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A Study of Regret and Rejoicing in Decision-Making and a New MCDM Method Xiaoting Wang and Evangelos Triantaphyllou College of Engineering and Department of Computer Science Louisiana State University Baton Rouge, LA INFORMS Annual Meeting, Seattle, WA Nov. 4 - 7, 2007

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Outline Introduction and problem description Previous studies on regret and rejoicing Some regret models The limitations of these regret models A new method to assess regret and rejoicing A new MCDM method based on regret and rejoicing Concluding remarks

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Introduction Multi-criteria decision-making (MCDM) A typical MCDM problem involves the evaluation of a finite number of alternatives in terms of a finite number of criteria which might be benefit or cost criteria. The Decision Matrix a ij : the performance value of the i-th alternative in terms of the j-th criterion w j : the weight of the j-th criterion C r i t e r i a C 1 C 2...C n (w 1 w 2...w n ) Alts.------------------------------------ A 1 a 11 a 12...a 1n A 2 a 21 a 22...a 2n... A m a m1 a m2...a mn Figure 1. A typical decision matrix

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Problem description The main factors in an MCDM problem The performance values of the alternatives under the decision criteria. The weights of the criteria. Also emotional factors, like feeling of regret and rejoicing. Why emotional factors matter? It comes from the fact that humans often base their choices on comparisons across the alternatives under consideration and relative to “what might have been” under another choice [Plous, 1993; Hastie and Dawes, 2001]. Example Given two alternatives A 1 and A 2 and some decision criteria Assume overall: A 1 > A 2 but a 1k is worse than a 2k for some criterion C k. Then the decision maker (DM) who chooses A 1 and forgoes A 2 may experience a certain level of regret because a 1k < a 2k. Sometimes, the regret feeling could be very strong and as result the DM may regret to have chosen A 1 instead of A 2. 4

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Problem description In order to avoid the above situation the DM would want to anticipate the regret feeling and consider it in the decision- making process by making some tradeoffs for a more balanced alternative. He/she can predict the emotional consequences of different decision outcomes in advance, and opt for the choices that minimize the possibility of negative emotions. The favorable feeling of rejoicing can be considered as the inverse of regret and analyzed in an analogous manner as regret. Note: because of time constraint, only regret is discussed in the next sections. Rejoicing can be analyzed in an analogous manner. 5

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Previous studies on regret and rejoicing Multi-attribute utility analysis (MAUA) One of the systematic MCDM method [Kirkwood, 1997]. The performance value of an alternative under each decision criterion is transferred into a utility value according to some utility functions. The utility (a value between [0, 1]) represents the preferability of the alternatives under each decision criterion. Usually, the total utility of each alternative can be computed as a weighted sum such as (1) One key assumption for the utility model DMs are “rational Individuals” which devoid of psychological influences or emotions [Luce, 1992]. Thus, DMs will always want to make choices that can maximize the utilities of the chosen alternatives. 6

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Previous studies on regret and rejoicing However, behavioral scientists have proved that it is not always appropriate to relate decision rationality to utility maximization [Allais, 1988; Ellsberg, 1961]. To broad the assumptions, some research has been made to incorporate behavioral issues into the decision making process, for example, regret and rejoicing. Regret is defined as “the painful sensation of recognizing that ‘what is’ compares unfavorably with ‘what might have been’.” [Sugden, 1985]. The converse experience of a favorable comparison between the two has been called “rejoicing”. Next, some of the main regret models in the literature will be introduced. 7

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8 Some regret models in the literature The minimax regret model [Savage, 1951] (2) The Regret Theory of Bell and Loomes and Sugden (RT-B/LS) [Loomes and Sugden, 1982; Bell, 1982 and 1985] (3) The Reference-Dependent Regret Model (RDRM) [Kujawski, 2005] (4) * G(.), the regret-building function.

