{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T09:11:24Z","timestamp":1749201084756,"version":"3.37.3"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T00:00:00Z","timestamp":1714694400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T00:00:00Z","timestamp":1714694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,6]]},"DOI":"10.1007\/s10957-024-02433-2","type":"journal-article","created":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T07:02:25Z","timestamp":1714719745000},"page":"1026-1062","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Variant of the Logistic Quantal Response Equilibrium to Select a Perfect Equilibrium"],"prefix":"10.1007","volume":"201","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3481-8930","authenticated-orcid":false,"given":"Yiyin","family":"Cao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yin","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chuangyin","family":"Dang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,5,3]]},"reference":[{"key":"2433_CR1","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719154","volume-title":"Introduction to Numerical Continuation Methods","author":"EL Allgower","year":"2003","unstructured":"Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. SIAM, Philadelphia (2003)"},{"key":"2433_CR2","first-page":"183","volume":"4","author":"FE Browder","year":"1960","unstructured":"Browder, F.E.: On continuity of fixed points under deformation of continuous mappings. Summa Brasiliensis Mathematicae 4, 183\u2013191 (1960)","journal-title":"Summa Brasiliensis Mathematicae"},{"key":"2433_CR3","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1007\/s10957-020-01792-w","volume":"188","author":"G Cai","year":"2021","unstructured":"Cai, G., Dong, Q.L., Peng, Y.: Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz operators. J. Optim. Theory Appl. 188, 447\u2013472 (2021)","journal-title":"J. Optim. Theory Appl."},{"key":"2433_CR4","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1007\/s10614-018-9847-0","volume":"54","author":"Y Chen","year":"2019","unstructured":"Chen, Y., Dang, C.Y.: A reformulation-based simplicial homotopy method for approximating perfect equilibria. Comput. Econ. 54, 877\u2013891 (2019)","journal-title":"Comput. Econ."},{"key":"2433_CR5","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.geb.2017.12.023","volume":"124","author":"Y Chen","year":"2020","unstructured":"Chen, Y., Dang, C.Y.: An extension of quantal response equilibrium and determination of perfect equilibrium. Games Econ. Behav. 124, 659\u2013670 (2020)","journal-title":"Games Econ. Behav."},{"key":"2433_CR6","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s10107-019-01422-y","volume":"185","author":"Y Chen","year":"2021","unstructured":"Chen, Y., Dang, C.Y.: A differentiable homotopy method to compute perfect equilibria. Math. Program. 185, 77\u2013109 (2021)","journal-title":"Math. Program."},{"key":"2433_CR7","volume-title":"Stability and Perfection of Nash Equilibria","author":"E van Damme","year":"2002","unstructured":"van Damme, E.: Stability and Perfection of Nash Equilibria, 2nd edn. Springer, Berlin (2002)","edition":"2"},{"key":"2433_CR8","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1287\/moor.16.1.148","volume":"16","author":"CY Dang","year":"1991","unstructured":"Dang, C.Y.: The $$D_{1}$$-triangulation of $$R^{n}$$ for simplicial algorithms for computing solutions of nonlinear equations. Math. Oper. Res. 16, 148\u2013161 (1991)","journal-title":"Math. Oper. Res."},{"key":"2433_CR9","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/BF01581250","volume":"59","author":"CY Dang","year":"1993","unstructured":"Dang, C.Y.: The $$D_2$$-triangulation for simplicial homotopy algorithms for computing solutions of nonlinear equations. Math. Program. 59, 307\u2013324 (1993)","journal-title":"Math. Program."},{"key":"2433_CR10","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1287\/moor.12.3.485","volume":"12","author":"TM Doup","year":"1987","unstructured":"Doup, T.M., Talman, A.J.J.: A continuous deformation algorithm on the product space of unit simplices. Math. Oper. Res. 12, 485\u2013521 (1987)","journal-title":"Math. Oper. Res."},{"key":"2433_CR11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01584975","volume":"3","author":"BC Eaves","year":"1972","unstructured":"Eaves, B.C.: Homotopies for the computation of fixed points. Math. Program. 3, 1\u201322 (1972)","journal-title":"Math. Program."},{"key":"2433_CR12","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1016\/S0165-1889(98)00073-6","volume":"23","author":"BC Eaves","year":"1999","unstructured":"Eaves, B.C., Schmedders, K.: General equilibrium models and homotopy methods. J. Econ. Dyn. Control 23, 1249\u20131279 (1999)","journal-title":"J. Econ. Dyn. Control"},{"key":"2433_CR13","first-page":"27","volume":"35","author":"AH van den Elzen","year":"1991","unstructured":"van den Elzen, A.H., Talman, A.J.J.: A procedure for finding Nash equilibriua in bi-matrix games. Zeitschrift fur Oper. Res. 35, 27\u201343 (1991)","journal-title":"Zeitschrift fur Oper. Res."},{"key":"2433_CR14","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1006\/game.1998.0688","volume":"28","author":"AH van den Elzen","year":"1999","unstructured":"van den Elzen, A.H., Talman, A.J.J.: An algorithmic approach towards the tracing procedure of Harsanyi and Selten. Games Econ. Behav. 28, 130\u2013145 (1999)","journal-title":"Games Econ. Behav."