the decision maker and whose occurrence
is often probabilistic in nature. These states of nature act to produce uncertain possibly different and
more or less favorable consequences of each alternative being considered. The sales of a seasonal
product like, e.g., ice cream depend on the weather which cannot be influenced by the producer and
some weather is more favorable for the ice cream sales than another one. Multiple conflicting objec-
tives are a typical feature of economic and political decision problems. Any entrepreneur planning
a new production plant searches for a location where the wages to be payed are as low as possible
and, at the same time, the quality of the personnel is as high as possible. A family house of a certain
category should be as cheap as possible and, at the same time, offer a maximum of convenience.

In this article, the general basic concepts which form the core of modern decision analysis as a sci-
entific discipline are presented. This presentation includes the basic structure by which, generally,
decision problems are characterized and the ingredients which have to be specified in the structural
stage of any practical application. Furthermore, the classical principles, methods, and rules to identify
the “best” solution in the decisional stage of a decision problem are developed. Thereby, the empha-
sis is put on decision making under uncertainty. Decision making with multiple objectives is also
touched upon but is treated more detailed in Multiple Criteria Decision Making. All of the concepts
will be introduced by providing simple classroom examples.



                                                    3
2.     Examples

2.1.    Example 1: Decision Problem Under Uncertainty

Assume that Connie is the owner of a bakery and every early Sunday morning she has to prepare some
cakes that will hopefully be sold during the day. The cakes contain a special kind of cream that does
not stay fresh for more than one day which means that at the end of the day the unsold cakes must
be thrown away. The selling price of a cake is $15.25 and the production cost of a cake is $5.75. Of
course, Connie does not know how many cakes will be asked by costumers on a particular Sunday,
but by experience she assumes that the demand will not exceed five cakes. If she wants to have a
chance of making any profit at all, she surely should prepare a few cakes. But on the other hand if
she prepares too many of them it may happen that there won’t be enough customers to buy them. The
question is how many cakes should she prepare?

This little example is clearly an instance of a decision problem as Connie must decide on the number
of cakes to prepare. As a first step, the verbal description of the problem is now represented by a
so-called decision matrix D defined as follows. Let x denote the number of cakes Connie is going to
prepare. Obviously, the value of x is an integer between 0 and 5. So there are six possible values of x,
called alternatives. Each alternative corresponds to a possible decision by Connie and is associated
with a row of the matrix D, i.e. the alternative consisting of making x cakes is associated with the
(x + 1)-th row of D. Of course, the matrix D has 6 rows.

On the other hand, let y denote the total number of cakes requested by the costumers on a particular
Sunday. Of course, y is also an integer between 0 and 5 and the value of y is a matter of chance. Each
possible value of y is called a state of nature and corresponds to a column of the matrix D. More
precisely, the state of nature y is associated with the (y + 1)-th column of D. So D is a square matrix
of dimension 6 × 6. For i = 1, . . . , 6 and j = 1, . . . , 6, let dij denote the element of the matrix D
located at the intersection of the i-th row and the j-th column. Then, by definition, the value of d ij is
Connie’s profit if she decides to make (i − 1) cakes and the demand of cakes is (j − 1). In this case,
it is easy to verify that

         15.25j − 5.75i − 9.5    if i ≥ j
dij =
         9.5(i − 1)              if i < j,

which is called the outcome function. This leads to the following decision matrix
                                                                      
              0.00      0.00      0.00       0.00       0.00      0.00
     −5.75             9.50      9.50       9.50       9.50      9.50 
     −11.50            3.75    19.00      19.00      19.00      19.00 
D=  −17.25
                                                                       .                             (1)
                     −2.00     13.25      28.50      28.50      28.50 
                                                                       
          −23.00      −7.75       7.50     22.75      38.00      38.00
          −28.75 −13.50           1.75     17.00      32.25      47.50
In this case, a solution of the decision problem consists of choosing the number of cakes to prepare,
which corresponds to the selection of a particular row of the decision matrix. This decision problem
obviously is a decision problem under uncertainty because the consequences of choosing any number
of cakes to prepare depends on the unknown total number of cakes requested by the costumers.

2.2.    Example 2: Multiple Criteria Decision Problem

Suppose that Brenda and her family want to move to her home city and she is, therefore, looking for
a house for her family. Her objectives are (1) sufficient living space, (2) an acceptable price, and a (3)

                                                   4
nice residential area not too far away from the center. Furthermore, the house should (4) not be very
old and be (5) in good condition. Assume that Brenda has examined the local daily newspaper and
compiled a list of 7 potential houses which seem to meet her objectives. The question is which house
she should choose?

As example 1, this example is clearly an instance of a decision problem as Brenda has the necessity
and the opportunity to choose between different houses. But, in this case, the problem presents itself
differently. Contrary to example 1, first, there is no uncertainty involved (assuming that the prices are
more or less fixed and the condition can be verified unequivocally). Second, Brenda is not interested
in just one criterion as Connie is in profit. Brenda obviously pursues 5, i.e. multiple objectives where,
e.g., price and quality surely are more or less conflicting. Finally, not all of these objectives are
already operationalized in a natural way as it is the case for “profit”. “living space”, e.g., can be
operationalized by the number of rooms of a house as well as by its habitable square meters, and how
the “condition” of a house should be operationalized is completely open.

In that example, as a first step of the structural stage of the decision making process, each of the more
or less latent variables corresponding to Brenda’s objectives has to be operationalized by a measur-
able criterion. In general, there are many different ways to operationalize a given latent variable.
The main goal when operationalizing any latent variable is to minimize the “discrepancy” between
that variable and its operationalization. When this discrepancy is “minimized” cannot generally be
answered but is a matter of intuition and critical reflection on the competing alternative possibilities
to operationalize the given latent variable.

Assume that Brenda has solved the problem of operationalizing all the latent variables corresponding
to her five objectives, and has, for each of the seven houses, collected all their “values”. Then this
information can be structured in a two-way table as follows. Let the seven alternatives between which
Brenda has to choose, denoted by ai (i = 1, . . . , 7), be arranged as the head column of that table, and
the five criteria, denoted by ci (i = 1, . . . , 5), as its head line. Then each house is associated with a
line of that table where the “values” of the house for the five criteria are summarized. Assume that
the information available in Brenda’s decision problem is given with table 1.


                               Table 1: Brenda’s house-buying problem


This table shows that, in general, the different criteria of a multiple criteria decision problem are
measured on different scale levels. The first criterion “number of rooms”, e.g., is measured on an
absolute scale, criteria (3) through (5) on a ratio scale, whereas the second criterion “condition” is
only measured on an ordinal scale. Furthermore, the criterion “condition” is not yet quantified by real
numbers. An admissible quantification is given by any order-preserving real-valued function, i.e., by
any real-valued function assigning real numbers to the “values” of that criterion such that their rank
order is respected.

After simply quantifying the condition variable by the first five natural numbers and after rescaling
the price variable by the factor 10000, Brenda’s decision problem as given by table 1 can, as Con-
nie’s decision problem, be represented by a decision matrix D. As in example 1, each alternative is
associated with a row of that matrix. Of course, the matrix D, now, has 7 rows. On the other hand,
each criterion c, now, corresponds to a column of the matrix D. So D is a matrix of dimension 7 × 5.
Then, by definition, the value of dij (i = 1, . . . , 7; j = 1, . . . , 5) of the matrix is the value of house i




                                                      5
on the criterion j.       The decision matrix of Brenda’s decision problem is given with the matrix
                                       
          10     4          4 26      5
         11     5          5 24      4
      7         2         15 20      7
                                       
D= 7            1         15 20      8.                                                                 (2)
      7         1         20 22      8
                                       
            9    3         10 24      6
          13     5          0 32      3
In this case, a solution of the decision problem consists of choosing a house which, as in example
1, corresponds to the selection of a particular row of the decision matrix. This decision problem
obviously is a multiple criteria decision problem because Brenda simultaneously pursues multiple
objectives. It is a decision problem under certainty because all the houses are evaluated as if this
evaluation were certain.

3.     General Concepts

3.1.    Decision Matrix

In general, for any kind of decision problem, the decision maker has to choose one alternative out of
a set of n mutually exclusive alternatives ai (i = 1, . . . , n). In the case of a decision problem under
uncertainty, the quality of the different alternatives depends on m 1 states of nature zj (j = 1, . . . , m1 )
which cannot be influenced by the decision maker and lead, for each alternative, to possibly different
and more or less favorable consequences. In general, given any state of nature, for each alternative
more than one consequence is considered, i.e., after suitable operationalization, the decision maker
pursues m2 criteria ck (k = 1, . . . , m2 ). Thereby, a criterion ck is a real-valued function defined on
the set of alternatives
A = {a1 , . . . , ai , . . . , an }
parameterized by the set of states of nature
Z = {z1 , . . . , zj , . . . , zm1 },
i.e. a criterion ck is specified, for every state of nature zj , by a mapping
ck,j : A −→ R.
Multiple criteria decision problems are, in other words, characterized by the fact that for each state
of nature zj the outcome of every alternative ai is characterized by a m2 -dimensional vector dij of
criteria values with
dij = (c1,j (ai ), . . . , ck,j (ai ), . . . , cm2 ,j (ai )).
This vector denotes the decision maker’s outcome if he or she chooses alternative a i and the state of
nature zj happens. The set of all these outcomes can be arranged in the so-called decision matrix
D = (dij )             (i = 1, . . . , n; j = 1, . . . , m1 )
having n rows and m1 columns of vector-valued elements. Each element dij of this matrix is a m2 -
dimensional vector of criteria values. Thereby, it is assumed that any operationalization problem of
latent variables already has been solved.

For reasons of simplicity of exposition, in this article, the cases of decision making under uncertainty
and with multiple criteria are treated separately. I.e., in the case of decision making under uncertainty

                                                                6
only one criterion is regarded, and in the case of multiple criteria decision making only one state of
nature is considered. This latter case is, therefore, called multiple criteria decision making under
certainty.

In the case of decision making under uncertainty with only one criterion, i.e. with m 2 = 1, the
decision matrix reduces to a n × m1 matrix where each element dij is just a single real number. This
number indicates the one-dimensional outcome of alternative a i when state zj occurs and is called
payoff in the sequel. In the case of multiple criteria decision making under certainty, i.e. m 1 = 1,
the decision matrix reduces to a n × 1 matrix where each element di1 is a m2 -dimensional vector of
criteria values. This means that also in the case of multiple criteria decision making under certainty
the decision matrix reduces to a matrix where each element is just a single real number, i.e. to the
n × m2 matrix where each element is a real number indicating the value of an alternative a i for a
certain criterion ck . This means that, in both cases of decision problems, the starting point for the
decisional stage of a decision analysis is a decision matrix D of dimension n × m with real elements,
i.e.
                                
         d11 · · · d1j · · · d1m
       .           .         . 
       .           .         . 
       .           .         . 
D=    di1 · · · dij · · · dim  .                                                                (3)
                                 
       .           .         . 
       . .         .
                    .         . 
                              .
         dn1 · · · dnj · · · dnm


A first important step at the structural stage of practical decision analyses is to structure the decision
problem in the sense of the decision matrix. Thereby, in general, alternatives, objectives, as well as
states of nature do not “fall from heaven” but have to be constructed or generated. This process can
be very time consuming.

3.2.   Generation of Alternatives and States of Nature

In some decision problems, the determination of the set of relevant alternatives is, as the two exam-
ples above show, no real problem. The alternatives are given in a “natural” way. However, in many
problems the alternatives are not readily available and their generation is an important part of the
structural stage of the decision analysis. They may be determined by a search process or by a suitable
alternative generating technique. Take, e.g., Brenda’s decision problem to buy a family home. In the
form the problem was presented in Section 2.2, Brenda has decided to buy an already built house and
determination of the set of alternatives, then, means to search for suitable opportunities. Another pos-
sible approach were to decide to construct a new house. Then, determination of the set of alternatives
means to work out, with the technical help of an architect, different construction plans. In both cases,
the problem arises when to stop the process of determination of alternatives. This “meta decision
problem” is a decision problem itself which can be very complicated.

In Example 2, Brenda treats her decision problem as a decision problem under certainty and no states
of nature whatsoever are considered. However, strictly speaking, there is no real certainty in practical
decision problems. The outcome of a decision always depends anyhow on some states of nature. E.g.,
the condition variable Brenda uses for the evaluation of the different houses can only be measured
up to a certain degree of trustworthiness. When constructing a decision matrix the decision maker
therefore, first, has to decide if he or she wants to treat his or her problem as a decision problem under
certainty or uncertainty. This problem, once more, is a kind of “meta decision problem” which has to
be solved. The advantage of treating a decision problem (approximately) as a problem under certainty

                                                   7
is the considerable simplification it brings about. But it is admissible only when uncertainty plays a
minor role.

