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preliminary version. Paper to be presented at the 52^{nd} meeting of the European Working Group MULTICRITERIA AID FOR DECISIONS. Vilnius, Lithuania, 5^{th} – 6^{th} October 2000. Multicriteria Decision Aid and Economic Choice Theory.SvenOlov Larsson MidSweden University/MAM, Östersund, and Centre for Transport Economics, Borlänge. Email: svenolov.larsson@mam.mh.se. Abstract: This paper searches for a relation between the outcomes of a decisions built upon an outranking process built upon Multicriteria decision aid, and what would be the outcome of an “ideal” decision situation making a complete cost benefit analysis possible. More generally comparisons are made between MCDA and economic choice theory (ECT). The comparisons are built partly upon the axioms underpinning ECT, and partly upon numerical examples. A conclusion is that we can expect similar outcomes in many cases. However it is easy to construct examples where the outcomes become different. The causes of these similarities and differences should be explored further. Résumé: Le MCDA est développé afin de prendre en compte beaucoup de problèmes de prise de décision: descriptions inexactes des possibilités, préférences incomplètes etc. Il n’existe normalement aucune alternative optimale pour prendre une décision. Le résultat de la procédure du sélection du MCDA peut être plus qu’une alternative suggérée. Quelle est la relation entre ce résultat et le choix optimal proposé par la théorie des choix économiques ? La théorie des choix en économie est un fondement important de la microéconomie. La théorie des consommateurs et la théorie des choix publics sont construites sur cette théorie des choix. L’analyse coût avantage (CBA), outil économique majeur dans l’évaluation ex ante et ex post, est également fondé sur cette théorie. L’avantage avec une base théorique solide est que les obstacles à la prise de décision devraient être liés à cette théorie, et le résultat des décisions réelles devrait être lié aux décisions théoriquement optimales. C’est aussi l’idée de ce document. Quelle est la différence entre le résultat obtenu dans des conditions optimales et des décisions construites à partir de MCDA? Ce document essaye de trouver une relation entre MCDA et la théorie des choix économiques (ECT). Le document comporte quatre parties. Le fondement de ECT est un nombre d’axiomes concernant la nature des préférences reliées aux conditions et procédures de MCDA, ou au moins une des méthodes de MCDA fréquemment utilisées, Electre III. Des exemples numériques sont utilisés pour illustrer les similitudes et les différences de résultat entre les deux approches. Les relations entre ECT et MCDA sont analysées au travers de ces similitudes et de ces différences. Les parties 4) et 5) donne quelques conclusions de cette analyse. Multicriteria Decision Aid (MCDA) is developed to take into account many of the problems of real decisionmaking: inexact description of the consequences of decision alternatives, incomplete and insecure preferences etc. There is normally no “optimal” alternative to decide about. The outcome of the outranking procedure of MCDA can be more than one suggested alternative. What is the relation between this outcome and an optimal choice suggested by economic choice theory? Choice theory in economics has a solid theoretical foundation in microeconomics. Consumer theory and public choice theory are built upon this theory of choice. Cost Benefit Analysis (CBA), the major tool for economic ex ante and ex post evaluation, is founded upon this choice theory. A demand for efficient solutions to public resource allocation problems has boosted the use of CBA. An advantage with a firm theoretical basis is that the obstacles of real decision making can be related to economic choice theory (ECT) and the outcomes of real decisions could be related to theoretically optimal decisions. How far from the outcome of the “ideal” decision conditions will decisions built upon MCDA take us? This paper tries to find a relation between MCDA and ECT. The paper consists of four parts. 1) A fundament of ECT is a number of axioms concerning the nature of preferences^{1}. These are related to the conditions and procedures of MCDA, or at least one of the most frequently used methods of MCDA, Electre III (Vincke 1992 p 64 ff.)^{2}. 2) Numerical examples are used to illustrate similarities and differences in outcome of the two approaches. 3) These similarities and differences are analysed in terms of the relation between ECT and MCDA. Part 4) draws some conclusions of this analysis. In the last part, this relation is discussed in more general terms. 