Well-Posedness of Weakly Cooperative Equilibria for Multi-objective Population Games

doi:10.1007/s40305-023-00526-9

Journal of the Operations Research Society of China ›› 2026, Vol. 14 ›› Issue (1): 293-308.doi: 10.1007/s40305-023-00526-9

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Well-Posedness of Weakly Cooperative Equilibria for Multi-objective Population Games

Tao Chen¹, Kun-Ting Chen¹, Yu Zhang²

1 School of Economics, Yunnan University of Finance and Economics, Kunming 650221, Yunnan, China;
2 School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, Yunnan, China

Received:2023-01-20 Revised:2023-11-09 Online:2026-03-30 Published:2026-03-16
Contact: Yu Zhang E-mail:zhangyu198606@sina.com
Supported by:
This work was supported by the National Natural Science Foundation of China (No.11901511),the Yunnan Fundamental Research Projects (No.202101AT070216),Yunnan Provincial Department of Education (No.2022J0601),Xingdian Talents Project Foundation for Young Scholar in Yunnan Province.

Abstract

Abstract: In this paper, we construct a bounded rationality function of weakly cooperative equilibrium point of multi-objective population games (WCEPOMPG) by virtue of nonlinear scalarization function. We investigate lower semicontinuity of the bounded rationality function and present a relationship between the level set of zero of the bounded rationality function and WCEPOMPG. By applying these properties, we first consider the generalized well-posedness for WCEPOMPG in the setting of multiplevalued assumption of solution set. Then, we obtain the uniqueness of solution set on a dense residual set with the stable assumption of vector payoff functions. Moreover, we investigate the well-posedness of WCEPOMPG on the dense residual set. As a special case, we obtain the generalized well-posedness and well-posedness of cooperative equilibria for single-objective population games, respectively.

Key words: Cooperative equilibria, Population game, Well-posedness, Uniqueness

CLC Number:

Tao Chen, Kun-Ting Chen, Yu Zhang. Well-Posedness of Weakly Cooperative Equilibria for Multi-objective Population Games[J]. Journal of the Operations Research Society of China, 2026, 14(1): 293-308.