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9 Limitations of these regret models The minimax model decides the selection of alternatives totally based on their regret values, it may lead to irrational choices. The RT/B-LS and the RDRM model They are based on the concept of utility It is not always convenient for DMs to find a proper utility function that can appropriately transform performance values with different units into unit-less and additive utility values. They quantify regret by using continuous functions. How to determine the customizing parameters as the R and D in the regret function? For different criteria, the DM may need to determine different values for the parameters. Regret does not always change as a continuous variable. Usually people feel a certain level of regret when the difference between two compared items is beyond a threshold value or when one of the compared performance values is below an “echelon” value. Example: consider three students taking an exam. Scores: 79, 70, 69. Grades: C, C, F. Then we may have R(69, 70) > R(70, 79)

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10 Limitations of using continuous regret functions Furthermore, decision criteria may be quantitative or qualitative. As emotional factors, regret and rejoicing are definitely qualitative aspects in decision problems. It would be more appropriate to assess people’s anticipated regret/rejoicing feelings by using some proper linguistic terms rather than continuous numbers. For qualitative aspects, the pairwise comparison approach proposed by Saaty (as part of the AHP method) [Saaty, 1994] has received widespread attention. In this approach, the DM is asked to select a linguistic statement from 9 statements that best describes his/her assessment for the comparison of two compared items (alternatives or criteria). Next, some consistency tests are performed as it is possible to have estimated values from DMs with high levels of inconsistency. * Why choose 9 as the upper limit? Psychological experiments have shown that most individuals cannot simultaneously compare more than seven objects (plus or minus two) [Miller, 1956].

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A new method to assess regret and rejoicing Here a similar pairwise approach is proposed First, a scale similar to the Saaty scale is built as in Table 1 (on the next slide). It has a finite set of linguistic choices with corresponding numerical values. the DM can use this scale to assess his/her anticipated regret value for choosing one alternative over another under a specific decision criterion. Example Two alternatives: A i and A j having a ik and a jk in terms of a benefit criterion C k. R(a ik, a jk ): the true (more accurate) anticipated regret value for choosing a ik and forgoing a jk. r(a ik, a jk ): the estimated regret value chosen by the DM from Table 1. R(a ik, a jk) = e ij k + r(a ik, a jk ), for any i, j = 1,2,3,…,n. (5) e ij k is the error coefficient which is expected to be close to 0 assuming the DM is consistent and reasonable. If a ik,> a jk, r(a ik, a jk ) =0. Otherwise, a linguistic expression need to be selected and the corresponding numerical value will be assigned to r(a ik, a jk ). 11

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Table 1 12 Linguistic ExpressionNumerical Value There is no distinguishable feeling of regret when choosing alternative A i over alternative A j. 0 The feeling of regret when choosing alternative A i over alternative A j is noticeable.2 The feeling of regret when choosing alternative A i over alternative A j is high.4 The feeling of regret when choosing alternative A i over alternative A j is very high.6 The feeling of regret when choosing alternative A i over alternative A j is as high as it can be. 8 The numerical values of 1, 3, 5, and 7 are used when the decision maker cannot choose between two successive linguistic expressions from the above list of choices. 1, 3, 5, 7

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Regret matrix Given a decision problem with n alternatives and m criteria, the DM compares each alternative with all the others under each single criterion to build a regret matrix in terms of each criterion. A regret matrix in terms of criterion C k is defined as follows: Alts. (A 1 A 2...A m ) Alts.------------------------------------ A 1 0 r 12...r 1m A 2 r 21 0...r 2m..... A m r m1 r m2...0 Figure 2. A regret matrix Some relations about the entries of a regret matrix. r(a ik, a ik ) = 0, for any i = 1, 2, 3, …, n. If a ik,> a jk, then r(a ik, a jk ) =0, for any i, j = 1,2,3, …,n.

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The additive transitivity property Suppose the performance values of three alternatives A i, A j, and A m under a benefit criterion C k, are a ik, a jk, and a m, assuming a ik > a jk > a mk. It is required that the anticipated regret values produced when compared them two at a time should satisfy an additive transitivity property defined as follows: (6) Since R(.) =r(.)+ e(.), then (7) where

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The additive transitivity property In terms of a specific criterion, if the DM is rational and consistent, then the error coefficients should be small. The objective is to minimize the sum of squares of the error coefficients under the additive transitivity constraints. Then, we have an optimization problem as follows: (8) subject to (7) where

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A new method to assess regret and rejoicing Solving the optimization problem to get the error coefficients which produce the minimum value, the adjusted regret values R(.) which are also regarded as the more accurate regret values. can also be used as a consistency criterion. The more consistent the estimated regret values are, the smaller this value will be. An acceptable value should be decided for the specific application at hand. We can also check some other statistic values about the error coefficients, such as their average, the maximum, and the standard deviation etc., to determine if there is any significant inconsistency within the estimated values. If the inconsistency between the estimated regret values is beyond a certain acceptable level, then the DM would be required to review his/her estimated regret values.

