In many theories and applications with uncertainty, using intervals to characterize uncertainty is simple and operable. It is crucial to choose a proper order for an interval method. In general, a total order is superior to a partial order for those applications in which we have to make a final decision. Motivated by this idea, we generalized interval-valued monotone comparative statics (MCS) with a partial order to interval-valued MCS with a total order, to be more precise, with an admissible order. The generalization was not trival. We obtained a necessary and sufficient condition for MCS by a series of new concepts such as an interval-valued quasi-super-modular function and an interval-valued single crossing property with an admissible order. The same condition was only sufficient in the existing literature. Furthermore, we illustrated the efficiency of the interval-valued MCS with the lexicographical orders and XY-order, which are well-known admissible orders. Finally, we applied our results in interval games with strategic complements to get the monotonity of the best response correspondence for player $ i $.
Citation: Xiaojue Ma, Chang Zhou, Lifeng Li, Jianke Zhang. Admissible interval-valued monotone comparative statics methods applied in games with strategic complements[J]. AIMS Mathematics, 2025, 10(2): 3160-3179. doi: 10.3934/math.2025146
In many theories and applications with uncertainty, using intervals to characterize uncertainty is simple and operable. It is crucial to choose a proper order for an interval method. In general, a total order is superior to a partial order for those applications in which we have to make a final decision. Motivated by this idea, we generalized interval-valued monotone comparative statics (MCS) with a partial order to interval-valued MCS with a total order, to be more precise, with an admissible order. The generalization was not trival. We obtained a necessary and sufficient condition for MCS by a series of new concepts such as an interval-valued quasi-super-modular function and an interval-valued single crossing property with an admissible order. The same condition was only sufficient in the existing literature. Furthermore, we illustrated the efficiency of the interval-valued MCS with the lexicographical orders and XY-order, which are well-known admissible orders. Finally, we applied our results in interval games with strategic complements to get the monotonity of the best response correspondence for player $ i $.
| [1] |
T. M. Costa, Y. Chalco-Cano, R. Osuna-Gomez, W. A. Lodwich, Interval order relationships based on automorphisms and their application to interval optimization, Inform. Sciences, 615 (2022), 731-742. https://doi.org/10.1016/j.ins.2022.10.020 doi: 10.1016/j.ins.2022.10.020
|
| [2] |
T. S. Du, T. C. Zhou, On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings, Chaos Soliton. Fract., 156 (2022), 111846. https://doi.org/10.1016/j.chaos.2022.111846 doi: 10.1016/j.chaos.2022.111846
|
| [3] |
Y. T. Liu, D. Y. Xue, Y. Yang, Two types of conformable fractional grey interval models and their applications in regional eletricity consumption prediction, Chaos Soliton. Fract., 153 (2021), 111628. https://doi.org/10.1016/j.chaos.2021.111628 doi: 10.1016/j.chaos.2021.111628
|
| [4] |
J. Ye, Y. Li, Z. Z. Ma, P. P. Xiong, Novel weight-adaptive fusion grey prediction model based on interval sequences and its applications, Appl. Math. Model., 115 (2023), 803-818. https://doi.org/10.1016/j.apm.2022.11.014 doi: 10.1016/j.apm.2022.11.014