1. ECT and MCDAThe set of axioms underpinning the preference or utility function, the core concept of the theory. To be able to compare choices according to ECT and the outcome of a MCDA process, it is necessary to refer to these axioms. First we should specify the utility function to use. If we are analysing a collective choice problem it is also necessary to aggregate the individual values into a social utility or welfare function^{3}: , where and where r represents all alternative baskets, s represents all the possible contents that can appear in the baskets. The social choice problem is to find the alternative giving the highest value of . To make this choice comparable to the outranking procedure of Electre III we have to develop our economic choice problem a little. Instead of talking about goods and services in the alternative baskets, let us talk about consequences of the alternatives. Then we can connect to appendix 2, where Electre III is summarised. If represents all the possible alternatives and consequences, where represent only the known alternatives and consequences. This means that . If a utility function exists, which covers these alternatives and consequences, it should be written . A choice made under this utility function is in accordance with a choice made from the comprehensive utility function only if the preferences expressed by are independent of the set of consequences not included in ^{4}. It should anyhow fulfil the axioms. The relation between these axioms and MCDA, especially the Electre III model, has been discussed in several earlier contributions and summarised and developed by Bouyssou (1996). This discussion will not be repeated here. The main results of the discussion can however give some ideas about what is found in the next section: COMPLETE preferences: One limitation is already included in the differences between and . The MCDA process also allows a development of preferences if the set of consequences or alternatives is not given from the beginning. The outranking procedure allows incomparability at least to some extent. REFLEXIVE preferences are a necessary condition for making comparisons between alternatives. In terms of the outranking index (appendix 2) reflexivity appears in the form of . If an alternative is compared to itself, the index is always = 0. TRANSITIVE preferences are not a condition to be fulfilled in the outranking process of Electre III. This depends upon the concordance and discordance principles allowing uncertainty in preferences. This means that we can find indifference between alternatives a and b, and between alternatives b and c and still think that a is better than c. If the three conditions above are not fulfilled, then we can not talk about CONTINUOUS preferences. NON SATIATION and STRICTLY CONVEX preferences are necessary conditions for achieving optimal outcomes in ECT, but these conditions are not necessary in the outranking process. On the other hand, we can not expect a single dominating alternative coming out of the process. Since Electre III, as well as other MCDA models, does not fulfil the axioms we can not expect the outranking procedure to come up with the same dominating alternatives as the optimal alternative built with a utility function. But how different can the outcomes be? 2. Numerical examples.A comparison of ECT and MCDA in terms of numerical examples can demonstrate differences and similarities between the two approaches. The example is a case including a choice of corridor for a new motorway. The numbers used is a summary from an experiment with a group of students given the task to choose a corridor for the motorway. From this starting point five more numerical examples are developed to illustrate some relations between MCDA and ECT. Two alternative goal or utility functions represent economic choice theory: A function of CobbDouglas form: A linear function: where = the consequence i of alternative j and = the weights of the consequences. Example 1: The consequences of the six alternatives are of six different kinds, which gives a 6 x 6 consequence matrix (appendix 2):
The first two characteristics are amounts of investment and yearly gains in traffic economics. The other characteristics are averages of comparisons of alternatives according to environmental impacts made by the members of the test group. In a first test no thresholds are used. The medians of the weights decided by the members are:
The outranking procedure in this example gives the following results: The alternatives 1, 2, and 3 are dominating. A choice between them should be made on other grounds than the data in the MCDA model. A first option is to go back to those data and see if they are the most relevant. Another way to make a difference could be to introduce thresholds in the analysis. Let us first have a look at the outcome if the two goal functions are used. To be able to apply them, it is necessary to transform the consequence matrix into normalised, positive numbers. The transformation is done with the following formula: This gives the following transformed consequence matrix:
We apply the goal functions on this problem, adopting the same weights as above, to find an optimal alternative from the transformed consequence matrix. The following values are found.