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A new method to assess regret and rejoicing Next, the overall regret value of alternative A i in terms of the k-th criterion is defined as the average of the regret values produced when alternative A i is compared with each of the other alternatives under the same criterion. (9) The overall regret value of alternative A i in terms of all the criteria is defined as: (10) Then, how to aggregate the regret and rejoicing values and the performance values of the alternatives together to get the final priority for each alternative? 17

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18 A new MCDM method based on regret and rejoicing Some previous studies had reported that some types of rank reversals may occur with some MCDM methods which use additive formulas to compute the final priorities of the alternatives, such as the AHP method and the revised AHP method [Dyer and Wendell, 1985; Triantaphyllou, 2001]. (11) A method known as the multiplicative AHP [Lootsma, 1999] is immune to those ranking problems. (12) (13) By using the multiplicative formula, no matter how the decision matrix is normalized, the ratios of the alternatives’ performance values will be kept the same because the normalization factor is cancelled off in the multiplicative formula.

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A new MCDM method based on regret and rejoicing Thus, in the new MCDM method, the multiplicative formula is used to compute the final priority for each decision alternative. Then, for decision making problems, when the set of alternatives is changed, all variation left is due to the regret and rejoicing effects and that could be justifiable as it would not be due to any mathematical artifacts. Based on the above points, assume only considering the anticipated regret for a given MCDM problem which has m alternatives and n benefit decision criteria. The formula for computing the final overall priority for each alternative by using the multiplicative formula and the benefit to cost approach to deal with conflicting criteria will be as follows: (14)

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20 A new MCDM method based on regret and rejoicing For a more complex decision problem m alternatives and n decision criteria with n 1 benefit criteria and (n-n 1 ) cost criteria Consider both the anticipated regret and rejoicing The equation to compute the overall priority for each alternative is as follows: (15) is the overall regret for choosing A i under the benefit criteria; is the overall rejoicing for choosing A i under the benefit criteria; is the overall performance value of A i under the benefit criteria; and have the similar meaning as the above ones but now in terms of the cost criteria.

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Why to satisfy the additive transitivity property? Given is a symmetric decision problem where x>y>z. Table 3. A symmetric decision problem Because of the symmetry of this decision problem, it is expected that the three alternatives should be ranked equivalently by a valid MCDM method when two of them are ranked at a time. However, some regret model like the RT-B/LS model ranks these three alternatives in a cyclic way [Kujawski, 2005]. That is, A 2 > A 1, A 3 > A 2, and A 1 > A 3. The new method should not allow cyclic preferences to happen. Criteria AlternativesC1C1 C2C2 C3C3 A1A2A3A1A2A3 zxyzxy yzxyzx xyzxyz Weights1/3

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Why to satisfy the additive transitivity property? Consider the task of ranking A 1 and A 2 The overall regret values for A 1 and A 2 are: (16) (17) The aggregated performance: The overall performance value for A 1 and A 2 are: (18) (19) Only if then That is, the additive transitivity property should be satisfied.

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23 A summary of the new method Step 1. Set up the scale and the linguistic terms for esimating the regret and rejoicing. Step 2. Ask the DM to choose the best linguistic term that can assess his/her anticipated regret and rejoicing feeling independently and then to fill the entries for the regret and rejoicing matrixes. Step 3. Build the objective function and the constraints which require the adjusted regret/rejoicing values to satisfy the additive transitivity property. Step 4. Solve the optimization problem and get the error coefficients and the adjusted regret and rejoicing values. Step 5. Check the value of Z * and the error coefficients to see if there is any significant inconsistency within the estimated regret and rejoicing values. If the answer is yes, go to Step 6, else go to Step 7. Step 6. Ask the DM to review the estimated regret and rejoicing values and repeat step 3-5 until there are no significant inconsistent estimated values. Step 7. Use the adjusted regret and rejoicing values to compute the final overall priority for each alternative and rank them.