If a decision problem is considered as a problem under uncertainty, the decision maker has to specify
different states of nature. In many cases, the relevant states of nature can be described by a combi-
nation of finitely many parameters which can be interpreted as values of certain state variables. Each
possible parameter combination is a conceivable state of nature and when specifying possible states
of nature the decision maker has to choose certain parameter combinations. Once more, a kind of
“meta decision problem” has to be solved. In Example 1, the relevant states of nature are described
by only one parameter, i.e. the number of cakes demanded. But in a more elaborate modelling of
Connie’s decision problem some more parameters influencing the possible demand like, e.g., the cake
prices in other bakeries should be considered.

A technique often applied in such cases for modelling uncertainty is the so-called scenario analysis.
Thereby, a scenario is a combination of parameters, i.e. a combination of values of different state
variables. In scenario analysis a small subset of all possible parameter combinations is selected and
their respective probabilities are determined. This subset is selected such that it presumably contains
the most realistic parameter combinations, i.e. those with the highest probabilities. Then this subset
is taken as the set of states of nature to build up the decision matrix. Then, the decision problem is
solved assuming that the “true” state of nature is among the scenarios implemented.

A similar technique often used in practice is the following quasi-certain scenario technique. Thereby,
in a first step, a quasi-certain scenario is determined as if the decision problem would be treated as
a problem under certainty. This scenario is, according to the decision maker’s experience, conceived
as “quasi-certain” in the sense that it represents (1) the “average” scenario of some set of possible
scenarios or (2) the scenario which, subjectively, is most likely to occur. In a second step, this scenario
is “embedded” in a certain “neighborhood” which usually means that some parameters of the quasi-
certain scenario are varied to some extent. Usually, the quasi-certain scenario is altered by augmenting
or reducing certain of its parameters by a small number of percentages, e.g., by ±5%, ±10%, and
±15%. Thereby, dependencies between the parameters varied have to be taken into account suitably.
For each of these scenarios, then, the different alternatives are evaluated. This means that, in a certain
sense, the “quasi-certain” evaluation of the alternatives undergoes what usually is called sensitivity
analysis. Then, once more, the quasi-certain scenario and its alterations are taken as the set of states
of nature to build up the decision matrix and the decision problem is solved assuming that the “true”
state of nature is among the scenarios implemented.

Finally, also as far as the objectives the decision maker pursues are concerned there is a “meta decision
problem” to be solved. Which objectives should be pursued because they are decisive and which ones
not? In practice one of the main goals of the decision analyst is to help the decision maker to get
clearness over his or her objectives.

Note that the result of the structural stage of a decision analysis is by no means unique. The same
decision situation can, in general, be represented in different ways. This holds for the set of alter-
natives as well as for the set of states of nature. Every representation of a decision problem is just
an approximation of the real problem. The two kinds of modelling uncertainty described above, e.g.,
always lead only to an approximation of the “true” set of states of nature. It is important to recognize
that the final decision taken depends on an approximation of the decision problem and is, therefore,
only conditionally “best”.

In both cases, decision making under uncertainty with only one criterion and multiple criteria decision


                                                    8
making under certainty, at the decisional stage of decision analysis the same basic concepts are used.
Therefore, in the sequel, for the presentation of these concepts, it is started off from any decision
matrix D with real-valued elements where D can be interpreted in both ways outlined above. For all
decision problems which can be represented by a decision matrix, several general principles have been
proposed to reach a solution. In this section, these principles will be presented in a general context
and then applied to the particular examples introduced above. Thereby, generally, it is assumed that
the decision maker wants to maximize all of the criteria involved. This can be assumed without loss
of generality because any criterion ck which he or she wants to minimize can be replaced by −ck
which, then, has to be maximized.

3.3.    Dominance and Efficiency

The problem in any kind of decision problem represented by a decision matrix D is to choose an
alternative, i.e. a row vector of that matrix, which is optimal in some sense. In general, it is difficult to
compare the row vectors of a decision matrix among themselves. In example 1 introduced above, e.g.,
it is not immediately clear which of the third or fourth row vector should be preferred, or, in example
2, the first and the second row vector are difficult to compare. In general, for some states of nature or
some criteria, i.e., for some column, a first row vector leads to a better outcome than a second one,
whereas for some other states the opposite holds.

Of course, when comparing a row vector

di = (di1 , . . . , dim )

with another row vector

dk = (dk1 , . . . , dkm ),

if dij ≤ dkj for all j = 1, . . . , m and dij < dkj for at least one j, then obviously the vector dk should
be preferred to the vector di . In this case the vector di is said to be dominated by the vector dk or dk
dominates di . Note that dominance satisfies the transitivity property, i.e. if d k dominates di and di
dominates dl then dk dominates dl .

For the simple case where only two states of nature or only two criteria are considered, i.e. for m = 2,
the set of all alternatives of any given decision problem can, as indicated in Figure 1, graphically be
represented in a two-dimensional cartesian product where the two dimensions indicate the two states
of nature or the two criteria, respectively. Every alternative a is represented by the corresponding
point (d1 (a), d2 (a)) where dl (a) denotes the value of a under the criterion cl or the payoff of a if the
state of nature zl occurs (l = 1, 2).


                   Figure 1: Graphical representation of a decision problem (m = 2).


For any alternative a, all alternatives represented by a point in the hatched quadrant in Figure 2 are
those which are dominated by this alternative.


                                         Figure 2: Dominance.


In a first step of the decision making process, obviously, all dominated alternatives can be neglected.
Any “best” alternative has to be searched for in the set of all non-dominated alternatives. In Figure 1,

                                                     9
this set is indicated by the bold line in the “north-east” part of the set of all alternatives. Usually, this
set is called the efficient boundary of the set of all alternatives.

In Connie’s decision problem, there is no dominated alternative as can be easily seen by looking at the
first and last column of the payoff matrix (1). For determining the dominated alternatives in Brenda’s
decision problem, first, the values in the last three columns of matrix (2) have to be negated since it is
assumed that all criteria are maximized. Then it can easily be checked that Beacon Avenue dominates
Forest Street, Cambridge Street dominates Davis Square, and Davis Square dominates Exeter Road.
Furthermore, by transitivity of the dominance relation, it follows that Cambridge Street dominates
Exeter Road. Since these are the only dominance relations, the efficient boundary consists of the
alternatives Ash Street, Beacon Avenue, Cambridge Street and Glen Road.

3.4.   Valuation Function

Examples 1 and 2 show that, in general, it cannot be expected that the elimination of dominated row
vectors leads to a single “best” row vector and, therefore, a single “optimal” alternative. The basic
decision analytical idea to approach that problem is to evaluate the row vectors of decision matrices by
real numbers such that the natural order of these real numbers bring the decision maker’s preferences
to light. If such an evaluation is found, the vector (and the corresponding alternative) with the highest
evaluation is “best” and should be chosen.

Let, for a more formal exposition, Rows(D) denote the set of all row vectors of a given decision
matrix D. Then the basic idea to come to an “optimal” solution of that decision problem, i.e. to be
able to choose a “best” alternative, is to specify a real-valued function V defined on Rows(B),

V : Rows(D) −→ R

such that the number V (di ) represents the overall value of the vector di in Rows(D), i.e. the overall
value of the corresponding alternative. In this article, every such function will be called valuation
function.

The most important and most widely used examples of a valuation function are the convex type valu-
ation functions where the overall value of an alternative characterized by a vector d i = (di1 , . . . , dim )
in Rows(D) is given with
                                     m                                       m
V (di ) = V (di1 , . . . , dim ) =         wj vj (dij )   where wj > 0 and         wj = 1 .                  (4)
                                     j=1                                     j=1

Thereby, vj (dij ) denotes the contribution of the j-th component of the vector d i = (di1 , . . . , dim ) to
its overall value and wj denotes the weight attached to that component.

In multiple criteria decision making under certainty, the functions v j are usually called partial value
functions. The partial value function vj characterizes the contribution of the j-th criterion to the
overall value of an alternative. The weight wj reflects the importance the DM attaches to the j-th
criterion. In decision making under uncertainty (with only one criterion), for all components of the
vector di = (di1 , . . . , dim ), usually the same function u is used, i.e., vj = u for all j = 1, . . . , m, and
is called utility function. The utility value u(dij ) characterizes the contribution of the component dij
to the overall utility of an alternative characterized by the vector d i = (di1 , . . . , dim ). As weight wj ,
the probability of occurrence of the state of nature zj or some transformation of it is used.

Given a valuation function, that alternative is chosen which is “best” in the sense of the valuation.

                                                             10
This means choosing the alternative ai∗ with the index

i∗ = i∗ (V (d1 ), . . . , V (dn ))

defined by

i∗ (V (d1 ), . . . , V (dn )) = argmax V (di ).                                                            (5)
                                i=1,...,n



Any valuation function V together with the prescription (5) defines a decision rule. A decision rule
is a procedure that, for any decision matrix, specifies an element i ∗ in {1, . . . , n} representing the
index of the alternative that should be selected by the decision maker because the overall value of its
corresponding vector di∗ is largest.

Note that this notion of a valuation function also holds for the more general case where the outcome
of every alternative ai when the state of nature zj occurs is characterized by a m2 -dimensional vector
dij of criteria values. In that case the row vectors of the decision matrix are vectors of m 2 -dimensional
vectors and the domain of a valuation function has an additional dimension, i.e. valuation functions
are then defined on matrices. In the case of decision making under certainty characterized by the
fact that only one state of nature is considered, regardless whether there are multiple criteria or just
one criterion, a valuation function usually is called value function (see: Multiple Criteria Decision
Making).

Now, the notion of a decision problem can be defined as follows: A decision problem consists of two
consecutive stages, a structural stage and a decisional stage. In the structural stage, the situation in
which the decision maker has to make a choice has to be structured in the sense that (1) a set A of
n alternatives ai (i = 1, . . . , n), (2) a set Z of m1 states of nature zj (j = 1, . . . , m1 ), and (3) a set
C of m2 criteria ck (k = 1, . . . , m2 ) has to be specified and arranged in a decision matrix. Given a
decision matrix, in the decisional stage, (4) a valuation function V has to be specified and then (5)
the alternative maximizing this valuation function has to be determined, i.e., a decision rule has to be
applied. Each of these five points can be quite difficult in practice.

In the following sections of this article, an overview is given on the most important classical concepts
developed for the decisional stage of decision problems under uncertainty. An overview on the most
important concepts developed for the decisional stage of multiple criteria decision problems is given
in Multiple Criteria Decision Making.

4.     Decision Making Under Uncertainty

4.1.    Uncertainty, Risk, and Partial Probability Information

As mentioned in the Introduction, uncertainty arises when the quality of the different alternatives of
a decision problem depends on states of nature which cannot be influenced by the decision maker
and which, for each alternative being considered, act to produce uncertain possibly different and
more or less favorable consequences. Many authors, for the decisional stage of decision problems,
traditionally make a difference between two cases of decision making under uncertainty. Given the
decision matrix structuring a decision problem, the first case is characterized by the fact that the
only information the DM has or assumes to have is that the true state of nature belongs to the set
of states integrated in that decision matrix. He or she does in that case, then called decision making
under uncertainty, not dispose of any probabilistic information whatsoever on the occurrence of these
different states. In the second case, usually called decision making under risk, the DM additionally

                                                      11
is provided with objective or subjective probabilities p j (j = 1, . . . , m) on the occurrence of the
different states of nature taken into account in the decision matrix.

Basically, this difference is not made in this article for two reasons. First, decision situations under
uncertainty in the proper sense are not very realistic cases. In Connie’s decision problem, e.g., it is
not realistic to assume that she does not have the slightest idea on the likelihood of occurrence of the
different states considered. After all, Connie presumably is the owner of her bakery not just since
yesterday and Sunday after Sunday she gains experience on the consumption behavior of her clients.
So, in practice, she will have some kind of probability information on the states of nature. Second, the
assumption that the DM does not even have a subjective probability distribution on the set of states
of nature is theoretically dubious because it contradicts the basic rationality axioms of the Subjective
Expected Utility (SEU) theory introduced in the next section. This paradigm is the decision theoretic
model for decision making under uncertainty. The rationality axioms this model assumes on the DM’s
preferences not only imply a certain decision rule but also that there exists a subjective probability
distribution on the states of nature taken into account. Such information has to be used in practice.

In general, the essential problem in practical decision situations under uncertainty is that the DM has
some probabilistic information on the likelihood of occurrence of the different states but this informa-
tion is too vague or fuzzy to be able to specify without difficulties a precise probability distribution,
at least at reasonable costs. The position of the advocates of the Subjective Expected Utility (SEU)
paradigm is, nevertheless, that the every DM should try to “measure” his or her subjective probabili-
ties on the states of nature. Otherwise he or she does not behave rationally because useful information
is left aside.

A more pragmatic position is not to try to measure probabilities at any cost but to try to integrate the
partial probability information readily available in a suitable decision rule. The best known academic
example of a decision problem under partial probability information is the so-called three-color-
problem introduced by Ellsberg in 1961. In this problem, an urn containing 90 balls is presented. 30
of these balls are known to be red. The remaining ones are known to be black or yellow, but with
unknown proportion. From this urn, exactly one ball is to be drawn randomly. The alternatives are
different bets on colors or pairs of colors, respectively. In this situation, obviously, the probability of
red is 1/3 and the probabilities of black or yellow are known to be between 0 and 2/3 but uncertain.
The probability information available is, therefore, only partial.

