{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T02:08:56Z","timestamp":1769825336533,"version":"3.49.0"},"reference-count":45,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12461054"],"award-info":[{"award-number":["12461054"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2025089"],"award-info":[{"award-number":["2025089"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Key Foundation of Guizhou Province","award":["12461054"],"award-info":[{"award-number":["12461054"]}]},{"name":"Science and Technology Key Foundation of Guizhou Province","award":["2025089"],"award-info":[{"award-number":["2025089"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Our work proves the existence of weakly Pareto\u2013Nash equilibrium (PNE) in multiobjective games (MGs) with infinitely many players. First, we demonstrate the existence of weakly PNE under compactness assumptions by using the intersection theorem. Then, we extend the intersection theorem to the non-compact case and obtain a new intersection theorem. Finally, we prove the existence of weakly PNE in MGs with infinitely many players without compactness assumptions.<\/jats:p>","DOI":"10.3390\/axioms14070517","type":"journal-article","created":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T06:03:13Z","timestamp":1751868193000},"page":"517","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Existence of Weakly Pareto\u2013Nash Equilibrium for Multiobjective Games with Infinitely Many Players"],"prefix":"10.3390","volume":"14","author":[{"given":"Huaxin","family":"Chen","sequence":"first","affiliation":[{"name":"State Key Laboratory of Public Big Data, School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5596-5091","authenticated-orcid":false,"given":"Wensheng","family":"Jia","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Public Big Data, School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"286","DOI":"10.2307\/1969529","article-title":"Non-cooperative games","volume":"54","author":"Nash","year":"1951","journal-title":"Ann. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/0304-4068(75)90002-6","article-title":"Equilibrium in abstract economies without ordered preferences","volume":"2","author":"Shafer","year":"1975","journal-title":"J. Math. Econ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"39","DOI":"10.2307\/1913732","article-title":"Markets with a continuum of traders","volume":"32","author":"Aumann","year":"1964","journal-title":"Econometrica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1006\/jeth.1997.2292","article-title":"On the existence of pure strategy equilibria in games with a continuum of players","volume":"76","author":"Khan","year":"1997","journal-title":"J. Econ. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.jmateco.2008.08.005","article-title":"Existence of Nash equilibria in large games","volume":"45","author":"Noguchi","year":"2009","journal-title":"J. Math. Econ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/0304-4068(83)90041-1","article-title":"Existence of maximal elements and equilibria in linear topological spaces","volume":"12","author":"Yannelis","year":"1983","journal-title":"J. Math. Econ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/S0362-546X(98)00055-8","article-title":"The study of equilibria for abstract economics in topological vector spaces\u2014A unified approach","volume":"37","author":"Yuan","year":"1999","journal-title":"Nonlinear Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11784-021-00927-4","article-title":"A unified approach to the Nash equilibrium existence in large games from finitely many players to infinitely many players","volume":"24","author":"Yang","year":"2022","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/pjm.1956.6.1","article-title":"An analog of the minimax theorem for vector payoffs","volume":"6","author":"Blackwell","year":"1956","journal-title":"Pac. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1002\/nav.3800060107","article-title":"Equilibrium points in games with vector payoffs","volume":"6","author":"Shapley","year":"1959","journal-title":"Nav. Res. Logist. Q."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/BF00940586","article-title":"Existence of a Pareto equilibrium","volume":"79","author":"Wang","year":"1993","journal-title":"J. Optim. Theory Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/S0898-1221(98)00053-4","article-title":"The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods","volume":"35","author":"Yu","year":"1998","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/S0893-9659(02)80006-4","article-title":"Essential components of the set of weakly Pareto-Nash equilibrium points","volume":"15","author":"Yang","year":"2002","journal-title":"Appl. Math. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.jmaa.2004.06.042","article-title":"The essential component of the set of equilibrium points for set-valued maps","volume":"300","author":"Yu","year":"2004","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Jia, W.S., Qiu, X.L., and Peng, D.T. (2020). An approximation theorem for vector equilibrium problems under bounded rationality. Mathematics, 8.","DOI":"10.3390\/math8010045"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2680","DOI":"10.1016\/j.na.2010.06.048","article-title":"On the stability of the solution for multiobjective generalized games with the payoffs perturbed","volume":"73","author":"Song","year":"2010","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1007\/s11590-021-01786-1","article-title":"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","volume":"16","author":"Hung","year":"2022","journal-title":"Optim. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/s10898-014-0178-y","article-title":"Existence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leader\u2013follower games","volume":"61","author":"Jia","year":"2015","journal-title":"J. Glob. Optim."