{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:51:53Z","timestamp":1760244713736,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"24","license":[{"start":{"date-parts":[[2022,12,11]],"date-time":"2022-12-11T00:00:00Z","timestamp":1670716800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (Portuguese Foundation for Science and Technology)","award":["PTDC\/MAT-APL\/31753\/2017","LA\/P\/0063\/2020","SFRH\/BD\/88742\/2012"],"award-info":[{"award-number":["PTDC\/MAT-APL\/31753\/2017","LA\/P\/0063\/2020","SFRH\/BD\/88742\/2012"]}]},{"name":"FCT","award":["PTDC\/MAT-APL\/31753\/2017","LA\/P\/0063\/2020","SFRH\/BD\/88742\/2012"],"award-info":[{"award-number":["PTDC\/MAT-APL\/31753\/2017","LA\/P\/0063\/2020","SFRH\/BD\/88742\/2012"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>In two-action generalized polymatrix games, Nash equilibria are support-type-symmetric, i.e., determined by supports for each type of player. We show that such a property does not generalize straightforwardly for games with at least three actions or where interaction weights have different signs (neither all positive nor negative). A non-trivial condition on interaction weights must be satisfied, which may go unnoticed as it is trivially satisfied for: (i) two-action games, (ii) conformity games, and (iii) congestion games. We derive this condition and the corresponding simplified analytic equation for mixed strategies.<\/jats:p>","DOI":"10.3390\/math10244696","type":"journal-article","created":{"date-parts":[[2022,12,12]],"date-time":"2022-12-12T04:05:22Z","timestamp":1670817922000},"page":"4696","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on Type-Symmetries in Finite Games"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8188-9475","authenticated-orcid":false,"given":"Renato","family":"Soeiro","sequence":"first","affiliation":[{"name":"INESC TEC, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2953-6688","authenticated-orcid":false,"given":"Alberto A.","family":"Pinto","sequence":"additional","affiliation":[{"name":"INESC TEC, Faculty of Sciences, University of Porto, Rua do Campo Alegre, s\/n, 4169-007 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"303","DOI":"10.3934\/jdg.2015007","article-title":"Externality effects in the formation of societies","volume":"2","author":"Soeiro","year":"2015","journal-title":"J. Dyn. Games"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.geb.2005.09.006","article-title":"Behavioral conformity in games with many players","volume":"57","author":"Wooders","year":"2006","journal-title":"Games Econ. Behav."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.jcss.2008.09.001","article-title":"Symmetries and the complexity of pure Nash equilibrium","volume":"75","author":"Brandt","year":"2009","journal-title":"J. Comput. Syst. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.dam.2013.12.005","article-title":"Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave","volume":"166","author":"Iimura","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1007\/s00182-015-0481-y","article-title":"Pure strategy equilibrium in finite weakly unilaterally competitive games","volume":"45","author":"Iimura","year":"2016","journal-title":"Int. J. Game Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1007\/s00182-020-00724-2","article-title":"Unilaterally competitive games with more than two players","volume":"49","author":"Iimura","year":"2020","journal-title":"Int. J. Game Theory"},{"key":"ref_7","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_8","first-page":"81","article-title":"On Symmetric Games","volume":"Volume I","author":"Kuhn","year":"1951","journal-title":"Contributions to the Theory of Games (AM-24)"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"121","DOI":"10.3934\/jdg.2014.1.121","article-title":"Dynamics of human decisions","volume":"1","author":"Soeiro","year":"2014","journal-title":"J. Dyn. Games"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1287\/mnsc.18.5.312","article-title":"Equilibria of Polymatrix Games","volume":"18","author":"Howson","year":"1972","journal-title":"Manag. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/BF01254291","article-title":"A note on polymatrix games","volume":"18","author":"Quintas","year":"1989","journal-title":"Int. J. Game Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s10957-006-9070-3","article-title":"Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games","volume":"129","author":"Audet","year":"2006","journal-title":"J. Optim. Theory Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.geb.2010.10.012","article-title":"Action-Graph Games","volume":"71","author":"Jiang","year":"2011","journal-title":"Games Econ. Behav."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2742345","article-title":"Hedonic Clustering Games","volume":"2","author":"Feldman","year":"2015","journal-title":"ACM Trans. Parallel Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Mertikopoulos, P., Papadimitriou, C., and Piliouras, G. (2018). Cycles in Adversarial Regularized Learning. Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Proceedings, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611975031.172"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Soeiro, R., and Pinto, A.A. (2021). Negative network effects and asymmetric pure price equilibria. Port. Econ. J.","DOI":"10.1007\/s10258-021-00199-3"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Aloisio, A., Flammini, M., Kodric, B., and Vinci, C. (2021). Distance Polymatrix Coordination Games. Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, International Joint Conferences on Artificial Intelligence Organization.","DOI":"10.24963\/ijcai.2021\/1"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.laa.2022.04.011","article-title":"A new subclass of Q0-matrix in linear complementarity theory","volume":"647","author":"Singh","year":"2022","journal-title":"Linear Algebra Its Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/1913820","article-title":"Independence of Irrelevant Alternatives","volume":"41","author":"Ray","year":"1973","journal-title":"Econometrica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1006\/jeth.1996.2203","article-title":"Equilibria in a Model with Partial Rivalry","volume":"72","author":"Konishi","year":"1997","journal-title":"J. Econ. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1006\/game.1997.0542","article-title":"Pure Strategy Nash Equilibrium in a Group Formation Game with Positive Externalities","volume":"21","author":"Konishi","year":"1997","journal-title":"Games Econ. Behav."},{"key":"ref_22","unstructured":"Fudenberg, D., and Tirole, J. (1991). Game Theory, MIT Press."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/24\/4696\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:38:06Z","timestamp":1760146686000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/24\/4696"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,11]]},"references-count":22,"journal-issue":{"issue":"24","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["math10244696"],"URL":"https:\/\/doi.org\/10.3390\/math10244696","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2022,12,11]]}}}