Decision rules under partial probability information will be presented at the end of this section as
well as in section 6. For the sake of completeness, in this section, first, a short review will be given
on the most important decision rules for decision situations under uncertainty where no probability
information is given at all.

4.2.   Decision Rules Under Uncertainty

A first “classical” decision rule tailored to decision situations under uncertainty where no probability
information is given at all reflects the attitude of a pessimist decision maker (DM) accentuating that
the worst possible outcome could occur. Thereby, each alternative characterized by a vector d i is
evaluated by the worst possible outcome
V (di ) = min dij .                                                                                     (6)
         j=1,...,m

On the basis of this valuation function, the DM selects the alternative for which this worst outcome
is the largest. This decision rule is called the maximin rule. Note that the decision obtained by
this rule is, in general, not necessarily unique. In the example given by the decision matrix (1), the

                                                    12
valuations V (di ) of the different alternatives ai (i = 1, . . . , n) are given by the first column vector
D1 = (d11 , . . . , dn1 ) . The optimal decision obtained by the maximin rule is to prepare zero cakes,
which is certainly in accordance with the attitude of a pessimist person. This rule has been criticized
for being overly cautious. To illustrate this, let’s consider the following decision matrix:

P =    15 15000
       17    16
According to the maximin rule, the second decision is optimal. Obviously, the maximin rule implies
that the DM misses the opportunity of making a big win.

A second decision rule is characteristic of an optimist DM accentuating that the best possible outcome
could occur. Thereby, each alternative is evaluated by the best possible outcome

V (di ) = max dij .
         j=1,...,m

Then the DM selects the alternative for which this best outcome is the largest. This decision rule is
called the maximax rule. Again, note that the decision obtained by this rule is, in general, not neces-
sarily unique. In the example given by the decision matrix (1), the valuations V (d i ) of the different
alternatives ai i = 1, . . . , n are given by the last column vector Dm = (d1m , . . . , dnm ) . The optimal
decision obtained by the maximax rule is to prepare five cakes, which is certainly in accordance with
the attitude of an optimist person. This rule has been criticized for being too optimistic. To illustrate
this let’s consider the following decision matrix

P =     15     16
        17 −15000

According to the maximax rule, the second decision is optimal in spite of the possibility of a big loss.

The two decision rules presented so far represent two extreme attitudes of the DM with respect to
uncertainty as one is completely pessimistic and the other is completely optimistic. It is possible to
define a rule reflecting an attitude that is somewhere between these two extremes. If α denotes a
number between 0 and 1, then define the valuation function

V (di ) = (α max dij ) + ((1 − α)       min dij ).                                                      (7)
              j=1,...,m                j=1,...,m

The corresponding decision rule is called the Hurwicz rule. If α is set to 0, then this rule is simply
the maximin rule, whereas if α is set to 1 the rule is the maximax rule. It is therefore quite natural
to call α an index of optimism. Once again, the decision resulting from the application of this rule
may not be unique. If, for the decision matrix (1), we take α = 0.5 then the optimal decision is also
to make 5 cakes. The problem with this rule is the choice of the value of α. Since most people are
rather pessimistic, α is usually chosen to be relatively small. Of course, it is possible to let α change
between two limits and observe how stable the optimal decision will be.

Now suppose that the DM is concerned about how his or her decision might be viewed in the future
after the state of nature is known with certainty. Here, the first step is to build the regret matrix from
the original decision matrix. For each state of nature j, the maximal decision that can be attained is

max j := max dij
          i=1,...,n

If the state zj indeed occurred, the regret of having made the decision ai is given by

rij := max j − dij .

                                                     13
The regret matrix is then formed of the various values of rij . Now the DM again assumes a pessimistic
attitude accentuating that whatever decision she makes, the state of nature leading to the largest regret
could occur. Then she selects the alternative for which this maximum regret is as small as possible.
Equivalently, this means that the valuation function

V (di ) = min −rij                                                                                     (8)
            j=1,...,m

together with the prescription (5) is used as decision rule. This method for selecting an alternative is
called the minimax regret rule and, once again, there may be more than one optimal decision according
to this rule. For the decision matrix (1), the regret matrix is
                                                                       
               0.00      9.50     19.00       28.50     38.00     47.50
              5.75      0.00       9.50      19.00     28.50     38.00 
             11.50      5.75       0.00        9.50    19.00     28.50 
R=   
                                                                        
             17.25     11.50       5.75        0.00      9.50    19.00 
                                                                        
              23.00     17.25     11.50         5.75      0.00     9.50
              28.75     23.00     17.25       11.50       5.75     0.00

and the fourth alternative is optimal because V (d4 ) = −19 = maxi=1,...,6 V (di ). So, according to this
rule, Connie should make 3 cakes.

4.3.   Decision Rules Under Risk

A “classical” decision rule tailored to the case of decision situations where probabilities of the differ-
ent states of nature are assumed to be available, i.e., to the case traditionally referred to as a decision
making under risk, is the expected monetary value (EMV) decision rule. If p j (j = 1, . . . , n) denotes
the probability of the state of nature zj the valuation function characterizing this rule is
             n
V (di ) =         pj dij .                                                                             (9)
            j=1


Of course, V (di ) is nothing but the mathematical expectation or the mean of the random payoff when
the alternative ai is chosen. This shows that applying the EMV rule means selecting the alternative
with maximal expected or mean payoff. The EMV decision rule sometimes is also called the Bayes
criterion to the honor of Thomas Bayes (1702-1761) who was one of the first decision theorists
in history and who was the first to advocate the expectation principle for decision making under
uncertainty.

In Connie’s decision problem characterized by the decision matrix (1), it can reasonably assumed that
Connie disposes of some experience on the consumption behavior of her clients. Assume that, based
on that experience, she is able to specify the probabilities p 1 = p5 = p6 = 0.1, p2 = p4 = 0.2 and
p3 = 0.3 for the corresponding states of nature. For each alternative a i (i = 1, . . . , 6), the values
attached to the row vectors di are given in the vector

(V (d1 ), . . . , V (d6 )) = (0, 7.975, 12.9, 13.25, 10.55, 6.325) .

The optimal decision according to the Bayes criterion obviously is to make 3 cakes.

In practical applications, the probabilities used may be objective or subjective in nature. In any case,
when a probability distribution over the states of nature is known or assumed to be known, this is
the result of some measurement or estimation procedure usually called probability assessment. In the
case of objective probabilities, the standard statistical estimations given by the relative frequencies

                                                       14
can be used to determine the probabilities. In the case of subjective probabilities, different techniques
have been developed to help the decision maker determine the probability assigned to each state of
nature according to his or her personal judgement. It has been found that these techniques give the
best results when they are applied during an interview between the analyst, i.e. the person trying to
obtain the probability distribution, and the decision maker.

A first technique called the probability method is composed of seven precise steps in which the analyst
asks questions to the DM. The probability method has the advantage of counteracting the observed
tendency of individuals to provide a range of possible values for the uncertain variable that is way
too short. In another method, called graph drawing, the DM is asked to select the graph of a proba-
bility distribution function that best represents his or her judgement among a collection of admissible
probability distribution graphs. In another technique called method of relative heights, first the most
probable state of nature is identified. Then, in a second step, the ratio between the probability of any
other state of nature and the probability of this most probable state of nature is assessed. In the last
step, the results are normalized to get a probability distribution.

During the assessment procedure, the analyst trying to identify the DM’s probability distribution is
confronted with two problems. The first problem is that the DM is not necessarily coherent in his
answers, e.g., he may well assert that a certain event E will occur with probability p and, at the
same time, assign probabilities that do not add up to 1 − p for the exhaustive and exclusive events
representing the negation of E. Another problem is that the DM may very well be inconsistent
in that two different probability distributions are obtained when two different assessment methods
are used. The information gained from incoherent or inconsistent behavior should be used for a
subsequent better probability distribution assessment. Of course, remains the question of the validity
of the assessed probability distribution, i.e. is it really the “true” distribution of the different states of
nature? In general, this question is very difficult to address.

4.4.   Decision Rules Under Partial Probability Information

As mentioned above, the prevailing problem in practical decision situations under uncertainty is that
the DM has some information on the likelihood of occurrence of the different states but this infor-
mation is too vague or fuzzy to specify a precise probability distribution. Such situations of partial
ignorance or ambiguity can be characterized by uncertainty about the probabilities of the different
states of nature and are, therefore, called decisions situations under partial probability information.

Generally, partial probability information can be covered by a subset P, with |P| > 1, of the set
PZ of all probability distributions p defined on the set of states of nature Z. Thereby, P is to be
interpreted as the set of all probability distributions compatible with the information available. In
Ellsberg’s three-color-problem introduced in section 4.1. , e. g., the partial probability information
can be covered by the set
                               1              2             2
P = {p = (p1 , p2 , p3 ) | p1 = , p2 = λ, p3 = − λ ; λ ∈ [0, ]}                                          (10)
                               3              3             3
where p1 is the probability of drawing a red ball, p2 a black, and p3 a yellow ball.

The most important practical cases of partial probability information are those where the decision
maker is able to (not necessarily completely) rank the states of nature according to their probability of
occurrence or where he disposes of interval-valued probabilities. It can be shown that in all of these
cases the probability information P is linear in the sense that it allows a description
P = {p = (p1 , . . . , pm ) ∈ PZ | Bp ≥ b} ,                                                             (11)

                                                     15
where B is a (k × m)-dimensional matrix and b is a k-dimensional vector. This is a convex polytope
of dimension m − 1 in Rm .

The “classical” decision rule tailored to the case of decision making under partial probability infor-
mation, is the MaxEmin decision rule. If P denotes the partial probability information on the states
of nature Z the valuation function characterizing this rule is
                   m
V (di ) = inf            pj dij .                                                                              (12)
            p∈P
                   j=1

Applying this decision rule means that the DM selects the alternative where the worst expected payoff
possible under the conditions given by P is maximal. Of course, finding V (d i ) is a linear program-
ming problem which can be solved for example by the simplex algorithm.

For practical applications it is important to note that, under weak conditions, the valuations of the
different alternatives are easily calculable because the linear function        pj dij takes on its infimum
in one of the finitely many vertices of the convex polytope (11). So to get this infimum for any
alternative, the DM has to calculate the values of the linear function         pj dij only for the extremal
points of P. Assume, e.g., that the DM is able to completely rank order the m different states of
nature according to their probabilities of occurrence, i.e., to specify, say, the ordering

z1     z2     z3       ...      zm

where zj zk means that the DM holds zj to be at least as probable as zk . In this case, the conditions
on p = (p1 , . . . , pm ) ∈ PZ describing P are m pj = 1, 0 ≤ pj ≤ 1 for all j = 1, . . . , m,
                                                       i=1
and p1 ≥ p2 ≥ . . . ≥ pm . Note that these conditions can easily be expressed in the form (11).
Then the extremal points of the corresponding P are given with the distributions p 1 = (1, 0, . . . , 0),
p2 = (1/2, 1/2, 0, . . . , 0), p3 = (1/3, 1/3, 1/3, 0, . . . , 0), . . . , pm = (1/m, . . . , 1/m). To get the
valuations (12) the DM has to calculate, for each alternative, the values of the function           pj dij only
for these distributions. This can be easily made by hand.

Considering Example 1, assume that Connie believes that a reasonable rank order of her six states of
nature is given with

z3     z4     z2       z5      z1    z6 .

Then for each alternative the valuation (12) can easily be calculated in the following way: (1) the
payoffs of each alternative characterized by the vector di have to be ordered according to the rank
order of the states of nature, which leads to the vector

di = (di1 , . . . , dim ),

(2) the values of the function              pj dij have to be calculated consecutively for the distributions

                                     p1 = (1, 0, . . . , 0),   p2 = (1/2, 1/2, 0, . . . , 0),

and so on, as long as these values can still fall.

The ordered payoffs for, e.g., alternative a3 are given with the vector

di = (19, 19, 3.75, 19, −11.50, 19)

and the minimum of pj dij is reached for the distribution p5 = (1/5, 1/5, 1/5, 1/5, 1/5, 0). For this
distribution, the value of pj dij is 9.85 and hence V (d3 ) = 9.85.

                                                                16
For each alternative ai (i = 1, . . . , 6), the values attached to the row vectors di according to the
MaxEmin decision rule (12) are given in the vector
(V (d1 ), . . . , V (d6 )) = (0, 6.45, 9.85, 10.20, 7.50, 1.75) .
The optimal decision according to the MaxEmin decision rule obviously is now to make 3 cakes.

A further decision rules tailored to the case of decision making under partial probability information
will be mentioned in section 6.

