{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:58:52Z","timestamp":1760147932642,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T00:00:00Z","timestamp":1678665600000},"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":["71961003","12061020","[2017] 7223","(2019) 49"],"award-info":[{"award-number":["71961003","12061020","[2017] 7223","(2019) 49"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Qian Ke He LH","award":["71961003","12061020","[2017] 7223","(2019) 49"],"award-info":[{"award-number":["71961003","12061020","[2017] 7223","(2019) 49"]}]},{"name":"Talent Introduction Foundation of Guizhou University","award":["71961003","12061020","[2017] 7223","(2019) 49"],"award-info":[{"award-number":["71961003","12061020","[2017] 7223","(2019) 49"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we try to study a class of biform games with the coalition function from the cooperation of players. For this purpose, we interpret the biform games as cooperative games by defining a characteristic function of minimax representation based on the coalition function and giving the core and Shapley value as cooperative solutions. The relations between the coalition function and the characteristic function are investigated in terms of additivity and convexity, and the properties associated with the characteristic function, such as individual rationalities and cores, are compared with the corresponding results. The relations among the solutions of the normal-form game, biform game, and cooperative game are discussed with several examples.<\/jats:p>","DOI":"10.3390\/axioms12030296","type":"journal-article","created":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T07:28:15Z","timestamp":1678692495000},"page":"296","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Cooperative Games Based on Coalition Functions in Biform Games"],"prefix":"10.3390","volume":"12","author":[{"given":"Chenwei","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8724-6449","authenticated-orcid":false,"given":"Shuwen","family":"Xiang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"},{"name":"College of Mathematical and Information Science, Guiyang University, Guiyang 550005, China"}]},{"given":"Yanlong","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China"}]},{"given":"Enquan","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Management, Guizhou University, Guiyang 550025, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1006\/game.1999.0729","article-title":"A Shapley value representation of potential games","volume":"31","author":"Ui","year":"2000","journal-title":"Game Econ. Behav."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1287\/mnsc.1060.0591","article-title":"Biform games","volume":"53","author":"Brandenburger","year":"2007","journal-title":"Manag. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s10479-019-03495-6","article-title":"On benefits of cooperation under strategic power","volume":"288","author":"Meca","year":"2020","journal-title":"Ann. Oper. Res."},{"key":"ref_4","unstructured":"Gillies, D. (1953). Some Theorems on n-Person Games. [Ph.D. Thesis, Princeton University]."},{"key":"ref_5","unstructured":"Shapley, L.S. (1953). Contributions to the Theory of Game, Princeton University Press. [2nd ed.]."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Chatterjee, K., and Samuelson, W. (2001). Game Theory and Business Applications, 189\u2013211, Kluwer Academic Publishers.","DOI":"10.1007\/b109681"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0899-8256(92)90010-P","article-title":"The hybrid solutions of an n-person game","volume":"4","author":"Zhao","year":"1992","journal-title":"Game. Econ. Behav."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1086\/261404","article-title":"The costs and benefits of ownership: A theory of vertical and lateral integration","volume":"94","author":"Grossman","year":"1986","journal-title":"J. Polit. Econ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1086\/261729","article-title":"Property rights and the nature of the firm","volume":"98","author":"Hart","year":"1990","journal-title":"J. Polit. Econ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.ijpe.2012.10.021","article-title":"Biform game: Reflection as a stochastic programming problem","volume":"142","author":"Summerfield","year":"2013","journal-title":"Int. J. Prod. Econ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104677","DOI":"10.1016\/j.eneco.2020.104677","article-title":"The promotion of community energy projects in chile and scotland: An economic approach using biform games","volume":"86","author":"Sauma","year":"2020","journal-title":"Energ. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"102072","DOI":"10.1016\/j.lrp.2021.102072","article-title":"When good deals need help getting done: Articulating side payment strategies","volume":"3","author":"Fox","year":"2021","journal-title":"Long. Range. Plann."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1090\/S0002-9947-1961-0127437-2","article-title":"The core of a cooperative game without side payments","volume":"98","author":"Aumann","year":"1961","journal-title":"T. Am. Math. Sco."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.2307\/1912051","article-title":"Endogenous formation of coalitions","volume":"51","author":"Hart","year":"1983","journal-title":"Econometrica"},{"key":"ref_16","unstructured":"Von Neumann, J., and Morgenstern, O. (2004). Theory of Games and Economic Behavior, Princeton University Press."},{"key":"ref_17","unstructured":"Myerson, R. (1991). Game Theory: Analysis of Conflict, Harvard University Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"194","DOI":"10.2307\/2525487","article-title":"A simplified bargaining model for the n-person cooperative game","volume":"4","author":"Harsanyi","year":"1963","journal-title":"Int. Econ. Rev."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Liu, C.W., Xiang, S.W., and Yang, Y.L. (2021). A biform game model with the shapley allocation functions. Mathematics, 9.","DOI":"10.3390\/math9161872"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/BF01753431","article-title":"Cores of convex games","volume":"1","author":"Shapley","year":"1971","journal-title":"Int. J. Game. Theory."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/0022-0531(81)90007-7","article-title":"Super-modularity: Applications to convex games and to the greedy algorithm for lp","volume":"25","author":"Ichiishi","year":"1981","journal-title":"J. Econ. Theory."},{"key":"ref_22","unstructured":"Osborne, M., and Rubinstein, A. (1994). A Course in Game Theory, MIT Press."},{"key":"ref_23","unstructured":"Branzei, R., Dimitrov, D., and Tijs, S. (2008). Models in Cooperative Game Theory, Springer. [2nd ed.]."},{"key":"ref_24","unstructured":"Narahari, Y. (1900). Game Theory and Mechanism Design, World Scientific Publishing."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF01719767","article-title":"Coincidence of and collinearity between game theoretic solutions","volume":"13","author":"Driessen","year":"1991","journal-title":"Oper. Res. Spektrum."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/S0165-4896(98)00017-1","article-title":"Individual rights and collective responsibility: The rights\u2013egalitarian solution","volume":"37","author":"Herrero","year":"1999","journal-title":"Math. Soc. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1353","DOI":"10.2991\/ijcis.d.191030.001","article-title":"A biform game approach to preventing block withholding attack of blockchain based on semi-cis value","volume":"12","author":"Du","year":"2019","journal-title":"Int. J. Comput. Int. 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