An Approximation Theorem and Generic Uniqueness of Weakly Pareto-Nash Equilibrium for Multiobjective Population Games

doi:10.1007/s40305-024-00565-w

Abstract

Abstract: We mainly study an approximation theorem and generic uniqueness of weakly ParetoNash equilibrium (PNE) for multiobjective population games. First, we define the concept of approximate weakly PNE of multiobjective population games. Then, an approximation theorem is proved under very mild conditions. Furthermore, we prove the generic uniqueness of weakly PNE for multiobjective population games by establishing a function space. Finally, we obtain the generic uniqueness of weakly PNE for multiobjective population games in the case of function perturbation. The above results are new and improve the related results of the recent literature.

Key words: Multiobjective population game, Approximation theorem, Weakly Pareto-Nash equilibrium

CLC Number:

Hua-Xin Chen, Wen-Sheng Jia. An Approximation Theorem and Generic Uniqueness of Weakly Pareto-Nash Equilibrium for Multiobjective Population Games[J]. Journal of the Operations Research Society of China, 2026, 14(2): 591-602.

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URL: https://www.jorsc.shu.edu.cn/EN/10.1007/s40305-024-00565-w

https://www.jorsc.shu.edu.cn/EN/Y2026/V14/I2/591

References

[1] Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. Springer, Berlin (2006)
[2] Anderlini, L., Canning, D.: Structural stability implies robustness to bounded rationality. J. Econ. Theory 101, 395–422(2001)
[3] 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, 196–206(2022)
[4] Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2005)
[5] Cirant, M.: Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. 103, 1294–1315(2015)
[6] Fort, M.K.: Points of continuity of semicontinuous functions. Publ. Math. Debrecen 2, 100–102(1951)
[7] Hofbauer, J., Sandholm, W.H.: Stable games and their dynamics. J. Econ. Theory 144(4), 1665–1693(2009)
[8] 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)
[9] Martinez-Piazuelo, J., Quijano, N., Ocampo-Martinez, C.: A payoff dynamics model for generalized Nash equilibrium seeking in population games. Automatica 140, 110227(2022)
[10] Miyazaki, Y., Azuma, H.: (λ , ε )-stable model and essential equilibria. Math. Social Sci. 65, 85–91(2013)
[11] Nash, J.: Noncooperative games. Dissertation, Princeton University. Princeton (1950)
[12] Rubinstein, A.: Modeling Bounded Rationality. MIT Press, Cambridge (1998)
[13] Sandholm, W.H.: Population Games and Evolutionary Dynamics. MIT Press, Cambridge (2010)
[14] Simon, H.A.: A behavioral model of rational choice. Q. J. Econ. 68, 99–118(1955)
[15] Simon, H.A., Behavior, A.: A study of decision-making processes in administrative organizations. Administrative Behavior (1957)
[16] Wang, G.L., Liu, Z.L., Yang, H., et al.: Existence of equilibrium solutions for multi-objective population games with fuzzy parameters. Fuzzy Sets Syst. 473, 108698(2023)
[17] Yang, Z., Zhang, H.: Essential stability of cooperative equilibria for population games. Optim. Lett. 13, 1573–1582(2019)
[18] Yang, Z., Zhang, H.: NTU core, TU core and strong equilibria of coalitional population games with infinitely many pure strategies. Theor. Decis. 87, 155–170(2019)
[19] 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)
[20] Yang, G.H., Yang, H., Song, Q.Q.: Stability of weighted Nash equilibrium for multiobjective population games. J. Nonlinear Sci. Appl. 9, 4167–4176(2016)
[21] Yu, C., Yu, J.: On structural stability and robustness to bounded rationality. Nonlinear Anal. TMA 65, 583–592(2006)
[22] Yu, C., Yu, J.: Bounded rationality in multiobjective games. Nonlinear Anal. TMA 67, 930–937(2007)
[23] Yu, J., Yang, H., Yu, C.: Structural stability and robustness to bounded rationality for non-compact cases. J. Glob. Optim. 44, 149–157(2009)
[24] Yu, J., Yang, Z., Wang, N.F.: Further results on structural stability and robustness to bounded rationality. J. Math. Econ. 67, 49–53(2016)
[25] Yu, J.: Bounded Rationality and the Stability of Equilibrium Point Set in Game Theory. Science Press, Beijing (2017)
[26] Yu, J.: Essential equilibria of n-person nocooperative games. J. Math. Econ. 31, 361–372(1999)