},{"key":"ref_19","first-page":"1689732","article-title":"Generic stability of the weakly Pareto-Nash equilibrium with strategy transformational barriers","volume":"2022","author":"Liu","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"112735","DOI":"10.1016\/j.cam.2020.112735","article-title":"A new class of generalized multiobjective games in bounded rationality with fuzzy mappings: Structural (\u03bb,\u03b5)-stability and (\u03bb,\u03b5)-robustness to equilibria","volume":"372","author":"Hung","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.fss.2021.11.012","article-title":"Essential stability of fuzzy equilibria for generalized multiobjective games with fuzzy constraint mappings","volume":"447","author":"Zhou","year":"2022","journal-title":"Fuzzy Sets Syst."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Li, W., Li, D.Y., Feng, Y.Q., and Zou, D. (2023). Fuzzy Weighted Pareto\u2013Nash Equilibria of Multi-Objective Bi-Matrix Games with Fuzzy Payoffs and Their Applications. Mathematics, 11.","DOI":"10.3390\/math11204266"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s10957-024-02471-w","article-title":"Existence and stability of fuzzy slightly altruistic equilibrium for a class of generalized multiobjective fuzzy games","volume":"203","author":"Tian","year":"2024","journal-title":"J. Optim. Theory Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4899","DOI":"10.1080\/00036811.2022.2147066","article-title":"Existence and stability of weakly Pareto-Nash equilibria for discontinuous multiobjective games","volume":"102","author":"Mou","year":"2023","journal-title":"Appl. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4167","DOI":"10.22436\/jnsa.009.06.59","article-title":"Stability of weighted Nash equilibrium for multiobjective population games","volume":"9","author":"Yang","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s11228-016-0391-6","article-title":"Stability of weakly Pareto-Nash equilibria and Pareto-Nash equilibria for multiobjective population games","volume":"25","author":"Yang","year":"2017","journal-title":"Set-Valued Var. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Yang, G.H., Li, C.C., Pi, J.X., Wang, C., Wu, W.J., and Yang, H. (2021). Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games. Mathematics, 9.","DOI":"10.3390\/math9010099"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Chen, T., Chang, S.S., and Zhang, Y. (2022). Existence and stability of weakly cooperative equilibria and strong cooperative equilibria of multi-objective population games. Axioms, 11.","DOI":"10.3390\/axioms11050196"},{"key":"ref_29","first-page":"1","article-title":"An Approximation Theorem and Generic Uniqueness of Weakly Pareto-Nash Equilibrium for Multiobjective Population Games","volume":"2024","author":"Chen","year":"2024","journal-title":"J. Oper. Res. Soc. China"},{"key":"ref_30","first-page":"3925","article-title":"Sur un th\u00e9or\u00e8me minimax","volume":"259","author":"Fan","year":"1964","journal-title":"CR Acad. Sci. Paris"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/0022-247X(69)90058-4","article-title":"On sets with convex sections","volume":"27","author":"Ma","year":"1969","journal-title":"J. Math. Anal. Appl."},{"key":"ref_32","first-page":"1840","article-title":"Intersection theorems and their applications to Berge equilibria","volume":"182","author":"Abalo","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_33","first-page":"1606","article-title":"The existence of Nash equilibrium in big games","volume":"30","author":"Yang","year":"2010","journal-title":"J. Syst. Sci. Math. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"473","DOI":"10.2140\/pjm.1985.119.473","article-title":"Noncompact sets with convex sections","volume":"119","author":"Shih","year":"1985","journal-title":"Pac. J. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3159","DOI":"10.1016\/j.na.2007.03.010","article-title":"A new generalization of the Yannelis-Prabhakar equilibrium existence theorem for abstract economies","volume":"68","author":"Hou","year":"2008","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"281","DOI":"10.37193\/CJM.2019.03.03","article-title":"Intersection theorems with applications in set-valued equilibrium problems and minimax theory","volume":"35","author":"Balaj","year":"2019","journal-title":"Carpath. J. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1007\/s10957-021-01927-7","article-title":"An intersection theorem for topological vector spaces and applications","volume":"191","author":"Fierro","year":"2021","journal-title":"J. Optim. Theory Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1017\/S0305004123000075","article-title":"Intersection theorems for finite general linear groups","volume":"175","author":"Ernst","year":"2023","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_39","unstructured":"Fudenberg, D., and Tirole, J. (1991). Game Theory, MIT Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1006\/jmaa.1996.0429","article-title":"Non-compact Pareto equilibria for multiobjective games","volume":"204","author":"Yuan","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"ref_41","unstructured":"Aubin, J.P., and Ekeland, I. (1984). Applied Nonlinear Analysis, Wiley."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF02191669","article-title":"Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions","volume":"81","author":"Tanaka","year":"1994","journal-title":"J. Optim. Theory Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1007\/BF01458545","article-title":"Some properties of convex sets related to fixed point theorems","volume":"266","author":"Fan","year":"1984","journal-title":"Math. Ann."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1017\/S0004972700011849","article-title":"A selection theorem and its applications","volume":"46","author":"Ding","year":"1992","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"301","DOI":"10.4064\/fm-15-1-301-309","article-title":"Sur les espaces complets","volume":"15","author":"Kuratowski","year":"1930","journal-title":"Fundam. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/517\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:04:31Z","timestamp":1760033071000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/517"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,4]]},"references-count":45,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070517"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070517","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,4]]}}}