Both goal functions favour alternative 2. Example 2: An introduction of indifference, preference, and veto thresholds changes the outranking picture considerably. The following numbers are chosen arbitrarily:
The result of the outranking is: The index numbers are now smaller depending upon the thresholds. There is still no single, dominating alternative. The alternatives 1 and 2 are ranked on the same level. The thresholds do not affect the outcomes of the goal functions. Example 3: The role of weights should also be discussed. Let the six kinds of characteristics have the same weights without thresholds:
The outranking procedure of this example gives a rather complicated picture: The alternatives 1, 2 and 3 are still firmly placed at the same ranking level. The goal functions give a common result:
Alternative 2 should be preferred. The equality of the weights of the alternatives obviously makes it more difficult to choose a preferred alternative in example 3. How do the thresholds change the picture with this set of weights? Example 4 demonstrates this: Now alternative 2 is the only dominating alternative. This result corresponds to the outcome of the goal functions. Example 5: Alternative 2 is among the dominating alternatives in all the four examples above. Let us now give it a more extreme set of characteristics by multiplying all its numbers in the consequences matrix by 1.2, i.e. an increase of positive as well as negative consequences by 20%. We use the weights of example 1. Consequence matrix:
The outcomes of the goal functions:
Alternative 1 obtains the highest goal value by both functions. Alternative 2 becomes a very low value by the CobbDouglas function. This is a result of its more extreme consequences. The outranking procedure, without thresholds, gives the following result: This outranking creates an interesting relation between the alternatives 1, 2, and 3. An “Arrow paradox” has appeared, and if we do not look at the size of the ranking index, or at other arguments, there is no dominating alternative. Compared to the results of the goal functions, the alternatives 2 and 3 remain of interest for choice. Let us then look at what happens if we introduce the thresholds of example 2 into example 6: The alternative 1 is preferred not only by the CobbDouglas and additive goal functions but also by the outranking procedure of Electre III. 4. Comparing the outcomes of MCDA and ECT.Some objections can of course be raised against the type of comparisons made in section 3. One objection is the use of weights. In the CobbDouglas function the weights are used as exponents in a multiplicative function. The weight can here be regarded as a kind of elasticity saying that if the transformed characteristic h_{ij } is increased with 1%, the value of the weight, in example 3, 0,167% gives the change in outcome of the goal function. If the numerical values of all the consequences increase with 1%, the value of the goal function will also increase with 1%, since the sum of the weights is 1. In the case of an additive goal function the weight can be regarded as a kind of “price” of the characteristic such as the case is in cost benefit analysis. A 1% increase of the value of all the transformed characteristics h_{ij } with 1 % increases the value of the goal function with 1%. In this aspect, the goal functions work in the same way. The difference lays in their way to handle alternatives deviating from the “middle” alternative. This difference is especially obvious in examples 1 and 5, where one of the dominating alternatives become rather low goal values with the CobbDouglas function but not with the linear function. A comparison between the outcomes of the goal functions and of the Electre III model concerning such changes is not directly possible because of the transformation of consequence matrix. 1% change in g_{ij} does not imply a 1% change of the relative number h_{ij}. However, such a transformation seems to be the only possible way to make numerical comparisons between the outcomes. The outranking procedure of the Electre III model used here responds to the changes in data of the four examples by changes of the ranking index. The introduction of thresholds reduces the index numbers. To some extent this could mean lesser differences between the alternatives, but the thresholds can also make differences more clear since only differences outside the thresholds matter (e.g. example 4 compared to 3). 5. Some conclusionsIn the examples 1 – 4 the goal functions select alternative 2. Electre III selects this alternative as the only dominating alternative in example 4. Without any thresholds (examples 1, 3, 5) the three alternatives, 1, 2 and 3 can not be separated. This can depend upon the fact that the alternatives are rather alike. That is why the introduction of thresholds makes the outranking procedure more selective. However, the use of thresholds adds a complication when comparing MCDA and ECT. For example, how does the indifference threshold relate to ECT built on certainty? Can similarity be treated the same way as equality?