5.     The Expected Utility Paradigm

5.1.      The St. Petersburg Paradox

In the beginning of the 18th century, the expectation principle for decision making under uncertainty
had been advocated by Thomas Bayes (1702-1761) and others. In 1738, based on ideas Cramer
already had communicated to him in a letter in 1728, Daniel Bernoulli (1700-1782) published a fa-
mous booklet entitled “Specimen theoriae novae de mensura sortis” . In this booklet Daniel Bernoulli
showed that there is an empirical problem with the Bayes criterion. To formulate that problem it
is useful to introduce the notion of a lottery. A lottery is a discrete random variable with possible
payoffs d1 , . . . , dm and corresponding probabilities p1 , . . . , pm . Lotteries will be denoted by
L((d1 ; p1 ), . . . , (dj ; pj ), . . . , (dm ; pm )).                                            (13)
According to the expected monetary value rule described in the previous subsection, it seems reason-
able to consider that the fair price for participating at a lottery (13) is the expected payoff
               m
E(L) =              pj d j .
              j=1

In consequence, every person who is given the opportunity to play the lottery for a price P less than
E(L) should decide to play the lottery.

However, D. Bernoulli considered a game in which a fair coin is tossed until “head” appears. If this
happens on trial j, then the player receives 2j dollars. This game is clearly a lottery. The possible
payoffs are
2, 22 , . . . , 2j , . . .
and the probability of the payoff 2j is 0.5j , i.e. this game can be represented by the lottery
L((2; 1/2), (22 ; 1/22 ), . . . , (2j ; 1/2j ), . . .).
The expected payoff of this lottery is
               ∞
                     1· j
E(L) =                  2 = + ∞.
              j=1
                     2j

So, according to the Bayes rule, no matter how much is requested for playing, any person should,
hypothetically, decide to play the lottery. This statement is quite conter-intuitive and, in fact, in
practice people are willing to pay only very limited prices. This phenomenon has later become known
as the so-called St. Petersburg paradox. To settle this problem, Bernoulli introduced the idea of
a person’s attitude towards risk. Different risk attitudes can be characterized by different so-called
certainty equivalents.

                                                          17
5.2.   Certainty Equivalent

For the presentation of the notion of a certainty equivalent and different risk attitudes let, in the sequel,
every lottery of the type
L((d1 ; 0.5), (d2 ; 0.5))
be denoted by L(d1 , d2 ). For any DM, the certainty equivalent of such a lottery is defined as the
certain payment CE that makes the DM indifferent between playing the lottery and receiving that
payment CE . For the lottery L(12000, −2000) with expectation $5000, e.g., this decision situation
can be represented as in Figure 3 where d denotes a possible certain payment received by the DM.


Figure 3: The decision problem between the lottery L(12000, −2000) and a certain payment d


If the certain payment d is close to $12000, any DM will certainly choose not to play the lottery and
take the certain payment of d, thereby avoiding the risk of losing $2000. On the other hand, if d is
close to $−2000, then he will certainly choose to play the lottery as he wants to seize the opportunity
of winning the $12000. As the value of the certain payment d goes down from its original value of
$12000, the certain payment d gets less and less attractive. It is reasonable to assume continuity in the
sense that, at some point, any DM is indifferent between playing the lottery and getting the certain
payment d. Then, this value of d is the DM’s certainty equivalent CE of the given lottery.

The certainty equivalent CE of a lottery will certainly not be the same for all decision makers and
the magnitude of the certainty equivalent can serve as an indicator for different risk attitudes. For
example, suppose that the CE of Rose is $3000. Then this value certainly indicates that Rose is a risk
averse person. Indeed, she is willing to accept any sure payment above $3000 and below the expected
value $5000 of the lottery for not playing the lottery because she does not want to take the risk of
losing $2000 by playing. Now suppose that Colin’s CE is $9000. Then he is certainly risk seeking
since he wants a sure payment of more than $9000, i.e. well above the expected value of the lottery,
for not playing the lottery where he expects to win the $12000. He must be paid a lot for not playing
and hence for not taking the risk of losing $2000. Finally, a DM is called risk neutral if his CE is the
same as the expected payoff of the lottery.

5.3.   Utility Function

To include these different attitudes towards risk in a decision principle, the notion of a utility function
has been proposed by Daniel Bernoulli. If the values in the real closed interval [a, b] correspond to
the possible payoffs in any given decision problem, then a utility function on [a, b] is an increasing
function
u : [a, b] −→ [0, 1]
such that u(a) = 0 and u(b) = 1 where, for every x ∈ [a, b], the value u(x) represents a decision
maker’s personal, subjective utility of the payoff x. A decision maker’s utility function can be derived
using the concept of a certainty equivalent.

For that purpose, in a first step, the DM is presented the lottery L(a, b) and he is asked to specify his
certainty equivalent CE (a, b) for this lottery. Since the DM is indifferent between playing the lottery
and the certainty equivalent, it is natural to assume that the utility of the certain payoff CE (a, b) is the
same as the expected value of the utilities u(a) and u(b), i.e. to set
u(CE (a, b)) = 0.5 u(a) + 0.5 u(b).

                                                     18
Since u(a) = 0 and u(b) = 1 we get u(CE (a, b)) = 0.5. Then the DM is required to specify the
certainty equivalent for the lottery L(a, CE (a, b)). Let CE 1 denote this value. Then, by the same
argument as above,

u(CE 1 ) = 0.5 u(a) + 0.5 u(CE (a, b)),

which implies that u(CE 1 ) = 0.25. Furthermore, the DM is required to specify his certainty equiva-
lent CE 2 for the lottery L(CE (a, b), b). Then

u(CE 2 ) = 0.5 u(CE (a, b)) + 0.5 u(b),

which implies that u(CE 2 ) = 0.75. In this way, arbitrary many further pairs (x, u(x)) can be deter-
mined by applying to different lotteries L(d1 , d2 ) the equivalence principle given by

u(CE ) = 0.5 u(d1 ) + 0.5 u(d2 ),

where CE is the certainty equivalent the DM has specified for the lottery L(d 1 , d2 ). Then the several
points obtained by this procedure can be connected by a smooth curve, which is considered to be the
graph of the decision maker’s utility function. A typical utility function is given in Figure 4.


                             Figure 4: A typical (concave) utility function.


Empirical evidence shows that the specification of a utility function is a quite demanding task. There-
fore, it is highly recommended to complete, in practice, this procedure by a consistency check in-
volving the certainty equivalents specified before. E.g., the DM, additionally, is required to specify
his certainty equivalent for the lottery L(CE 1 , CE 2 ). This certainty equivalent should be equal to
CE (a, b).

An inspection of this method to determine a utility function shows that the basic idea is to compare
a certain reference lottery L = ((xmin ; p), (xmax ; 1 − p)) with its certainty equivalent CE(L). Obvi-
ously, there are four parameters involved in that comparison and, in fact, given any three of them, the
forth parameter can be asked for. Above, the three parameters p = 0.5 and x min and xmin were given
and the certainty equivalent had to be determined by the DM. Alternatively, three other parameters in-
cluding the certainty equivalent can be given and, then, asked for the forth one. This shows that there
are different possibilities to determine a utility function. Methods asking for certainty equivalents are
called certainty equivalent methods, others asking for probabilities probability equivalent method.

On the basis of a given utility function, it is now possible to determine the DM’s attitude towards risk
by looking at the shape of his or her utility function. As it was explained above, risk-seeking behavior
is characterized by the fact that the certainty equivalent of any lottery is larger than the expected
payoff of that lottery. Risk-averse behavior is characterized by the fact that the certainty equivalent of
any lottery is smaller than the expected payoff of that lottery. It follows from above that for lotteries
of the type L(a, b), given a utility function u, the certainty equivalent CE (a, b) is given by the inverse
value u−1 (0.5). The expected payoff

E(a, b) := 0.5 a + 0.5 b

of the lottery L(a, b) is given by the inverse value D −1 (0.5) of the diagonal D connecting the points
(a, 0) and (b, 1). Figure 5 shows that for any concave utility function holds CE (a, b) < E(a, b), and


                           Figure 5: A concave and a convex utility function.

                                                    19
for any convex utility function CE (a, b) > E(a, b). I.e., concave utility functions imply risk averse-
ness and convex utility functions risk-seeking behavior.

Of course, if the utility function is both convex and concave, i.e. linear, then the decision maker is
called risk-neutral because the certainty equivalent of any lottery is the same as the expected payoff
of that lottery.

In summary, a risk-averse DM has a concave, a risk-seeking DM a convex and a risk-neutral DM a
linear utility function. Of course, the more concave (resp. convex) is the utility function, the more
risk-averse (resp. risk-seeking) is the DM. If the DM is known to be risk-neutral, then of course his
or her utility function is a straight line passing through the two points (a, 0) and (b, 1).

5.4.    Expected Utility Principle

Now, the concept of a utility function can be used to incorporate the DM’s risk attitude into a decision
rule for decision making under uncertainty. Let D be a given payoff matrix representation of any
decision problem and let a be the minimum of the elements dij of D and b their maximum:

a :=         min             dij ,   b :=         max             dij .
       i=1,...,n;j=1,...,m                  i=1,...,n;j=1,...,m



The basic idea, now, is to transform all the payoffs of the decision matrix by the DM’s utility function
such that his or her preferences on the set of all alternatives are represented by the expected utilities
of these alternatives. I.e., given probabilities pj (j = 1, . . . , m) of the states of nature and the DM’s
utility function u defined on the interval [a, b], each row vector d i is evaluated by the valuation function
             m
V (di ) =          pj u(dij ).                                                                         (14)
             j=1

which is, obviously, the expectation of the possible utility values of alternative a i . Therefore, the
valuation function (14) is called expected utility of alternative a i .

In fact, by equation (14) a whole family of valuation functions is defined, i.e. for every particular
utility function, an example of that family results. For that reason, the decision rule corresponding to
(14) is called expected utility (EU) principle. According to that principle the DM should choose the
alternative maximizing the expected value of his or her personal utility.

It is important to note that this decision rule leads to the same optimal decision if the utility function u
is replaced by a positive affine transformation of u, i.e. if u is replaced by a mapping v : [a, b] −→ R
such that, for all x ∈ [a, b], v(x) = µ · u(x) + ν for some µ > 0 and some ν ∈ R. Indeed, if V v
denotes the valuation function corresponding to v, i.e.
              m
Vv (di ) =         pj v(dij ),
             j=1

then

argmax V (di ) = argmax Vv (di )                                                                       (15)
i=1,...,n                i=1,...,n




                                                                  20
because

                n                     n
Vv (di ) =            pj v(dij ) =         pj (µu(dij ) + ν)
                j=1                  j=1
                 n                         n                 n                      n
            =         (pj µu(dij )) +           (pj ν) = µ         pj u(dij ) + ν         pj
                j=1                       j=1                j=1                    j=1
            = µV (di ) + ν


and hence

 max Vv (di ) = max (µV (di ) + ν) = µ( max V (di )) + ν,
i=1,...,n             i=1,...,n                        i=1,...,n

which implies (15). This means that the optimal decision a i∗ does not change when the utility function
u is replaced by positive affine transformation v, i.e. it is invariant under positive affine transforma-
tions. This property implies that, in practice, origin and measurement unit of utility functions can be
chosen arbitrarily.

Now consider Connie’s decision problem represented by the decision matrix (1). Empirical studies
have shown that a decision maker is typically risk seeking for losses and risk averse for gains. Suppose
that Connie is a person exhibiting this type of risk attitude and that her (non-normalized) utility
function on the payoff domain [a, b] = [−28.75, 47.5] is given by
               √
        1 + 2 x
                        if x > 0
v(x) = 0                 if x = 0
        
        
          −2 − 4 |x| if x < 0

for all x ∈ [a, b]. The graph of the function v is given in Figure 6.


                                  Figure 6: Graphs of the utility functions v and u.


The transformation of Connie’s decision matrix (1) by the function v leads to the matrix
                                                                                     
                          0.00        0.00      0.00       0.00       0.00       0.00
                     −11.59          7.16      7.16       7.16       7.16       7.16 
                     −15.56          4.87      9.72       9.72       9.72       9.72 
V (D) = (v(dij )) = 
                     −18.61
                                                                                      .
                                   −7.66       8.28      11.68     11.68       11.68 
                                                                                      
                        −21.18 −13.14           6.48      10.54     13.33       13.33
                        −23.45 −16.70           3.65       9.25     12.36       14.78

Using the same probabilities as in section 4.3., i.e. p1 = p5 = p6 = 0.1, p2 = p4 = 0.2 and p3 = 0.3,
this permits to compute the vector of EU -values

(V (d1 ), . . . , V (dn )) = (0.00, 5.29, 6.22, 3.76, 1.97, −0.03)

and hence, according to the expected utility decision rule (14), Connie should bake 2 cakes.

Note that, of course, the same optimal decision is obtained if the normalized utility function u :
[a, b] −→ R given by u(x) = v(x)−v(a) is used instead of v (the graph if u is given Figure 6). This
                              v(b)−v(a)


                                                                   21
function u is indeed normalized because u(a) = 0 and u(b) = 1, and the same optimal decision is
obtained because, obviously, u is a positive affine transformation of v.

