Strict uncertainty implies a complete lack of knowledge about the probabilities of possible future states of the world. However, there is complete information about the set of alternatives under consideration, the set of future states, and the scalar evaluation of choosing every alternative if a given state occurs. The principle of insufficient reason by Laplace, the maximin rule by Wald, the Hurwicz criterion, or the minimax regret criterion by Savage are examples of decision rules under strict uncertainty. Within the context of strict uncertainty, moderate pessimism implies the existence of a decision-maker who cautiously assumes that the most favorable state will not occur when the action has been taken with no conjecture being made about the other states. The criterion of moderate pessimism proposed by Ballestero implies the use of the inverse of the range of evaluation for each state as a weight system. In this paper, we extend the notion of moderate pessimism under strict uncertainty to solve some of its limitations. First, we propose a new domination analysis that avoids removing dominated alternatives that are still relevant in the final ranking of alternatives. Second, we propose additional score functions using the inverse of the standard deviation and the mean absolute deviation instead of the range of evaluations for each future state to reduce the impact of the possible existence of outliers in the decision table. This partial result is later generalized through the concept of average deviation of a given order. Finally, we show that all the mentioned decision rules are special cases of a general ranking method based on the Minkowski distance function. We illustrate the use of distance-based decision rules through an application in the context of portfolio selection.
Citation: Francisco Salas-Molina, David Pla-Santamaria, Maria Luisa Vercher-Ferrandiz, Ana Garcia-Bernabeu. New decision rules under strict uncertainty and a general distance-based approach[J]. AIMS Mathematics, 2023, 8(6): 13257-13275. doi: 10.3934/math.2023670
Strict uncertainty implies a complete lack of knowledge about the probabilities of possible future states of the world. However, there is complete information about the set of alternatives under consideration, the set of future states, and the scalar evaluation of choosing every alternative if a given state occurs. The principle of insufficient reason by Laplace, the maximin rule by Wald, the Hurwicz criterion, or the minimax regret criterion by Savage are examples of decision rules under strict uncertainty. Within the context of strict uncertainty, moderate pessimism implies the existence of a decision-maker who cautiously assumes that the most favorable state will not occur when the action has been taken with no conjecture being made about the other states. The criterion of moderate pessimism proposed by Ballestero implies the use of the inverse of the range of evaluation for each state as a weight system. In this paper, we extend the notion of moderate pessimism under strict uncertainty to solve some of its limitations. First, we propose a new domination analysis that avoids removing dominated alternatives that are still relevant in the final ranking of alternatives. Second, we propose additional score functions using the inverse of the standard deviation and the mean absolute deviation instead of the range of evaluations for each future state to reduce the impact of the possible existence of outliers in the decision table. This partial result is later generalized through the concept of average deviation of a given order. Finally, we show that all the mentioned decision rules are special cases of a general ranking method based on the Minkowski distance function. We illustrate the use of distance-based decision rules through an application in the context of portfolio selection.
| [1] |
E. Ballestero, Strict uncertainty: A criterion for moderately pessimistic decision makers, Decision Sci., 33 (2002), 87–108. https://doi.org/10.1111/j.1540-5915.2002.tb01637.x doi: 10.1111/j.1540-5915.2002.tb01637.x
|
| [2] | P. S. Laplace, A philosophical essay on probabilities, New York: Dover, 1952. |
| [3] | A. Wald, Statistical decision functions, New York: John Wiley, 1950. |
| [4] | L. Hurwicz, Optimality criteria for decision making under ignorance, Cowles Commission Discussion Paper, Statistics, 370 (1951). |