|
| [5] |
L. F. Li, Q. J. Luo, Interval-valued quasisupermodular function and monotone comparativ statics, Fuzzy Set. Syst., 476 (2024), 108772. https://doi.org/10.1016/j.fss.2023.108772 doi: 10.1016/j.fss.2023.108772
|
| [6] |
Z. S. Xu, R. R. Yager, Some geometric aggregation operators based on intuitionistic fuzzy sets, Int. J. Gen. Syst., 35 (2006), 417-433. https://doi.org/10.1080/03081070600574353 doi: 10.1080/03081070600574353
|
| [7] |
H. Bustince, J. Fernandez, A. Kolesárová, R. Mesiar, Generation of linear orders for intervals by means of aggregation functions, Fuzzy Set. Syst., 220 (2013), 69-77. https://doi.org/10.1016/j.fss.2012.07.015 doi: 10.1016/j.fss.2012.07.015
|
| [8] |
H. Bustince, M. Galar, B. Bedregal, A. Kolesárová, R. Mesiar, A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications, IEEE T. Fuzzy Syst., 21 (2013), 1150-1162. https://doi.org/10.1109/TFUZZ.2013.2265090 doi: 10.1109/TFUZZ.2013.2265090
|
| [9] |
L. De Miguel, H. Bustince, J. Fernandez, E. Induráin, A. Kolesárová, R. Mesiar, Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making, Inform. Fusion, 27 (2016), 189-197. https://doi.org/10.1016/j.inffus.2015.03.004 doi: 10.1016/j.inffus.2015.03.004
|
| [10] |
L. De Miguel, M. Sesma-Sara, M. Elkano, M. Asiain, H. Bustince, An algorithm for group decision making using n-dimensional fuzzy sets, admissible orders and OWA operators, Inform. Fusion, 37 (2017), 126-131. https://doi.org/10.1016/j.inffus.2017.01.007 doi: 10.1016/j.inffus.2017.01.007
|
| [11] |
F. Santana, B. Bedregal, P. Viana, H. Bustince, On admissible orders over closed subintervals of [0, 1], Fuzzy Set. Syst., 399 (2020), 44-54. https://doi.org/10.1016/j.fss.2020.02.009 doi: 10.1016/j.fss.2020.02.009
|
| [12] |
H. Bustince, C. Marco-Detchart, J. Fernandez, C. Wagner, J. M. Garibaldi, Z. Takácˇ, Similarity between interval-valued fuzzy sets taking into account the width of the intervals and admissible orders, Fuzzy Set. Syst., 390 (2020), 23-47. https://doi.org/10.1016/j.fss.2019.04.002 doi: 10.1016/j.fss.2019.04.002
|
| [13] |
J. V. Riera, S. Massanet, H. Bustince, J. Fernandez, On admissible orders on the set of discrete fuzzy numbers for application in decision making problems, Mathematics, 9 (2021), 95. https://doi.org/10.3390/math9010095 doi: 10.3390/math9010095
|
| [14] |
N. Zumelzu, B. Bedregal, E. Mansilla, H. Bustince, R. Díaz, Admissible orders on fuzzy numbers, IEEE T. Fuzzy Syst., 30 (2022), 4788-4799. https://doi.org/10.1109/TFUZZ.2022.3160326 doi: 10.1109/TFUZZ.2022.3160326
|
| [15] |
D. C. Li, Y. Leung, W. Z. Wu, Multiobjective interval linear programming in admissible-order vector space, Inform. Sciences, 486 (2019), 1-19. https://doi.org/10.1016/j.ins.2019.02.012 doi: 10.1016/j.ins.2019.02.012
|
| [16] |
L. F. Li, Optimality conditions for nonlinear optimization problems with interval-valued objective function in admissible orders, Fuzzy Optim. Decis. Making, 22 (2022), 247-265. https://doi.org/10.1007/s10700-022-09391-2 doi: 10.1007/s10700-022-09391-2
|
| [17] |
U. Bentkowska, H. Bustince, A. Jurio, M. Pagola, B. Pekala, Decision making with an interval-valued fuzzy preference relation and admissible orders, Appl. Soft Comput., 35 (2015), 792-801. https://doi.org/10.1016/j.asoc.2015.03.012 doi: 10.1016/j.asoc.2015.03.012
|
| [18] |