So far the decision maker was assumed to be a single person, but in general more than one person is
involved in a decision making process. When there are many decision makers, the problem is how to
incorporate everybody’s preferences and attitudes in a single decision rule. In group decision making
in a competitive environment, i.e. when several people are involved in the decision-making process
having conflicting interests, some members of the group may cooperate and form a coalition against
the other members of the group in order to improve their individual situation. This type of questions
is considered and analyzed in the literature under the name cooperative game theory.

Another important aspect of decision making is time. Indeed, it can happen that every alternative in
a particular decision problem is composed of a sequence of actions to be made at different moments
in time. After the optimal decision, i.e. the optimal sequence of actions, has been determined by the
decision maker and the first few actions have be implemented, some additional relevant knowledge
may become available and the previously implemented decisions may no longer be optimal under the
new circumstances. These various aspects of decision making are discussed in Decision Problems
and Decision Models.

5.5.    Expected Utility Theory

Every valuation function V is conceived as a function representing a preference relation on the DM’s
set of alternatives. Indeed, if the DM prefers an alternative a i to another one aj then this preference
should be reflected by the corresponding V -values but, on the other hand, if the value V (a i ) of an
alternative ai is at least as large as the value V (aj ) of an alternative aj then then DM should prefer ai
to aj , i.e.

V (ai ) ≥ V (aj )         ⇔       ai      aj                                                               (16)

Thereby, ai       aj means that, for the DM, the alternative ai is at least as “good” as aj , or, that he
(weakly) prefers ai to aj . The direction “⇒” of this equivalence shows that every valuation function
V implies a preference relation on the set A of all alternatives. For a deeper understanding of a given
V , it is important to know the properties of the preference relation implied by it. All decision theoretic
efforts serve the goal to gain deeper insight into the structure and properties of certain empirically
observed or normatively prescribed decision behaviors. To gain a deeper insight into the preferences
implied by the expected utility principle, this principle has to be considered in a much more general
framework than in the simple decision matrix setting with finite alternatives and states of nature.

Let, for this purpose, F = {x1 , . . . , xn } be any finite subset of any real domain X = [a, b] of possible
payoffs. Then a probability distribution over F is a mapping p : X −→ [0, 1] such that p(x) = 0 for
x ∈ X − F and n p(xj ) = 1. If B denotes the Borel σ-algebra restricted to the interval X, then,
                    j=1
for every probability distribution p, there is an associated probability measure P = P (p) on X given
by the mapping P : B −→ [0, 1] with P (S) = xj ∈F ∩S p(xj ).

So far, an alternative ai in a decision problem under uncertainty was characterized by the correspond-
ing row vector

di = (di1 , . . . , dij , . . . , dim )

of the payoff matrix D. By introducing probabilities p j of the states of nature, every alternative ai
can be considered as a probability distribution p = pi over the finite set F = {di1 , . . . , dij , . . . , dim }

                                                      22
defined by

             pj     if x = dij ,
pi (x) =                                                                                              (17)
             0      otherwise.

If Pi denotes the probability measure on X associated with this probability distribution, then, by going
one step further, one can identify the probability measure P i with the alternative ai . It is then natural
to say that, in general, an alternative is a probability measure on the set of payoffs X = [a, b], and that
the (theoretical) set of alternatives is given by the set M(X) of all probability measures on X.

Now, for generalizing the expected utility valuation function (14) to this general case, suppose that
the decision maker disposes of an utility function

u : [a, b] −→ [0, 1]

representing his or her attitude towards risk. If Pi denotes the probability measure on X associated
with an alternative ai , then equation (14) can be rewritten as a Lebesgue-Stieljes integral in the form

V (ai ) =         u(x) dPi .
             X

This equation can now easily be generalized to arbitrary probability measures on X. In general, the
expected utility (EU) valuation function V is given with the mapping V : M(X) −→ R, where for
any alternative P ∈ M(X), its value (“EU value”) is defined by

V (P ) =          u(x) dP.                                                                            (18)
            X



5.6.     Rationality Axioms

Now, the question is which preference properties the expected utility valuation function V implies.
First, as it holds for any valuation function, the preference relation defined by V fulfills the ordering
axiom (O). This axiom requires that a preference relation is a weak order, i.e. fulfills

•      completeness, i.e., for all P, Q ∈ M(X) holds P Q or Q P and
•      transitivity, i.e., for all P, Q, R ∈ M(X), if P Q, Q R then P R.

Completeness signifies that, for any pair of alternatives, the DM is able to compare them and to state
which of the two alternatives he (weakly) prefers. Transitivity means that if the DM prefers any
alternative P to any other one Q and at the same time prefers Q to an arbitrary third one R he or she,
then, prefers also P to R. That any preference relation defined by a valuation function V fulfills this
axiom follows from the fact that the reals as the range of V posses these properties.

A first important property the EU valuation function implies is that this function is linear in probabil-
ities, i.e., for all P, Q ∈ M(X) and for all λ ∈ [0, 1] holds

V (λP + (1 − λ)Q) = λV (P ) + (1 − λ)V (Q).                                                           (19)

Equation (19) means that the EU value for any convex combination of two probability distributions
results from the corresponding convex combination of the EU values of the two probability distribu-
tions involved. Thereby, the convex combination λP + (1 − λ)Q of two probability measures P and

                                                    23
Q denotes a certain compound probability measure, i.e., the probability measure where P “happens”
with probability λ and Q with probability (1 − λ).

As far as further properties of the preference relation = EU implied by the EU valuation function
(18) are concerned, it has been proved by several authors that, additionally to completeness and
transitivity, it satisfies the following two axioms. The first axiom, called the Archimedean axiom
(AR), requires that for all P, Q, R ∈ M(X) with P Q R there exist λ, µ in ]0, 1[ such that

λP + (1 − λ)R        Q    and Q        µP + (1 − µ)R.

Thereby, the relation     denotes the asymmetric part of   , i.e.   is defined by

P    Q    ⇔     (P       Q) and ¬ (Q     P ).

The third axiom, called the independence axiom (I), requires that for all P, Q, R ∈ M(X) and for all
λ ∈]0, 1],

P    Q ⇒ λP + (1 − λ)R          λQ + (1 − λ)R.

The Archimedean axiom (AR) means that, for any three alternatives P, Q, R ∈ M(X) where P
is best and R is least preferred, it is possible to find a parameter λ to get the probability measure
compounded of P and R better than Q, and to find a parameter µ to get this measure worse than
Q. I.e., depending on the weights for the best and for the least preferred measure respectively, the
probability measure compounded of P and R is more or less preferred than the “medium” measure Q.
The independence axiom (I) means that the preference between any two probability measures should
not change if both probability measures are compounded with another arbitrary one. After a short
reflection all the axioms implied by the EU valuation function appear to be very plausible.

It seems, therefore, very natural to assume that any rational DM accepts that the preference relation
on his or her set of alternatives possesses the properties these axioms require. I.e., it seems obvious
to consider these three axioms as basic rationality axioms which should be fulfilled by any rational
decision maker. Then the question arises which valuation functions V represent, in the sense of (16),
a preference ordering possessing these properties. This is what normative decision research is all
about: Given some axioms, considered as basic rationality axioms which should be fulfilled by any
rational decision maker, certain principles for decision making are developed to which, then, a rational
decision maker has to adhere if he or she wants to reach the “best” decision.

Assume now that, on M(X), there is given a preference relation satisfying the three axioms (O),
(AR) and (I). Then it can be shown that these axioms are sufficient to guarantee the existence of a
linear valuation function representing the preference relation , i.e. it can be shown that these axioms
imply the existence of a function V : M(X) −→ R fulfilling

P    Q    ⇔     V (P ) ≥ V (Q).

and satisfying the linearity property (19). This function V is unique up to positive affine transforma-
tions, i.e. another function V ∗ : M(X) −→ R represents if and only if there exist real constants
a > 0 and b such that V ∗ (P ) = aV (P ) + b for all P ∈ M(X).

Second, note that any linear valuation function V defined on M(X) generates a (non-normalized)
utility function u : [a, b] −→ R defined by u(x) := V (δx ) where δx denotes the one-point Dirac
probability measure at x ∈ [a, b]. If, now, Pi denotes the probability measure associated with the


                                                  24
probability distribution p over the finite set F = {d i1 , . . . , dij , . . . , dim } defined by equation (17),
then it follows from the linearity property (19) of V that
            m                      m
V (Pi ) =         pj V (δdij ) =         pj u(dij ).
            j=1                    j=1

Herewith, one of the major results of normative decision theory is achieved: In decision situations
under uncertainty, any DM should follow the EU principle, i.e., he or she has to choose the alterna-
tive maximizing the EU valuation function (14). This classical normative position is mathematically
derived from the assumption that any rational DM, for decision situations under uncertainty, accepts
the axioms (O), (AR), and (I) as basic rationality axioms.

This normative position, first, has been mathematically derived in the second edition (of 1947) of the
famous book “Theory of Games and Economic Behavior” first published by J. v. Neumann and O.
Morgenstern in 1944. Thereby, they started from “rationality” axioms similar to the three axioms (O),
(AR), and (I). Later on, several other authors have developed comparable but more elegant axiomatic
foundations of the EU principle. It should be emphasized that these foundations were mainly coined
to situations where, on the set of states of nature, objective probabilities are given.

In Foundations of Target-Based Decision Theory, Bordley reconsiders the common utility function
interpretation of the v. Neumann & Morgenstern decision theoretic axioms and suggests a so-called
target-based interpretation of the axioms in which an individual is interested in meeting certain needs.

The first to set forth a axiomatic foundation of the EU principle for the case of uncertainty where
no objective probabilities are available was J. L. Savage. In his famous book “Foundations of Statis-
tics” first published in 1954, he started off from a certain general set of alternatives endowed with a
preference relation . He assumed that this preference relation possesses certain plausible properties
similar to those mentioned above regarding these as basic rationality axioms every DM should be
willing to accept. Then he shows that under this condition there exist a subjective probability dis-
tribution p and a utility function u such that the valuation function (14) represents in the sense of
(16).

The main difference between Savage’s approach and the other approaches mentioned above is that,
as Savage does not assume objective probabilities on the states of nature, his rationality axioms are
formulated without any reference to probabilities. Therefore, his axiomatic theory of decision making
under uncertainty is called subjective expected utility (SEU) theory. In this theory, the axiom corre-
sponding to the independence axiom (I) is the famous sure thing principle saying that the preference
between two alternatives should not depend on states where these alternatives lead to the same results.

For any practical application of the EU principle in decision situations under uncertainty, first, the
DM’s probability distribution on the set of states of nature as well as his utility function have to be
provided. Both ingredients, in general, are not readily available in real decision situations and have to
be determined in a suitable way. By many practitioners, this often is seen as the major drawback of
the EU principle. In the decades since the fifties of last century, however, the EU principle has come
under attack mainly because it became evident that the so-called “rationality” axioms underlying this
principle are less convincing as their advocates thought.

5.7.   Empirical Results

In 1952, the French Economist Maurice Allais invited many of the leading decision theorists including
Savage to a colloquium to Paris. There he confronted them with the two following pairs of lotteries:

                                                       25
the first pair consists of lottery L1 bringing about $ 500,000 with certainty, i.e. L1 = L(500000; 1),
and lottery L2 leading to $ 2,500,000 with probability 0.1, to $ 500,000 with probability 0.89, and
to $ 0 with probability 0.01, i.e. L2 = L((2500000; 0.1), (500000; 0.89), (0; 0.01)). The second pair
consists of lottery L3 leading to $ 500,000 with probability 0.11, and to $ 0 with probability 0.89, i.e.
L3 = L((500000; 0.11), (0; 0.89)), and of lottery L4 leading to $ 2,500,000 with probability 0.1 and
to $ 0 with probability 0.9, i.e. L4 = L((2500000; 0.1), (0; 0.9)). People had to decide, first, which
of the lotteries L1 and L2 they prefer, and then which of the lotteries L3 and L4 . Allais’s lottery pairs
are represented in Figure 7.


                                Figure 7: Allais’s decision problems.


Up to these days, hundreds of individuals in and outside classrooms have been confronted with these
two pairs of lotteries. Everybody is invited to reflect on his preferences. A significant majority
of individuals has L1       L2 and, at the same time, L4       L3 , thereby violating the independence
axiom (I). Incidentally, even Savage himself demonstrated these preferences when first confronted
with them by Allais in Paris in 1952. This behavior later has been called the Allais Paradox. That
these preferences, in fact, violate the independence axiom (I) can easily be seen when the four lotteries
are presented as in Table 2.