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24 Concluding remarks The properties of the new MCDM method are: Besides the usual benefit and cost criteria, this method is able to incorporate the effects of regret and rejoicing for DMs who value these emotional factors in MCDM situations. The determination of regret and rejoicing effects is more reasonable and consistent with usual human behavior. For the new method, rank reversals may occur only as result of readjusting the effects of regret and/or rejoicing when the set of the alternatives is altered. The new method is able to deal with qualitative and quantitative criteria expressed in different units of measurement. The new method does not allow cyclic preference to happen to a symmetric decision problem. By using the new method, decision makers will have the flexibility to decide their own regret/rejoicing levels and the importance of these emotional factors in their decision-making process.

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25 References 1.Allais, M., (1988), The general theory of random choices in relation to the invariant cardinal utility function and the specific probability function, Risk, Decision and Rationality, in Munier, editor, pp. 231-291, Dordrecht, The Netherlands. 2. Bell, D.E., (1982), “Regret in Decision Making under Uncertainty,” Operations Research, Vol.30, pp. 961-981 3. Bell, D.E., (1985), “Disappointment in decision making under uncertainty,” Operations Research, Vol. 33, pp.1-27. 4. Dyer, J.S., and R.E. Wendell, (1985), “A Critique of the Analytic Hierarchy Process,” Technical Report 84/85-4-24, Department of Management, the University of Texas at Austin, Austin, TX, U.S.A. 5. Ellsberg, D. (1961), “Risk, ambiguity and the Savage axioms,” Quarterly Journal of Economics, Vol. 75, pp. 643-649. 6. Hastie, R., and R.M. Dawes, (2001), Rational choice in an uncertain world, Sage Publications, Thousand Oaks, CA, USA. 7. Kirkwood, C.W., (1997), Strategic Decision Making, Duxbury Press, Wadsworth Publishing Company, Belmont, Calif., USA. 8. Kujawski, E., (2005), “A reference-dependent regret model for deterministic tradeoff studies,” Systems Engineering, Vol. 8, pp. 119-137. 9. Loomes, G., and R. Sugden, (1982), “Regret Theory: an Alternative Theory of Rational Choice Under Uncertainty,” The Economic Journal, Vol. 92, pp.805-824.

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26 References (Cont’d) 10. Lootsma, F.A., (1999), "Multi-Criteria Decision Analysis via Ratio and Difference Judgment." Kluwer Academic Publishers, Applied Optimization Series, Vol. 29, Dordrecht, The Netherlands. 11. Plous, S., (1993), The psychology of judgment and decision making, McGraw-Hill, New York, NY, U.S.A. 12. Saaty, T.L., (1980), The Analytic Hierarchy Process, McGraw-Hill, New York, NY, U.S.A. 13. Saaty, T.L., (1994), Fundamentals of Decision Making and Priority Theory with the AHP, RWS Publications, Pittsburgh, PA, U.S.A. 14. Savage, L.J., (1951), “The theory of statistical decision,” Journal of American Statistical Association, Vol. 46, pp.55-67. 15. Stam, A., and A.P.D. Silva, (1997), “Stochastic Judgments in the AHP: the measurement of rank reversal probabilities,” Decision Sciences, Vol.28, pp.655-688. 16. Triantaphyllou, E., (2000), Multi-Criteria Decision Making Methods: A Comparative Study, Kluwer Academic Publishers, Boston, MA, USA. 17. Triantaphyllou, E., (2001), “Two New Cases of Rank Reversals when the AHP and Some of its Additive Variants are Used that do not Occur with the Multiplicative AHP,” Multi-Criteria Decision Analysis, (May 2001 issue) Vol. 10, pp. 11-25. 18. Luce, R.D., (1992), “Where does subjective expected utility fail descriptively?” Journal of Risk and Uncertainty, Vol. 5, pp. 5-27 19. Mellers, B.A., (2000), “Choice and the relative pleasure of consequences”, Psychological bulletin, Vol. 126, pp. 910–924. 20. Miller, C.A., (1956), “The magic number seven plus or minus two: some limits on our capacity for processing information,” Psychological Review, Vol. 13, pp. 81-97.

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