P. D. Liu, Y. Y. Li, P. Wang, Social trust-driven consensus reaching model for multiattribute group decision making: Exploring social trust network completeness, IEEE T. Fuzzy Syst., 31 (2023), 3040-3054. https://doi.org/10.1109/TFUZZ.2023.3241145 doi: 10.1109/TFUZZ.2023.3241145
|
| [19] |
P. D. Liu, Y. Li, X. H. Zhang, W. Pedrycz, A multiattribute group decision-making method with probabilistic linguistic information based on an adaptive consensus reaching model and evidential reasoning, IEEE T. Cybern., 53 (2023), 1905-1919. https://doi.org/10.1109/TCYB.2022.3165030 doi: 10.1109/TCYB.2022.3165030
|
| [20] |
J. Derrac, F. Chiclana, S. García, F. Herrera, Evolutionary fuzzy k-nearest neighbors algorithm using interval-valued fuzzy sets, Inform. Sciences, 329 (2016), 144-163. https://doi.org/10.1016/j.ins.2015.09.007 doi: 10.1016/j.ins.2015.09.007
|
| [21] |
S. Zeraatkar, F. Afsari, Interval-valued fuzzy and intuitionistic fuzzy-KNN for imbalanced data classification, Expert Syst. Appl., 184 (2021), 115510. https://doi.org/10.1016/j.eswa.2021.115510 doi: 10.1016/j.eswa.2021.115510
|
| [22] |
T. da Cruz Asmus, J. A. Sanz, G. P. Dimuro, B. Bedregal, J. Fernández, H. Bustince, N-dimensional admissibly ordered interval-valued overlap functions and its influence in interval-valued fuzzy-rule-based classification systems, IEEE T. Fuzzy Syst., 30 (2022), 1060-1072. https://doi.org/10.1109/TFUZZ.2021.3052342 doi: 10.1109/TFUZZ.2021.3052342
|
| [23] |
X. X. Wu, H. Tang, Z. Y. Zhu, L. T. Liu, G. R. Chen, M. S. Yang, Nonlinear strict distance and similarity measures for intuitionistic fuzzy sets with applications to pattern classification and medical diagnosis, Sci. Rep., 13 (2023), 13918. https://doi.org/10.1038/s41598-023-40817-y doi: 10.1038/s41598-023-40817-y
|
| [24] | M. Pagola, A. Jurio, E. Barrenechea, J. Fernández, H. Bustince, Interval-valued fuzzy clustering, In: Proceedings of the 2015 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology, 2015, 1288-1294. https://doi.org/10.2991/ifsa-eusflat-15.2015.182 |
| [25] | P. Sussner, L. C. Carazas, An approach towards image edge detection based on interval-valued fuzzy mathematical morphology and admissible orders, In: Proceedings of the 11th Conference of the European, Society for Fuzzy Logic and Technology, 2019,690-697. https://doi.org/10.2991/eusflat-19.2019.96 |
| [26] | P. Milgrom, C. Shannon, Monotone comparative statics, Econometrica, 62 (1994), 157-180. |
| [27] |
D. M. Topkis, Minimizing a submodular function on a lattice, Oper. Res., 26 (1978), 209-376. https://doi.org/10.1287/opre.26.2.305 doi: 10.1287/opre.26.2.305
|
| [28] |
V. Xavier, Nash equilibrium with strategic complementarities, J. Math. Econ., 19 (1990), 305-321. https://doi.org/10.1016/0304-4068(90)90005-T doi: 10.1016/0304-4068(90)90005-T
|
| [29] | P. Milgrom, J. Roberts, The economics of modern manufacturing: Technology, strategy, and organization, Am. Econ. Rev., 80 (1990), 511-528. |
| [30] | P. Milgrom, J. Roberts, Rationalizability, learning and equilibrium in games with strategic complementarities, Econometrica, 58 (1990), 1255-1277. |
| [31] | P. Milgrom, J. Roberts, Comparing equilibria, Am. Econ. Rev., 84 (1994), 441-459. |
| [32] |
S. Athey, Monotone comparative statics under uncertainty, Q. J. Econ., 117 (2002), 187-223. https://doi.org/10.1162/003355302753399481 doi: 10.1162/003355302753399481
|
| [33] |