       Table 2: Matrix Representation of Allais’ Decision Problem (payoffs in 100’000$)


Imagine, for each lottery Li an urn with hundred tickets where on each ticket a prize is noted accord-
ing to the schedule shown in Table 2. Playing any of the four lotteries then simply means drawing
randomly a ticket from the corresponding urn. The number on that ticket, then, indicates the state of
nature which has been occurred. Table 2 makes clear that in both decision problems a choice has to
be made between lotteries that can be perceived as compound lotteries.

In the first decision problem, a choice has to be made between the compound lotteries L 1 = λP1 +(1−
λ)Q and L2 = λP2 +(1−λ)Q where the lotteries P1 = L(500000; 1) and P2 = ((0; 11 ), (2500000; 11 ))
                                                                                    1              10

are compounded with the sure payoff 500000, i.e. with the degenerate lottery Q = L(500000; 1), at
a rate of λ = 100 . In the second decision problem, a choice has to be made between the com-
                  11

pound lotteries L3 = λP1 + (1 − λ)R and L4 = λP2 + (1 − λ)R where, once more, the lotteries
P1 = L(500000; 1) and P2 = ((0; 11 ), (2500000; 10 )) are compounded with a sure payoff but this
                                      1
                                                    11
time with the sure payoff 0, i.e. with the degenerate lottery R = L(0; 1), and that at the same rate of
λ = 100 .
      11



This shows that in both decision problems, the choice, basically, has to be made between the same
pair of lotteries, P1 and P2 . In the first problem, these lotteries are compounded with one and the same
sure payoff 5, and in the second one with one and the same sure payoff 0. Therefore, the preferences
L1      L2 and L4        L3 violate the independence axiom (I) because this axiom requires that the
preference between two lotteries has to be independent of any other lottery they are compounded
with.


                 Table 3: Matrix Representation of Ellsberg’s Decision Problem


One particularity of Allais’ decision situation is that objective probabilities are given. The question
arises if violations of the independence requirement also happen in cases where no probabilities are

                                                   26
readily available. The first to contrive a decision situation where a majority of people violates the
independence axiom of SEU theory, i.e., the sure thing principle, was Ellsberg in 1961. His three color
problem already has been introduced in subsection 4.1 as an example where only partial probability
information is available. Given Ellsberg’s urn containing 90 balls, 30 of which are known to be red
and the remaining ones black or yellow, exactly one ball is drawn randomly. Again, two pairs of
alternatives are presented. The first pair consists of alternatives a 1 : “bet on red” and a2 : “bet on
black”. The second pair consists of alternatives a3 : “bet on red or yellow” and a4 : “bet on black or
yellow”. People have to choose, first, between a1 and a2 and then between a3 and a4 .

Also in this case, a significant majority of individuals shows preferences violating the independence
requirement. People usually prefer a1 to a2 and, at the same time, a4 to a3 . That these preferences
violate the sure thing principle can easily be seen when the four lotteries are presented as in Table
3. The alternatives a1 and a2 , as well as the alternatives a3 and a4 lead to the same payoff if state z3
occurs. However, restricted to the states z1 and z2 , the alternatives a1 and a3 , and the alternatives a3
and a4 are identical. Therefore, the preferences a1 a2 and a4 a3 violate the sure thing principle
because this axiom requires that the preference between two alternatives has to be independent of
states where these alternatives lead to same payoffs. Furthermore, it can immediately be shown that
there is no probability distribution compatible with those observed preferences.

Since Allais’ and Ellsberg’s studies, a huge amount of experimental studies have been conducted
                                                                                        ı
mainly by psychologists and experimental economists. They investigated not only if (na¨ve) decision-
makers adhere to the rationality axioms of the expected utility paradigm but, going much further,
empirically investigated how these really make their decisions. Thereby, a lot of important empirical
findings have been made.

Generally speaking, it has been found that DMs, be they laypersons or experts, are prone to a lot of
cognitive biases when making evaluations or judgments or taking decisions. Thereby, bias usually is
defined as a systematic and predictable error as opposed to unsystematic or random errors. One goal
of decision analysis is to help DMs to avoid biases. A certain knowledge of possible biases serves
this goal. It should be noted however that there is an ongoing discussion if certain observed behaviors
can be labelled as “biased” at all. Take, e.g., the observed violations of the independence axioms in
Allais’ and Ellsberg’s “paradoxes”.

The presentations of Allais’ decision situation in Table 2 and of Ellsberg’s decision situation in Table 3
clearly confirm the position that some independence axiom should be part of any definition of rational
behavior under uncertainty. If, in Allais’ problem, one of the tickets numbered from 12 through 100 is
drawn from the urn, obviously it will not matter, in either decision problem, which gamble is chosen.
Therefore, attention has to be given to the tickets numbered from 1 through 11 but for these cases the
decision problems 1 and 2 are identical. As a consequence, a DM preferring L 1 to L2 must also prefer
L3 to L4 as the independence axiom requires. On the basis of that reasoning, Savage classified his
spontaneously uttered preferences in the Allais situation as “error”. If, in Ellsberg’s problem, state z 3
occurs, it will not matter, in either decision problem, which bet is chosen. However, restricted to the
states z1 and z2 , the decision problems 1 and 2 are identical. Therefore, every DM preferring a 1 to a2
must also prefer a3 to a4 .

Allais himself never accepted this argumentation. He emphasized that Savage’s representation of
his decision situation as in Table 2 is a reformulation of the original problem represented in Figure
7 which, in fact, “changes the nature of the problem completely”. Since then, empirical evidence
has been gathered indicating that, indeed, it cannot be proceeded from the assumption that DMs
generally are willing to accept this argumentation. In any case, the high rate of violation of the


                                                   27
independence axioms causes doubt about whether people are sufficiently consistent in their beliefs to
allow assessment of subjective probabilities.

Arguing that Savage’s representation of his decision situation is a reformulation changing the nature
of the problem, Allais was maybe the first to point to a problem now known as framing effect. This
effect is the finding that different descriptions of formally identical problems can result in different
preferences. This, above all, concerns the structural stage of the decision making process: It has
been demonstrated by many experiments that the DM’s preferences depend on his or her perception
of the alternatives and their outcomes, on the wording and the presentation of the problem. More
specifically, describing outcomes as gains or losses relative to a reference point leads to different risk
attitudes (see: Framing effects in theory and in practice).

                                                                      ı
Another important finding of empirical decision research is that na¨ve DMs usually do not tend to
represent realistic non-lottery decision problems in lottery form. In most non-lottery decision prob-
lems, DMs even do not seem to be actively interested in probability information. They rather try to
diffuse the risks involved by employing risk diffusing operators, i.e., actions performed additionally
to a specific alternative with the intention of decreasing a risk of that alternative. “Risk diffusing
behavior” is in more detail treated in Risk diffusing behavior.

As mentioned above, one of the essential problems when applying the EU principle is that the DM’s
probability distribution on the set of states of nature as well as his utility function are not readily
available in real decision situations and have to be determined in a suitable way. In subsection 5.3.,
one method of determining a utility function has been presented. It has been pointed out that there are
different possibilities to determine a utility function. Unfortunately, empirical evidence has shown
that the determination of utility functions is prone to systematic biases. Depending on the method
used, different utility functions result. Therefore, in practical applications, it is always important to
check utility functions for consistency in sense indicated in subsection 5.3.

Another important finding of experimental decision research is that the probabilistic competencies of
DMs, in general, leave much to be desired. Systematic biases of different kinds have been observed.
The most important contribution in this field is the book published by Kahneman, Slovic, and Tversky
in 1982. They diagnose a lot of deficiencies when humans are making probability judgments. DMs,
to reach such judgments, usually use simple judgmental heuristics, i.e., strategies relying on natural
assessments that are carried out routinely to produce an estimation or prediction.

An important example of such a heuristic is the so-called representativeness heuristic where subjec-
tive probability judgments are made on the basis of an assessment of the degree of correspondence
between an outcome and a model. This heuristic reliably leads to what is called conjunction fallacy
in the literature: people, under certain conditions, rate the probability of a conjunction A ∩ B higher
than one of its constituents, A and B. Another important example of a heuristic is the so-called avail-
ability heuristic where people estimate the probability of a specific event according to the frequency
of instances stored in mind Risk diffusing behavior.

Such behavior is qualified as “biased” because there is a theory to which they are contradictory, i.e.,
probability theory. However, even in this case where the theory is well established since genera-
tions, there is an ongoing discussion if certain behaviors can really be labelled as “biased” at all.
The psychologist G. Gigerenzer is the most prominent defenders of the position that some of these
strategies DMs use, even when they contradict probability theory, are not erroneous at all but simply
“reasonable” in a different sense.



                                                   28
Additionally, it has to be stressed that, as in the case of the determination of a utility function, different
elicitation methods can be used for determining probabilities. Unfortunately, also the probabilities
estimated may depend on the elicitation method used. Furthermore, it should be mentioned that DMs
tend to overweight small and to underweight high probabilities. In the light of this tendency and the
above biases it is the more important that, in practical applications, the estimated probabilities are
checked for consistency, i.e., that it is checked that they satisfy the axioms of probability.

5.8.   Extensions of Expected Utility

Finally, based on the empirical findings discussed above, highly interesting descriptive psychological
theories of decision behavior have been developed in Behavioral Decision Theory. The most remark-
able descriptive decision theory, today, is the so-called prospect theory proposed by Kahneman and
Tversky in 1979. Mathematically, it is a generalization of EU theory which is characterized by the
following three features.

In prospect theory it is assumed that the DM, in the structural stage of the decision problem, first edits
his or her alternatives. The goal of this editing phase is to organize and reformulate the alternatives so
as to simplify subsequent evaluation and choice. Thereby, e.g., dominated alternatives are eliminated.
Then a value function is determined. The main difference between this function and the EU utility
function is that it evaluates outcomes relative to a certain reference point. The possibility to account
for reference points is one of the most important features of prospect theory. Typically, DMs choose
their actual wealth as reference point and regard the possible payoffs as gains or losses relative to that
reference point. The value function, typically, is concave for gains and convex for losses. See the
function v(·) represented in Figure 6 for an example of such a value function.

Furthermore, the fact mentioned above that DMs tend to weigh probabilities is taken into account
by a probability weighting function π = π(p(x)). As mentioned on page 280 of the book published
by Kahneman and Tversky in 1979, this function transforms the probabilities such that the resulting
decision weights “measure the impact of events on the desirability of prospects and not merely the
perceived likelihood of these two events”. It indicates the weight the DM gives to the different prob-
abilities and is, typically, monotonously increasing with discontinuities at 0 and 1, has π(p) > p for
small probabilities, and some more specific properties. A typical example of a probability weighting
function is shown in Figure 8.


                       Figure 8: Prospect theory probability weighting function.


Finally, the valuation function used is basically of the form
          m
V (p) =         u(dj )π(p(xj )).                                                                         (20)
          j=1

This valuation function can, as the EU valuation function, be derived from some axioms on the DM’s
preferences. These include besides completeness, transitivity, and an Archimedean axiom an inde-
pendence axiom which is weaker than the independence axiom (I) criticized above.

The reader is referred to Expected Utility Theory and Alternative Approaches for a more detailed
presentation of prospect theory and other important alternative approaches to expected utility theory
like rank dependent expected utility theories.


                                                     29
6.     The Risk-Value Approach

6.1.        General Characterization

A different approach to decision making under uncertainty is taken in the risk-value approach. Unlike
the approaches considered in sections 4 and 5, in this type of approach to decision making under
uncertainty, it is assumed that the preference for an alternative is explicitly and exclusively determined
by its “riskiness” and its “value” (or “worth”). Thereby, the value of an alternative only means the
location of the corresponding payoff distribution without taking into account any risk considerations.
It should not be confounded with the “overall value” of an alternative resulting from any valuation
function representing the DM’s preferences. Risk refers to the riskiness of an alternative and is a
matter of perception of the variability of the corresponding payoff distribution.

Within risk-value models, value as well as risk are treated as primitives, and therefore it is assumed
that people can provide judgements about value and risk directly through introspection or some other
means. Then the decision problem is viewed as a problem of choosing among possible risk-value
combinations where riskiness and value of each alternative are numerically represented by a risk
and a value measure, respectively. These risk and value measures, separately, are, in general, real-
valued functions of certain pointwise transformations of the payoffs characterizing an alternative and
of another pointwise transformations of the corresponding probabilities. The final valuation is a
function defined on the set of all possible risk-value combinations.

Hence, risk-value models are generally characterized by the following five preference assumptions:


•      there is a value ordering W on the decision maker’s set of alternatives A, numerically repre-
       sentable by a real-valued function W , i.e.
       ai    W   aj ⇔ W (ai ) ≥ W (aj ),
       ai W aj meaning that alternative ai is at least as valuable as alternative aj ;
•      there is a risk ordering R on A, numerically representable by a real-valued function R, i.e.
       ai    R   aj ⇔ R(ai ) ≥ R(aj ),
       ai R aj meaning that alternative ai is at least as risky as alternative aj ;
•      the quality of each alternative a is completely described by its value W (a) and its risk R(a);
•      value W is “good”, i.e. other things being equal, more W is preferred to less;
•      risk R is “bad”, i.e. other things being equal, less R is preferred to more.

