A Study of Regret and Rejoicing in Decision-Making and a New MCDM Method

1
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

2
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

3
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
• aij the performance value of the i-th
  alternative in terms of the j-th criterion
• wj the weight of the j-th criterion
• 
  C r i t e r i a
• C1 C2 ... Cn
• (w1 w2 ... wn)
• Alts. ------------------------------------
• A1 a11 a12 ... a1n
• A2 a21 a22 ... a2n
• . . .
• . . .
• Am am1 am2 ... amn
• Figure 1. A typical
  decision matrix

4
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 A1 and A2 and some
  decision criteria
• Assume overall A1 gt A2 but a1k is worse than a2k
  for some criterion Ck.
• Then the decision maker (DM) who chooses A1 and
  forgoes A2 may experience a certain level of
  regret because a1k lt a2k.
• Sometimes, the regret feeling could be very
  strong and as result the DM may regret to have
  chosen A1 instead of A2.

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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.

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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.

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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.

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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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) gt R(70, 79)

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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 peoples
  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 Ai and Aj having aik and ajk
  in terms of a benefit criterion Ck.
• R(aik, ajk) the true (more accurate) anticipated
  regret value for choosing aik and forgoing ajk.
• r(aik, ajk) the estimated regret value chosen by
  the DM from Table 1.
• R(aik, ajk) eijk r(aik, ajk), for any i, j
  1,2,3,,n. (5)
• eijk is the error coefficient which is expected
  to be close to 0 assuming the DM is consistent
  and reasonable.
• If aik,gt ajk, r(aik, ajk) 0. Otherwise, a
  linguistic expression need to be selected and the
  corresponding numerical value will be assigned to
  r(aik, ajk).

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Table 1

Linguistic Expression Numerical Value
There is no distinguishable feeling of regret when choosing alternative Ai over alternative Aj. 0
The feeling of regret when choosing alternative Ai over alternative Aj is noticeable. 2
The feeling of regret when choosing alternative Ai over alternative Aj is high. 4
The feeling of regret when choosing alternative Ai over alternative Aj is very high. 6
The feeling of regret when choosing alternative Ai over alternative Aj 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 Ck is
  defined as follows
• Alts.
• (A1 A2 ... Am)
• Alts. ------------------------------------
• A1 0 r12 ... r1m
• A2 r21 0 ... r2m
• . . . . .
• Am rm1 rm2 ... 0
• Figure 2. A
  regret matrix
• Some relations about the entries of a regret
  matrix.
• r(aik, aik) 0, for any i 1, 2, 3, , n.
• If aik,gt ajk, then r(aik, ajk) 0, for any i, j
  1,2,3, ,n.

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The additive transitivity property

• The additive transitivity property
• Suppose the performance values of three
  alternatives Ai, Aj, and Am under a benefit
  criterion Ck, are aik , ajk, and am, assuming
  aik gt ajk gt amk.
• 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 Ai
  in terms of the k-th criterion is defined as the
  average of the regret values produced when
  alternative Ai is compared with each of the other
  alternatives under the same criterion.
• (9)
• The overall regret value of alternative Ai 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?

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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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A new MCDM method based on regret and rejoicing

• For a more complex decision problem
• m alternatives and n decision criteria with n1
  benefit criteria and (n-n1) 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 Ai under
  the benefit criteria
• is the overall rejoicing for choosing Ai
  under the benefit criteria
• is the overall performance value of Ai
  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
  xgtygtz.
• 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, A2gt A1, A3 gt A2, and
  A1gt A3.
• The new method should not allow cyclic
  preferences to happen.

Criteria Criteria Criteria
Alternatives C1 C2 C3
A1 A2 A3 z x y y z x x y z
Weights 1/3 1/3 1/3
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Why to satisfy the additive transitivity property?

• Consider the task of ranking A1 and A2
• The overall regret values for A1 and A2 are
• (16)
• (17)
• The aggregated performance
• The overall performance value for A1 and A2 are
• (18) (19)
• Only if
  then
• That is, the additive transitivity property
  should be satisfied.

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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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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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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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References (Contd)

• 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. 910924.
• 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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Thanks!