B. H. Strulovici, T. A. Weber, Monotone comparative statics: Geometric approach, J. Optim. Theory Appl., 137 (2008), 641-673. https://doi.org/10.1007/s10957-007-9339-1 doi: 10.1007/s10957-007-9339-1
|
| [34] |
A. C. Barthel, T. Sabarwal, Directional monotone comparative statics, Econ. Theory, 66 (2018), 557-591. https://doi.org/10.1007/s00199-017-1079-3 doi: 10.1007/s00199-017-1079-3
|
| [35] | T. Sabarwal, A unified approach to games with strategic complements and substitutes, In: Monotone games, Chan: Palgrave Pivot, 2021. |
| [36] | D. M. Topkis, Supermodularity and complementarity, Princeton: Princeton University Press, 1998. |
| [37] |
E. P. Klement, R. Mesiar, E. Pap, Archimax copulas and invariance under transformations, C. R. Math. Acad. Sci. Paris Math., 340 (2005), 755-758. https://doi.org/10.1016/j.crma.2005.04.012 doi: 10.1016/j.crma.2005.04.012
|
| [38] |
J. J. Arias-Garcia, B. De Baets, On the lattice structure of the set of supermodular quasi-copulas, Fuzzy Set. Syst., 354 (2019), 74-83. https://doi.org/10.1016/j.fss.2018.03.013 doi: 10.1016/j.fss.2018.03.013
|
| [39] |
C. L. Luo, X. Y. Zhou, B. Lev, Core Shapley value, nucleolus and Nash bargaining solution: A survey of recent developments and applications in operations management, Omega, 110 (2022), 102638. https://doi.org/10.1016/j.omega.2022.102638 doi: 10.1016/j.omega.2022.102638
|
| [40] | R. Branzei, D. Dimitrov, S. Tijs, Shapely-like values for interval bankruptcy games, Econ. Bull., 3 (2002), 1-8. |
| [41] |
A. Chakeri, F. Sheikholeslam, Fuzzy Nash equilibriums in crisp and fuzzy games, IEEE T. Fuzzy Syst., 21 (2013), 171-176. https://doi.org/10.1109/TFUZZ.2012.2203308 doi: 10.1109/TFUZZ.2012.2203308
|
| [42] |
J. C. Fígueroa-Garcia, A. Mehra, S. Chandra, Optimal solutions for group matrix games involving interval-valued fuzzy numbers, Fuzzy Sets Syst., 362 (2019), 55-70. https://doi.org/10.1016/j.fss.2018.07.001 doi: 10.1016/j.fss.2018.07.001
|
| [43] |
K. R. Liang, D. F. Li, K. W. Li, J. C. Liu, An interval noncooperative-cooperative biform game model based on weighted equal contribution division values, Inform. Sciences, 619 (2023), 172-192. https://doi.org/10.1016/j.ins.2022.11.016 doi: 10.1016/j.ins.2022.11.016
|
| [44] |
X. Wang, K. L. Teo, Generalized Nash equilibrium problem over a fuzzy strategy set, Fuzzy Set. Syst., 434 (2022), 172-184. https://doi.org/10.1016/j.fss.2021.06.006 doi: 10.1016/j.fss.2021.06.006
|
| [45] |
Q. C. X. Zhang, L. Shu, B. C. Jiang, Moran process in evolutionary game dynamics with interval payoffs and its application, Appl. Math. Comput., 446 (2023), 127875. https://doi.org/10.1016/j.amc.2023.127875 doi: 10.1016/j.amc.2023.127875
|
| [46] |
R. Branzei, S. Z. A. Gok, O. Branzei, Cooperative games under interval uncertainty: On the convexity of the interval undominated cores, Cent. Eur. J. Oper. Res., 19 (2011), 523-532. https://doi.org/10.1007/s10100-010-0141-z doi: 10.1007/s10100-010-0141-z
|
| [47] | B. A. Davey, H. A. Priestley, Introduction to lattices and order, Cambridge: Cambridge University Press, 2002. |
| [48] |
M. Boczek, L. S. Jin, M. Kaluszka, Interval-valued seminormed fuzzy operators based on admissible orders, Inform. Sciences, 574 (2021), 96-110. https://doi.org/10.1016/j.ins.2021.05.065 doi: 10.1016/j.ins.2021.05.065
|
| [49] |
P. Sussner, L. C. Carazas, Construction of $K_{\alpha}$-orders including admissible ones on classes of discrete intervals, Fuzzy Set. Syst., 480 (2024), 108857. https://doi.org/10.1016/j.fss.2024.108857 doi: 10.1016/j.fss.2024.108857