It is characteristic for risk-value approaches that every decision problem under uncertainty is trans-
formed into a special case of a multiple-criteria decision problem under certainty. The n×m-decision
matrix (3) of the problem under uncertainty is, by means of the two criteria c 1 (·) = W (·) and
c2 (·) = R(·), reduced to the corresponding n × 2-decision matrix of a two-criteria decision problem
under certainty. Then, the general concepts presented in sections 3.3 and 3.4 including the valuation
function concept are applied to the rows of this latter matrix.

6.2.        Risk-value Dominance

On the basis of a risk-value model, the set of all alternatives of any given decision problem can, as for
any two-criteria decision problem under certainty, graphically be represented in a two-dimensional

                                                     30
cartesian product where the first dimension indicates the “risk” of these alternatives and the second
one their “value”. Every alternative a is represented by the corresponding point (R(a), W (a)). See
Figure 9 for illustration.

          Figure 9: Risk-value representation of a decision problem under uncertainty.

If more “value” is better than less and less “risk” better than more then, of course, in a first step
of the decision making process, obviously, all dominated alternatives can be neglected. Any “best”
alternative has to be searched for in the efficient boundary of the set of all non-dominated alternatives.
In Figure 9, this set is indicated by the bold line in the “north-west”-part of the set of all alternatives.
Note that in this case where the second criterion has to be minimized the alternatives dominated by a
given alternative a are in the “south-east” quadrant of a.

The best known example of a risk-value model is the classical (µ, σ)-approach also called mean-
variance approach in the literature. Thereby, the value of an alternative a i is measured by the mathe-
matical expectation or the mean µ of its payoffs dij , i.e. by
                  n
µ = E(ai ) =           dij pj ,                                                                        (21)
                 j=1

and the riskiness of the alternative by their standard deviation σ, i.e. by
                              n
σ = StdDev(ai ) =                  (dij − µ)2 pj .                                                     (22)
                             j=1

Note that in the name of that special case of a risk-value model, traditionally, the order of “risk” and
“value” is changed.

Assume, e.g., that in decision problem (1) of how many cakes to bake Connie’s preferences are
exclusively determined by the riskiness and the value of the different alternatives. Assume further that,
for the different states of nature, probabilities are given and that “risk” and “value” of the different
alternatives can be measured by standard deviation and mean of their payoffs. Using, as in section 4.3.,
for the six different states of nature, the probabilities p 1 = p5 = p6 = 0.1, p2 = p4 = 0.2 and p3 = 0.3,
then, for each possible alternative i = 1, . . . , 6, the (µ, σ)-values attached to the row vectors d i are
given in the first two columns of Table 4.

                        Table 4: Mean-risk values of Connie’s decision problem.

All the six (µ, σ)-pairs corresponding to alternatives 1 through 6 of Connie’s decision problem are
graphically represented in Figure 10. This figure shows that, obviously, alternatives 5 and 6 are
dominated, e.g., by the alternative 4. Therefore, the efficient boundary is given with the alternatives 1
through 4.

                   Figure 10 (µ, σ)-representation of Connie’s decision problem.

As it is generally the case, also in this example, dominance does not lead to a satisfying solution of
the decision problem. Which of the non-dominated alternatives on the efficient boundary should be
taken as the final decision? Basically, there are two different approaches to solve that problem and to
find a “best” solution to the decision problem.

                                                     31
6.3.   Compensatory and Lexicographic Approaches

According to how the problem to find a “best” solution is treated, risk-value models can be classified
into compensatory and lexicographic approaches. In compensatory risk-value models, it is assumed
that high risks can be traded off by high values and, accordingly, preference comparisons between
alternatives are made by means of a function V reflecting the trade-off between value and riskiness.
In other words, in compensatory risk-value models it is assumed that there is a valuation function V
such that

ai     aj ⇔ V (W (ai ), R(ai )) ≥ V (W (aj ), R(aj ))

where ai     aj denotes the decision maker’s preference relation on the set of alternatives, i.e., it is
assumed that the decision maker’s preferences can be numerically represented by a valuation function
V defined on the set of all possible risk-value combinations. Thereby, according to the preferential
assumptions of risk-value models listed above, V is assumed to be increasing in W , decreasing in R,
continuous, and quasi-convex.

A typical trade-off map between value and risk characterized by three indifference curves is shown
in Figure 11. Thereby, each indifference curve is the locus of all risk-value combinations between
which the decision maker is indifferent. Quasi-convexity of V means by definition that all indifference
curves of V are curved upwards. Upward curvature of the indifference curves implies that higher risks
have to be traded off by higher values, i.e., that the decision maker is risk averse. The curvature of
the indifference curves in a risk-value map is an indicator of the decision maker’s degree of risk
averseness. The steeper the indifference curves increase with increasing risk the more risk averse the
decision maker is.


     Figure 11: Indifference curves of a compensatory risk-value model with risk averseness.


Assume now, e.g., that Connie is a risk averse decision maker and that her indifference curves are as
indicated in Figure 12. Then Connie’s optimal decision according to the (µ, σ)-approach obviously
is to choose alternative 3, i.e. to make 2 cakes. Remember that Connie’s optimal solution according
to the Bayes criterion (see section 4.3.) was alternative 4. Figure 12 shows that, in fact, this decision
is the alternative with the highest mean payoff. However, the mean payoff of that alternative is not
sufficiently higher than the mean payoff of the alternative to bake just two cakes to trade-off the
considerably higher risk of that alternative.


          Figure 12: Linear compensatory (µ, σ)-approach to Connie’s decision problem.


The valuation function V corresponding to the indifference curves in Figure 12 is of the linear form

V (W (a), R(a)) = W (a) + λR(a),                                                                    (23)

with a parameter λ ∈ R, λ < 0. In general, the amount |λ| > 0 indicates how much the value
of an alternative must increase to compensate an additional unit of risk in order to keep the same
level of value. In the special case of the (µ, σ)-approach, it specifies how much the expectation of an
alternative must increase to compensate an additional unit of standard deviation. The parameter λ is
an indicator for the decision maker’s degree of risk averseness. The greater |λ| the more risk averse
the decision maker is.

                                                    32
In practice, it is not necessary to specify a precise value for the parameter λ. Generally, to come to
a final decision, it is sufficient to check if it can reasonably be assumed that λ belongs to a certain
interval. For Connie’s decision problem, e.g., it can easily be calculated that for λ = λ 1 = −0.0682
the alternatives 3 and 4 are equally good because they lie on the same indifference curve. For the
same reason, for λ = λ2 = −0.889 the alternatives 2 and 3, and for λ = λ3 = −1.74 the alternatives
1 and 2 are equally good. I.e., for each λ ∈ [−0.0682; 0] alternative 4 is optimal, for each λ ∈
[−0.889; −0.0682] alternative 3, for each λ ∈ [−1.74; −0.889] alternative 2, and for each λ < −1.74
alternative 1. I.e., to each of the four non-dominated alternatives corresponds an interval of λ-values
in the sense that every λ-value in that interval renders this alternative optimal.

Now, to come to a final decision, Connie has to clarify in which of these intervals her λ-value falls.
As these intervals represent four different categories of degrees of risk-averseness she, in other words,
has to clarify to which category of risk-averse decision maker she belongs. If she is hardly risk-averse,
i.e. if λ > −0.0682, she should choose alternative 4. If she is very risk-averse, i.e. if λ < −1.74, she
should chose alternative 1. If both of these cases appear to be too extreme Connie has to check if a
λ-value in the interval [−0.889; −0.0682], as in Figure 12, reasonably characterizes her risk attitude.
If this is the case then Connie should choose alternative 3. Whereas, for λ ∈ [−1.74; −0.889] she
should choose alternative 2.

It should be noted that compensatory risk-value models are, as the EU paradigm, sufficiently flexible
to integrate other risk attitudes than risk averseness. In Figure 13, the cases of a risk seeking and a
risk neutral decision maker are illustrated, respectively, by four corresponding indifference curves. A
risk seeker is willing to trade off value to get a higher risk. A risk neutral decision maker just looks at
the mean of the payoffs of an alternative, i.e., a risk neutral decision maker uses the Bayes criterion
introduced in section 4.3. which, therefore, can be regarded as a special case of a risk-value model. A
risk neutral decision maker prefers an alternative with a higher value to every other one with a lower
value irrespective of its risk. In the linear case of a valuation function, a risk seeking decision maker
is characterized by a parameter λ > 0, and a risk neutral one by λ = 0.

 Figure 13: Indifference curves of a compensatory risk-value model with risk seeking or risk
                                         neutrality.

In lexicographic risk-value models, high risks or low values cannot be compensated by high values
or low risks, respectively. It is assumed that there is an ordering on the two criteria c 1 (·) = W (·)
and c2 (·) = −R(·), and either c1 (·) is assumed to be more important than c2 (·) or vice versa. An
alternative ai is preferred to another alternative aj if and only if it is better for the more important
criterion or, if both alternatives are equally good with respect to that criterion, if it is better with
respect to the second one, formally
ai   aj ⇔ ck (ai ) > ck (aj ) ∨ [(ck (ai ) = ck (aj )) ∧ (c3−k (ai ) > c3−k (aj ))]                   (24)
with k = 1 or k = 2.

If, e.g., Connie ranks value higher than risk then Connie’s optimal decision according to the lexi-
cographic (µ, σ)-approach obviously is to choose alternative 4. Remember that this is the optimal
solution according to the Bayes criterion (see section 4.3). If she ranks risk higher than value then her
optimal decision is to make zero cakes. Thereby, of course, it is assumed that Connie is a risk-averse
decision maker. If she should be a risk seeker then her optimal decision is to make five cakes.

The lexicographic risk-value approach can be criticized because of its unconditional maximization of
the first criterion. Therefore, in the literature, modifications of this approach have been proposed. The

                                                       33
basic idea of these approaches is that the importance of the first criterion is weakened by introducing
an aspiration level. An alternative ai is preferred over aj if aj does not meet that aspiration level and
if, in addition, ai is better than aj with respect to the more important criterion. If both, a i and aj , are
equally good with respect to the more important criterion or meet the aspiration level, a i is preferred
over aj if ai is better than aj with respect to the second criterion. I.e., in such a weak lexicographic
risk-value approach, an alternative may be preferred to another one although it is worse with respect
to the more important criterion as long as it reaches the aspiration level.

6.4.   Alternative Risk-Value Models

In this section, only the classical (µ, σ)-approach where value is quantified by the mean of the payoffs
and risk by their standard deviation has been explicitly treated. Further compensatory or lexicographic
risk-value approaches, including approaches tailored to the case of partial probability information
treated in section 4.4., result when value and risk are quantified by other measures. As far as value
is concerned only three measures are important, namely mean, median, and mode of the payoff dis-
tribution of an alternative. As far as risk is concerned, unfortunately, there is little consensus on its
definition and on how to measure it. There is a large list of possible candidates of risk measures. In
Measurement of Risk, a detailed review is given on more naive risk measures as well as on recently
developed economic or psychological theories of perceived risk which rely on suitable axioms.

In empirical studies, typically, two dimensions which appear to determine perceived risk have been
identified: amount of potential loss and probability of occurrence of loss. The risk of an alternative
increases if the probability of loss increases or if the amount of potential loss increases. Thereby,
losses are defined with reference to a certain target payoff. This target payoff may be the zero payoff,
status quo, a certain aspiration level, as well as the best result attainable in a certain situation. A
payoff is regarded as a loss if and only if it falls below the target payoff.

These empirical studies have shown that there is no risk model which is clearly superior to all oth-
ers. Nevertheless, there are some results which appear to be pretty stable across different empirical
studies. Traditionally, the risk of an alternative has primarily been associated with the dispersion of
the corresponding monetary payoffs. Then, it is reasonable to measure the riskiness of an alternative
by its variance σ 2 or its standard deviation σ. However, one important result is that perceived risk
generally is not represented by variance. There are examples of different lotteries with constant vari-
ance, where people consistently judge some of them riskier than others or are even able to order them
according to their riskiness.

In colloquial language but also in economics and other fields of research, risk is understood as the
possibility of injury or loss attached to a given alternative or action or simply as the chance of some-
thing bad happening. In this vein, risk is associated with an payoff that is worse than some specific
target payoff and its probability. Within the risk measures tailored to this notion of risk are the lower
semivariance

LSV (ai ) =              (dij − µ)2 pj .                                                                (25)
              j:dij <µ

or the probability of loss

PL(ai ) =              pj .                                                                             (26)
            j:dij <0

Assume, e.g., that in decision problem (1), once more, Connie’s preferences are exclusively deter-
mined by the riskiness and the value of the different alternatives and that, for the different states of

                                                     34
nature, probabilities are given. Now, however, assume that, according to Connie’s perception, risk is
measured by lower semivariance and probability of loss, respectively. Using, as in section 4.3., for
the six different states of nature, the probabilities p1 = p5 = p6 = 0.1, p2 = p4 = 0.2 and p3 = 0.3,
then, for each possible alternative i = 1, . . . , 6, the corresponding LSV - and PL-values are given in
the last two columns of Table 4.