^{5} We might have to compare MCDA to expected utility theory for further discussions. The outcomes of the outranking procedures do however deliver some results directly contradicting the “ideal” selection provided by the two goal functions. The goal functions in the cases used here deliver one alternative with the highest goal value. This gives no reason for a second thought on which one is the “best”. The decision process as such is neglected when the outcome of a cost benefit analysis is supposed to be taken for granted. The selection process of MCDA could be a more acceptable form of selection of a “best” alternative from the decisionmaker’s point of view. But still the question remains; how far from a decision built upon the outcome of economic choice theory does Multicriteria Decision Aid lead the decisionmaker? This paper only presents some glimpses of similarities and differences of outcomes. Is this difference a problem or is it just an illustration of something Arrow wrote long ago? ”… almost any human action involves choice: the eternal environment delimits a range of possible actions at any given moment but does not usually reduce that range to a single alternative. The formulation of a theory of human action in some sphere as a theory of choice means its presentation as a functional relation associating with each possible range of alternatives a chosen one among them. Economic theory is a clear illustration; the attitudes expressed in polls are again examples of choices from a given range of alternatives, which in this case are verbal expressions. One might perhaps draw a distinction between choices and decisions. … economists are not clear in this matter…” (Arrow, 1958) Appendix 1. Economic choice theoryThe consumer or decisionmaker is supposed to have values or preferences making it possible to choose among different baskets of goods and services presented to him. He is supposed to rank the baskets and select the mot preferred one. The preferences are assumed to have the following properties (following Varian (1984) p 111 ff.) ^{6}.
With a utility function fulfilling these axioms it is always possible for the decisionmaker to make an optimal choice of basket within the limited resources available. Appendix 2. Electre III.A set of alternative actions is given: Each action has a set of consequences: A consequence or impact matrix is thus defined as: The alternatives are compared according to each of the consequences in order to establish differences and similarities. These comparisons are used to find an outranking relation between each pair of alternatives. This relation can be more or less credible, which is expressed in terms of thresholds:
Weights of the consequences are needed: w_{i}, where . The outranking relation is built upon arguments in advantage of a preference relation, the concordance index: This gives the index: The arguments against the preference relation are summarised in a discordance index: which gives the index: The degree of outranking is defined by a ranking index: The credibility of the preference ordering is shown by . To find the most preferable alternative(s) we look for and the preference orderings not too far from the maximum. ReferencesArrow, K. (1958) ”Utilities, Attitudes, Choices: A Review Note.” Operations Research, 5 (1958): 765774. Reprinted in Arrow 1984 (p 55 ff.). Arrow, K. (1984) Individual Choice under Certainty and Uncertainty. Collected papers of Kenneth J. Arrow. Oxford: Basil Blackwell. Bouyssou, D. (1996) ”Outranking relations: Do They Have Special Properties?” Journal of MultiCriteria Decision Analysis. Vol 5, 99  111 Lipman, B.L. (1999) “Decision Theory without Logical Omniscience: Toward an Axiomatic Framework for Bounded Rationality”. Review of Economic Studies. Vol. 66, 339 – 361. Rubinstein, A. (1996) Lectures on Modelling Bounded Rationality. Core lecture series, Core foundation. Lovainlaneuve: Universitй Catholique de Louvain. Sen, A. (1986) ”Social Choice Theory” in Handbook of Mathematical Economics, Vol. III edited by Kenneth J. Arrow and Michael D. Intriligator. Amsterdam: New Holland. Varian, H.R. (1984) Microeconomic Analysis. Second edition. New York: WW Norton & Company. Vincke, P. (1992) Multicriteria Decisionaid.. Chichester: John Wiley & sons. 1 The axioms are summarised in appendix 1. 2 The fundamentals of Electre III are summarised in appendix 2. 3 This aggregation contains theoretical problems that we will not deal with here. 4 This assumption is also used for Arrow’s ”Social Welfare Function” under the name of ”independence of irrelevant alternatives” (Sen (1986) s 1077). 5 Rubinstein (1996) discusses such a problem in p 2:8 ff. He shows that it is possible that choices based on similarities are consistent with the “Rational man” paradigm, in this context, the conditions of ECT, however with narrow limits on the types of preferences. 6 There is a development to find an axiomatic framework also for bounded rationality (Lipman (1999)). This development has so far not come to a state making a comparison with MCDA meaningful. 