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References

[1] Nash, J.: Noncooperative games. Dissertation, Princeton University, Princeton (1950)
[2] Sandholm, W.H.: Population Games and Evolutionary Dynamics. MIT Press, Cambridge (2010)
[3] Hofbauer, J., Sandholm, W.H.: Stable games and their dynamics. J. Econ. Theory 44, 1665-1693(2009)
[4] Jia, W.S., Qiu, X.L., Peng, D.T.: The generic uniqueness and well-Posedness of Nash equilibria for stable population games. J. Oper. Res. Soc. China 9, 455-464(2021)
[5] Jia,W.S., Xiang, S.W., He, J.H., Yang, Y.L.: Existence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leader-follower games. J. Global Optim. 61, 397-405(2015)
[6] Lin, Z.: Essential components of the set of weakly Pareto-Nash equilibrium points for multiobjective generalized games in two different topological spaces. J. Optim. Theory Appl. 124, 387-405(2005)
[7] Scalzo, V.: Hadamard well-posedness in discontinuous non-cooperative games. J. Math. Anal. Appl. 360, 697-703(2009)
[8] Scalzo, V.: Continuity properties of the Nash equilibrium correspondence in a discontinuous setting. J. Math. Anal. Appl. 473, 1270-1279(2019)
[9] Yang, Z., Zhang, H.Q.: Essential stability of cooperative equilibria for population games. Optim. Lett. 13, 1573-1582(2019)
[10] Kajii, A.: A generalization of Scarf’s theorem: an α-core existence theorem without transitivity or completeness. J. Econ. Theory 56, 194-205(1992)
[11] Yang, Z., Zhang, H.Q.: NTU core, TU core and strong equilibria of coalitional population games with infinitely many pure strategies. Theor. Decis. 87, 155-170(2019)
[12] Zhang, H.Q.: The stability of cooperative equilibria for population games under bounded rationality. Acta. Math. Appl. Sin. 45, 433-447(2021)
[13] Yang, G.H., Yang, H.: Stability of weakly Pareto-Nash equilibria and Pareto-Nash equilibria for multiobjective population games. Set-valued Var. Anal. 25, 427-439(2017)
[14] Chen, T., Chang, S.S., Zhang, Y.: Existence and stability of weakly cooperative equilibria and strong cooperative equilibria of multi-objective population games. Axioms 11, 196(2022)
[15] Hung, N.V., Keller, A.A.: Existence and generic stability conditions of equilibrium points to controlled systems for n-player multiobjective generalized games using the Kakutani-Fan-Glicksberg fixed-point theorem. Optim. Lett. 16, 1477-1493(2022)
[16] Zhang, Y., Sun, X.K.: On the α-core of set payoffs games. Ann. Oper. Res. (2022). https://doi.org/10.1007/s10479-022-05090-8
[17] Hung, N.V., Tam, V.M., O’Regan, D., Cho, Y.J.: A new class of generalized multiobjective games in bounded rationality with fuzzy mappings: structural (λ, ε)-stability and (λ, ε)-robustness to ε- equilibria. J. Comput. Appl. Math. 372, 112735(2020)
[18] Yang, Z., Ju, Y.: Existence and generic stability of cooperative equilibria for multi-leader-multifollower games. J. Global Optim. 65(3), 563-573(2016)
[19] Zhang, W.Y., Zeng, J., Hu, R.T.: Well-posedness and existence for the weak multicriteria Nash equilibrium of multicriteria games. Optimization (2022). https://doi.org/10.1080/02331934.2022.2054340
[20] Chen, J.W., Cho, Y.J., Wan, Z.: The existence of solutions and well-posedness for bilevel mixed equilibrium problems in Banach spaces. Taiwan J. Math. 17, 725-748(2013)
[21] Chen, J.W., Wan, Z.P., Cho, Y.J.: Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems. Math. Method. Oper. Res. 77, 33-64(2013)
[22] Chen, J.W., Wang, G.G., Ou, X.Q., Zhang, W.Y.: Continuity of solutions mappings of parametric set optimization problems. J. Ind. Manag. Optim. 16, 25-36(2020)
[23] Chen, J.W., Wan, Z.P., Yuan, L.Y.: Existence of solutions and α-well-posedness for a system of constrained set-valued variational inequalities. Numer. Algebra Contr. Optim. 3, 567-581(2013)
[24] Hung, N.V.: LP well-posed controlled systems for bounded quasi-equilibrium problems and their application to traffic networks. J. Comput. Appl. Math. 401, 113792(2022)
[25] Hung, N.V., Dai, L.X., Köbis, E., Yao, J.C.: The generic stability of solutions for vector quasiequilibrium problems on Hadamard manifolds. J. Nonlinear Var. Anal. 4, 427-438(2020)
[26] Li, S.J., Li, M.H.: Levitin-Polyak well-posedness of vector equilibrium problems. Math. Method. Oper. Res. 69, 125-140(2009)
[27] Li, S.J., Zhang, W.Y.: Hadamard well-posed vector optimization problems. J. Global Optim. 46, 383-393(2010)
[28] Zeng, J., Li, S.J., Zhang, W.Y., Xue, X.W.: Hadamard well-posedness for a set-valued optimization problem. Optim. Lett. 7, 559-573(2013)
[29] Yang, Z., Meng, D.W.: Hadamard well-posedness of the α-core. J. Math. Anal. Appl. 452, 957-969(2017)
[30] Li, Y.B., Jia, W.S.: Existence and well-posedness of the α-core for generalized fuzzy games. Fuzzy Set Syst. 458, 108-117(2023)
[31] Yu, J.: The Continued Study on Game Theory and Nonlinear Analysis. Science Press, Beijing (2011)
[32] Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
[33] Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc., New York (1991)
[34] Fort, M.K., Jr.: Points of continuity of semicontinuous functions. Publ. Math. Debrecen 2, 100-102(1951)
[35] Chen, C.R., Li, M.H.: Holder continuity of solutions to Parametric vector euilibrium problems with nonlinear scalarization. Numer. Func. Anal. Optim. 35, 685-707(2014)
[36] Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297-320(1990)

Well-Posedness of Weakly Cooperative Equilibria for Multi-objective Population Games

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