| [5] | L. J. Savage, The theory of statistical decision, J. Am. Stat. Assoc., 46 (1951), 55–67. |
| [6] |
E. Ballestero, Selecting textile products by manufacturing companies under uncertainty, Asia Pac. J. Oper. Res., 21 (2004), 141–161. https://doi.org/10.1142/s0217595904000102 doi: 10.1142/s0217595904000102
|
| [7] |
E. Ballestero, M. Günther, D. Pla-Santamaria, C. Stummer, Portfolio selection under strict uncertainty: A multi-criteria methodology and its application to the Frankfurt and Vienna Stock Exchanges, Eur. J. Oper. Res., 181 (2007), 1476–1487. https://doi.org/10.1016/j.ejor.2005.11.050 doi: 10.1016/j.ejor.2005.11.050
|
| [8] |
M. Bravo, D. Pla-Santamaria, Evaluating loan performance for bank offices: a multicriteria decision-making approach, Inform. Syst. Oper. Res., 50 (2012), 127–133. https://doi.org/10.3138/infor.50.3.127 doi: 10.3138/infor.50.3.127
|
| [9] |
D. Pla-Santamaria, M. Bravo, J. Reig-Mullor, F. Salas-Molina, A multicriteria approach to manage credit risk under strict uncertainty, Top, 29 (2021), 494–523. https://doi.org/10.1007/s11750-020-00571-0 doi: 10.1007/s11750-020-00571-0
|
| [10] |
C. P. Chambers, Ordinal aggregation and quantiles, J. Econ. Theory, 137 (2007), 416–431. https://doi.org/10.1016/j.jet.2006.12.002 doi: 10.1016/j.jet.2006.12.002
|
| [11] |
M. Rostek, Quantile maximization in decision theory, Rev. Econ. Stud, 77 (2010), 339–371. https://doi.org/10.1111/j.1467-937x.2009.00564.x doi: 10.1111/j.1467-937x.2009.00564.x
|
| [12] |
H. Sadeghi, F. Moslemi, A multiple objective programming approach to linear bilevel multi-follower programming, AIMS Math., 4 (2019), 763–778. https://doi.org/10.3934/math.2019.3.763 doi: 10.3934/math.2019.3.763
|
| [13] | M. Zeleny, Compromise programming, Multiple criteria decision making, University of South Carolina Press, Columbia, (1973), 262–301. |
| [14] | R. E. Steuer, Multiple criteria optimization. Theory, Computation, and Application, New York: John Wiley, 1986. |
| [15] | E. Ballestero, C. Romero, Multiple criteria decision making and its applications to economic problems, Dordrecht: Kluwer Academic Publishers, 1998. |
| [16] | S. French, Decision theory: an introduction to the mathematics of rationality, New York: John Wiley and Sons, 1986. |
| [17] | J. Von Neumann, O. Morgenstern, Theory of games and economic behavior, Princeton: Princeton University Press, 1944. |
| [18] |
Z. Wu, D. L. St-Pierre, G. Abdul-Nour, Decision making under strict uncertainty: case study in sewer network planning, Int. J. Comput. Infor. Eng., 11 (2017), 815–823. https://doi.org/10.5281/zenodo.1131355 doi: 10.5281/zenodo.1131355
|
| [19] |
V. Ulansky, A. Raza, Generalization of minimax and maximin criteria in a game against nature for the case of a partial a priori uncertainty, Heliyon, 7 (2021), e07498. https://doi.org/10.1016/j.heliyon.2021.e07498 doi: 10.1016/j.heliyon.2021.e07498
|
| [20] |
M. Özkaya, B. İzgi, M. Perc, Axioms of Decision Criteria for 3D Matrix Games and Their Applications, Math., 10 (2022), 4524. https://doi.org/10.3390/math10234524 doi: 10.3390/math10234524
|
| [21] |
G. Balakina, E. Kibalov, M. Pyataev, System assessment of the Kyzyl-Kuragino railroad project as an element of Russia-Mongolia-China transport corridors, Transpor. Res. Proc., 63 (2022), 1817–1825. https://doi.org/10.1016/j.trpro.2022.06.199 doi: 10.1016/j.trpro.2022.06.199
|
| [22] | M. Ehrgott, Multicriteria optimization, Berlin: Springer, 2005. |
| [23] |
C. Romero, A note on distributive equity and social efficiency, J. Agr. Econ., 52 (2001), 110–112. https://doi.org/10.1111/j.1477-9552.2001.tb00928.x doi: 10.1111/j.1477-9552.2001.tb00928.x
|
| [24] |
J. González-Pachón, C. Romero, Bentham, Marx and Rawls ethical principles: In search for a compromise, Omega, 62 (2016), 47–51. https://doi.org/10.1016/j.omega.2015.08.008 doi: 10.1016/j.omega.2015.08.008
|
| [25] | J. Bentham, An introduction to the principles of morals and legislation, London: Payne and Son, 1789. |
| [26] | J. Rawls, A theory of justice, Oxford: Oxford University Press, 1973. |
Francisco Salas-Molina, David Pla-Santamaria, Maria Luisa Vercher-Ferrandiz, Ana Garcia-Bernabeu. New decision rules under strict uncertainty and a general distance-based approach[J]. AIMS Mathematics, 2023, 8(6): 13257-13275. doi: 10.3934/math.2023670
DownLoad: