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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 2 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) 38 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. 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