|
| [50] |
L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Set. Syst., 161 (2010), 1564-1584. https://doi.org/10.1016/j.fss.2009.06.009 doi: 10.1016/j.fss.2009.06.009
|
| [51] | M. Hukuhara, Intégration des applications measurables dont la valeur est un compact convexe, Funkcialaj Ekvacioj, 10 (1967), 205-223. |
| [52] | L. Stefanini, On the generalized LU-fuzzy derivative and fuzzy differential equations, In: Proceedings of the 2007 IEEE International Conference on Fuzzy Systems, 2007,710-715. https://doi.org/10.1109/FUZZY.2007.4295453 |
| [53] |
Y. Chalco-Cano, A. Rufian-Lizana, H. Roman-Flores, M. D. Jiménez-Gamero, Calculus for interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Set. Syst., 219 (2013), 49-67. https://doi.org/10.1016/j.fss.2012.12.004 doi: 10.1016/j.fss.2012.12.004
|
| [54] |
L. Stefanini, M. Arana-Jiménez, Karush-Kuhn-Tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability, Fuzzy Set. Syst., 362 (2019), 1-34. https://doi.org/10.1016/j.fss.2018.04.009 doi: 10.1016/j.fss.2018.04.009
|
| [55] |
R. Osuna-Gómez, T. M. Costa, Y. Chalco-Cano, B. Hernández-Jiménez, Quasilinear approximation for interval-valued functions via generalized Hukuhara differentiability, Comp. Appl. Math., 41 (2022), 149. https://doi.org/10.1007/s40314-021-01746-6 doi: 10.1007/s40314-021-01746-6
|
| [56] |
D. Qiu, Y. Yu, Some notes on the switching points for the generalized Hukuhara differentiability of interval-valued functions, Fuzzy Set. Syst., 453 (2023), 115-129. https://doi.org/10.1016/j.fss.2022.04.004 doi: 10.1016/j.fss.2022.04.004
|
| [57] |
P. Roy, G. Panda, D. Qiu, Gradient-based descent linesearch to solve interval-valued optimization problems under gH-differentiability with application to finance, J. Comput. Appl. Math., 436 (2024), 115402. https://doi.org/10.1016/j.cam.2023.115402 doi: 10.1016/j.cam.2023.115402
|
| [58] |
A. F. Veinott, Representation of general and polyhedral subsemilattices and sublattices of product spaces, Linear Algebra Appl., 114-115 (1989), 681-704. https://doi.org/10.1016/0024-3795(89)90488-6 doi: 10.1016/0024-3795(89)90488-6
|
| [59] |
L. Bruttel, M. Bulutay, C. Cornand, F. Heinemann, A. Zylbersztejn, Measuring strategic-uncertainty attitudes, Exp. Econ., 26 (2023), 522-549. https://doi.org/10.1007/s10683-022-09779-2 doi: 10.1007/s10683-022-09779-2
|
| [60] |
F. Alvarez, F. Lippi, P. Souganidis, Price setting with strategic complementarities as a mean field game, Econometrica, 91 (2023), 2005-2039. https://doi.org/10.3982/ECTA20797 doi: 10.3982/ECTA20797
|
| [61] |
Q. Li, B. Pi, M. Y. Feng, J. Kurths, Open data in the digital economy: An evolutionary game theory perspective, IEEE T. Comput. Soc. Sys., 11 (2024), 3780-3791. https://doi.org/10.1109/TCSS.2023.3324087 doi: 10.1109/TCSS.2023.3324087
|
| [62] |
M. Y. Feng, B. Pi, L. J. Deng, J. Kurths, An evolutionary game with the game transitions based on the Markov process, IEEE T. Sys. Man Cy-S., 54 (2024), 609-621. https://doi.org/10.1109/TSMC.2023.3315963 doi: 10.1109/TSMC.2023.3315963
|
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Xiaojue Ma, Chang Zhou, Lifeng Li, Jianke Zhang. Admissible interval-valued monotone comparative statics methods applied in games with strategic complements[J]. AIMS Mathematics, 2025, 10(2): 3160-3179. doi: 10.3934/math.2025146
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