All the six (µ, LSV )-pairs (represented by dots) as well as all the six (µ, PL)-pairs (represented
by crosses) corresponding to alternatives 1 through 6 of Connie’s decision problem are graphically
represented in Figure 14. This figure shows that, as in the (µ, σ)-approach, also in the (µ, LSV )-
approach alternatives 5 and 6 are dominated by any of the alternatives 3 or 4. Therefore, the efficient
boundary, once more, is given with the alternatives 1 through 4. In the case of the (µ, PL)-approach,
alternative 2 is dominated by alterative 3 and alternatives 5 and 6 by alternative 4. In that case, the
efficient boundary is given with the alternatives 1, 3, and 4.


        Figure 14: (µ, LSV )- and (µ, PL)-representations of Connie’s decision problem.


Note that Figure 14 shows that the (µ, LSV )-approach is, in that case, very similar to the (µ, σ)-
approach. This comes from the fact that, in this example, the payoff distributions of the different
alternatives are almost symmetric around the mean. In general, when the payoff distributions are
asymmetric around the mean the two approaches can be quite different. This may make an important
difference in cases where the decision maker is, above all, interested in potential losses and not just
in variation about the mean. Note also that the risk measure PL discriminates less between the
alternatives than variance or semivariance do. This is due to the fact that PL takes into account
only the probability of potential losses and not their amount.

As it is generally the case with risk-value approaches, also in these examples, dominance does not lead
to a satisfying solution of the decision problem. Analogous to the (µ, σ)-model, a compensatory or a
lexicographic approach can be used to come to a final choice. For a compensatory approach, assume,
e.g., that Connie is risk averse with a valuation function of the linear type (23) with negative λ. As in
the case of the (µ, σ)-approach considered above, for both approaches, it is not necessary to specify
a precise value for the parameter λ because to each of the non-dominated alternatives of both ap-
proaches corresponds an interval of λ-values in the sense that every λ-value in that interval renders this
alternative optimal. To come to a final decision, Connie, according to her degree of risk-averseness,
has to clarify in which of these intervals her λ-value falls. If, e.g., λ ∈ [−0.0857; −0.00553] for the
(µ, LSV )-approach or if λ ∈ [−129; −1.75] for the (µ, PL)-approach then Connie, in both cases,
should choose alternative 3 and bake two cakes.

As this example shows, if a risk-value model is chosen to find the optimal solution of a decision
problem under risk, this solution depends on the risk measure and the valuation function used. In
practice, it is always more or less uncertain which risk measure and which valuation function should
be chosen and it is, therefore, recommended to try several approaches. Then, the decision maker is
on the safe side if the optimal solution of her decision problem is invariant against the choice of risk
measure and valuation function.




                                                   35
7.     Graphical Representation of Decision Problems

7.1.    Decision Trees

In practice, it is quite common that a decision problem is characterized by the fact that a sequence of
decisions has to be made. Imagine, e.g., that Jill is considering to open a new laundry service in town.
Assume that she can buy one or two laundry machines from the beginning, each machine costing
$5000. Or she can buy only one machine now and postpone her decision to buy a second one to six
months from now, for the higher price of $10000. Assume further that, during the first six months,
the demand can be high or low, being low with probability 0.6 and high with probability 0.4. Jill’s
planning horizon is two years, and in the last 18 months the demand may also be either high of low.
The probability of a high demand following a high demand is 0.8, whereas the probability of a low
demand following a low demand is 0.6. During the first 6 months, the revenues from the business are
estimated to be $3000 if the demand is high and she has bought two machines, and $1500 otherwise.
During the last 18 months, the revenues are estimated to be $9000 if she has a total of two machines
and the demand is high, whereas it is $4500 if she has a total of two machines and the demand is low.
If during the last 18 months she has a total of only one machine, then the revenues for that period will
also be $4500, regardless of the level of the demand. Should Jill buy two machines now or should she
just by one? In the latter case, after 6 months, should she buy a second machine or should she stick
with her unique machine? In this problem, a first decision has to be made, then, after some chance
event happened, a second decision has to be taken.

In general, decision problems where the problem consists of taking a sequence of decisions are called
multistage decision problems. A very useful tool to solve such multistage decision making problems
are decision trees. A decision tree is a graphical representation of a multistage decision problem by a
tree containing two kinds of nodes: decision nodes and chance nodes. The so-called decision nodes,
represented by rectangles, correspond to stages where a decision has to be made, and the so-called
chance nodes, represented by circles, correspond to stages where random events happen. An edge
going from node A to node B means that A preceeds B in time. Each path from the root of the tree
to a leaf is called a scenario and each sequence of decisions is called a strategy. Strategies now play
the role of the various alternatives. Once a decision tree has been constructed, a simple and efficient
evaluation algorithm can be used to find the optimal strategy in the sense of the rule of Bayes, i.e. to
find the strategy that leads to the maximal expected payoff.

For a decision tree representation of Jill’s problem, let M1 denote the number of machines Jill buys
at the beginning, so that the domain of M1 is {one, two}, and let M2 denote the total number of
machines she has during the last 18 months, so that the domain of M 2 is {one, two}. Also, let D1
denote the demand level during the first 6 months, whereas D 2 denotes the demand level during the
last 18 months. Of course, the domain of both D1 and D2 is {h, l}, where h stands for high and l
stands for low. The decision tree representation of Jill’s problem is given in Figure 15.


                     Figure 15: The decision tree for Jill’s decision problem.


The left most node (root of the tree) corresponds to the decision about the value of M 1 and the next
two circular nodes corresponds to the level of demand during the first 6 months. The next two decision
nodes corresponds to the choice of the value of M2 , whereas the remaining chance nodes correspond
to the level of demand during the last 18 months. The numbers attached to the right most nodes, i.e.
the leaves of the tree, represent the payoffs corresponding to the different scenarios. These payoffs
are computed from the data of the problem concerning the prices of the machines and the revenues

                                                  36
they bring. The numbers on the edges emanating from chances nodes correspond to the probabilities
mentioned in the verbal description of the problem. Note that the probabilities on the edges coming
out of chance nodes representing the level of demand during the last 18 months are indeed conditional
probabilities.

To solve Jill’s decision problem, first the possible strategies and the different states of nature have to be
identified to establish the decision matrix. Obviously, there are three strategies, i.e. buy one machine
first and stick with it for the whole time, buy one machine first and buy a second one after 6 months,
and buy two machines right away. The first can be characterized by the vector a 1 = (one, one), the
second one by a2 = (one, two) and the third one by a3 = (two, two). The different states of nature
for Jill’s problem, obviously, are given by the four vectors

z1 = (h, h), z2 = (h, l), z3 = (l, h), z4 = (l, l),

where the first component of each vector expresses the demand level during the first 6 months and the
second one during the remaining 18 months. Therefore, the payoff matrix of Jill’s decision problem
is of dimension 3 × 4. Taking for each pair (ai , zj ) the corresponding payoff from the decision tree,
Jill’s payoff matrix is hence given by

         1000 1000 1000 1000
D=      −4500 −9000 −4500 −9000             .                                                          (27)
         2000 −2500   500 −4000
In a second step, it is necessary to compute the probability for each of the four states of nature. The
probability of the first state of nature is computed as

P ((D1 , D2 ) = (h, h)) = P (D2 = h|D1 = h)P (D1 = h) = 0.8 · 0.4 = 0.32 .

The probability of the other states of nature can be computed in a similar way, which yields the
probability vector for the four states of nature given by

(0.32, 0.08, 0.24, 0.36).

Now, in a third step, the Bayes criterion introduced in section 4.3. can be used to determine the best
strategy. For each possible strategy ai (i = 1, 2, 3), the expected payoffs attached to the corresponding
row vectors di are given in the vector

(V (d1 ), V (d2 ), V (d3 )) = (1000, −6480, −880) .

Therefore, according to the Bayes criterion, the optimal strategy is a 3 , i.e. to buy one machine and
then stick with that machine after six months.

Instead of computing the payoff matrix, there is a widely used general algorithm to find the optimal
strategy given a decision tree. It is called the average out and fold back algorithm and works as
follows. Starting from the leaves of the tree, all nodes are successively evaluated: a chance node is
evaluated by computing the expectation of the values placed at the end of its outgoing edges, and a
decision node is evaluated by computing the maximum of the values placed at the end of its outgoing
edges. After a decision node has been evaluated, only the edge with maximal value is retained and
the other edges (and their attached subtree) or deleted from the original tree. In this way, at the end of
the procedure, i.e. after the root of the tree has been evaluated, the remaining tree graphically displays
the optimal strategy. Figure 16 shows the result of this algorithm applied to the decision tree given in
Figure 15. Of course, the optimal solution is the same as the one determined above, i.e. a 3 = (1, 1)
and the expected payoff of the optimal solution is 1000.

                                                      37
                            Figure 16: Evaluation of Jill’s decision tree.


Within the framework of decision trees it is also possible to take into account the decision maker’s
risk attitude by applying the EU principle. Indeed, simply replace the payoffs on the leaves of the
decision tree by the DM’s utility values and apply the average out and fold back algorithm with these
values. The optimal strategy obtained in this way is the one maximizing the DM’s expected utility.

7.2.   Influence Diagrams

Clearly, when a multistage decision problem gets larger, the corresponding decision tree can easily
become very large and can hardly be drawn on a simple sheet of paper. Then there is a need for a
more concise representation of the problem. An influence diagram representation provides an answer
to this need. An influence diagram is a pair composed of a an oriented graph and a set of numerical
values, where these numerical values are usually not displayed on the graph. The graph contains three
kinds of nodes: decision nodes (represented by rectangles), chance nodes (represented by circles), and
utility nodes (represented by diamonds). Furthermore, the arcs connecting different kinds of nodes
have different meanings. E.g., an arc going from one chance node to another one means that the
conditional distribution of the random variable represented by the second chance node is known for
each possible value of the random variable represented by the first chance node. Finally, it makes an
important difference if there is an arc entering a node or not.

As example of an influence diagram, the graph of the influence diagram representation of Jill’s de-
cision problem treated in subsection 7.1. is given in Figure 17. The set of numerical values of this
influence diagram is given below.


                          Figure 17: The graph of the influence diagram.


The two chance nodes D1 and D2 represent the respective stochastic demand levels and the two
decision nodes M1 and M2 the decisions to be taken at the respective decision stages. The utility node
U stands for a variable expressing the possible final payoffs that can be attained. These payoffs are
given in the payoff matrix (27). Of course, if the decision maker’ attitudes toward risk are considered,
then the payoffs must be replaced by the corresponding utilities. The arcs placed in the graph represent
the qualitative information available about the decision problem. The first information is that the
payoffs depend on the D1 , D2 , M1 and M2 , which is represented by the four arcs entering the utility
node U . The second information is that the conditional distribution of the random variable D 2 given
each of the two possible values of D1 is known, i.e.

P (D2 = h|D1 = h) = 0.8      P (D2 = l|D1 = h) = 0.2                                               (28)

and

P (D2 = h|D1 = l) = 0.4      P (D2 = l|D1 = l) = 0.6 .                                             (29)

The arc going from D1 to D2 means that these conditional distributions are known and the fact that
there is no arc entering the chance node D1 means that the (unconditional) distribution of D1 must be
known. In this case, this distribution is

P (D1 = h) = 0.4     P (D1 = l) = 0.6 .                                                            (30)


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Finally, some time information is available in the decision problem. First, the value of D 1 is known at
the time the decision about M2 has to be made. This is represented by the arc going from D1 to M2 .
Second, the decision about M1 is known at the time the decision about M2 has to be made, which is
represented by the arc going from M1 to M2 . In general, the arcs representing this kind of constraints
are called information arcs. The set of numerical values of the influence diagram representation of
Jill’s decision problem is given by the conditional probabilities (28) and (29), the probabilities (30)
and the payoffs (27).

For a general presentation of influence diagrams, the reader is referred to Decision Trees and Influence
Diagrams. The algorithm used to find the optimal strategy in an influence diagram, i.e. the one
maximizing the expected utility, is the so-called arc reversal algorithm. This is a fairly complicated
algorithm that is described in Decision Trees and Influence Diagrams.

Bibliography

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[This book provides a thorough discussion of the different axioms underlying expected utility.]
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tiarum imperialis Petropolitanae. 5, 175–192. [In this historical paper the idea of expected utility has
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Luce, R. D., Raiffa, H. (1957). Games and Decisions. New-York. [This book is one of the “classic”
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Neumann, J. von, Morgenstern, O. (1947). Theory of Games and Economic Behavior (second edi-
tion). Princeton. [This book provides the first axiomatic foundations of expected utility theory.]
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Tversky A., Kahneman D. (1992). Advances in Prospect Theory: Cumulative Representation of
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zation of the at present prevailing behavioral theory of decision making under uncertainty.]




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