},{"key":"2433_CR15","volume-title":"Finite-Dimensional Variational Inequalities and Complementarity Problems","author":"F Facchinei","year":"2003","unstructured":"Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)"},{"key":"2433_CR16","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1609\/aaai.v31i1.10571","volume":"31","author":"G Farina","year":"2017","unstructured":"Farina, G., Gatti, N.: Extensive-form perfect equilibrium computation in two-player games. Proceedings of the AAAI Conference on Artificial Intelligence 31, 502\u2013508 (2017)","journal-title":"Proceedings of the AAAI Conference on Artificial Intelligence"},{"key":"2433_CR17","volume-title":"Introduction to Sensitivity and Stability Analysis in Nonlinear Programming","author":"AV Fiacco","year":"1983","unstructured":"Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York (1983)"},{"key":"2433_CR18","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s001820200113","volume":"31","author":"S Govindan","year":"2002","unstructured":"Govindan, S., Klumpp, T.: Perfect equilibrium and lexicographic beliefs. Int. J. Game Theory 31, 229\u2013243 (2002)","journal-title":"Int. J. Game Theory"},{"key":"2433_CR19","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0022-0531(03)00005-X","volume":"110","author":"S Govindan","year":"2003","unstructured":"Govindan, S., Wilson, R.: A global Newton method to compute Nash equilibria. J. Econ. Theory 110, 65\u201386 (2003)","journal-title":"J. Econ. Theory"},{"key":"2433_CR20","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1023\/A:1017964015910","volume":"112","author":"DR Han","year":"2002","unstructured":"Han, D.R., Lo, H.K.: New alternating direction method for a class of nonlinear variational inequality problems. J. Optim. Theory Appl. 112, 549\u2013560 (2002)","journal-title":"J. Optim. Theory Appl."},{"key":"2433_CR21","doi-asserted-by":"crossref","unstructured":"Hansen, K.A., Miltersen, P.B., S$$\\phi $$rensen T.B.: The computational complexity of trembling hand perfection and other equilibrium refinements. SAGT 2010, 198\u2013209 (2010)","DOI":"10.1007\/978-3-642-16170-4_18"},{"key":"2433_CR22","volume-title":"A General Theory of Equilibrium Selection in Games","author":"JC Harsanyi","year":"1988","unstructured":"Harsanyi, J.C., Selten, R.: A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge (1988)"},{"key":"2433_CR23","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1007\/s001990050024","volume":"15","author":"PJJ Herings","year":"2000","unstructured":"Herings, P.J.J.: Two simple proofs of the feasibility of the linear tracing procedure. Econ. Theory 15, 485\u2013490 (2000)","journal-title":"Econ. Theory"},{"key":"2433_CR24","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s00199-009-0441-5","volume":"42","author":"PJJ Herings","year":"2010","unstructured":"Herings, P.J.J., Peeters, R.J.A.P.: Homotopy methods to compute equilibria in game theory. Econ. Theory 42, 119\u2013156 (2010)","journal-title":"Econ. Theory"},{"key":"2433_CR25","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/PL00004129","volume":"18","author":"PJJ Herings","year":"2001","unstructured":"Herings, P.J.J., Peeters, R.J.A.P.: A differentiable homotopy to compute Nash equilibria of $$n$$-person games. Econ. Theory 18, 159\u2013185 (2001)","journal-title":"Econ. Theory"},{"key":"2433_CR26","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.2307\/1912320","volume":"54","author":"E Kohlberg","year":"1986","unstructured":"Kohlberg, E., Mertens, J.F.: On the strategic stability of equilibria. Econometrica 54, 1003\u20131037 (1986)","journal-title":"Econometrica"},{"key":"2433_CR27","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1007\/BF01588226","volume":"17","author":"G van der Laan","year":"1979","unstructured":"van der Laan, G., Talman, A.J.J.: A restart algorithm for computing fixed points without an extra dimension. Math. Program. 17, 74\u201384 (1979)","journal-title":"Math. Program."},{"key":"2433_CR28","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1287\/moor.12.3.377","volume":"12","author":"G van der Laan","year":"1987","unstructured":"van der Laan, G., Talman, A.J.J., van der Heijden, L.: Simplicial variable dimension algorithms for solving nonlinear complementarity problems on a product of unit simplices using a general labeling. Math. Oper. Res. 12, 377\u2013397 (1987)","journal-title":"Math. Oper. Res."},{"key":"2433_CR29","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1137\/0112033","volume":"12","author":"CE Lemke","year":"1964","unstructured":"Lemke, C.E., JrJT, Howson: Equilibrium points in bimatrix games. SIAM J. Appl. Math. 12, 413\u2013423 (1964)","journal-title":"SIAM J. Appl. Math."},{"key":"2433_CR30","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s10107-014-0806-9","volume":"153","author":"G Li","year":"2015","unstructured":"Li, G., Mordukhovich, B.S., Pham, T.S.: New fractional error bounds for polynomial systems with applications to H\u00f6lderian stability in optimization and spectral theory of tensors. Math. Program. 153, 333\u2013362 (2015)","journal-title":"Math. Program."},{"key":"2433_CR31","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1137\/0804021","volume":"4","author":"XD Luo","year":"1994","unstructured":"Luo, X.D., Luo, Z.Q.: Extension of Hoffman\u2019s error bound to polynomial systems. SIAM J. Opti. 4, 383\u2013392 (1994)","journal-title":"SIAM J. Opti."},{"key":"2433_CR32","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/BF01580239","volume":"6","author":"A Mas-Colell","year":"1974","unstructured":"Mas-Colell, A.: A note on a theorem of F. Browder. Math. Program. 6, 229\u2013233 (1974)","journal-title":"Browder. Math. Program."},{"key":"2433_CR33","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1006\/game.1995.1023","volume":"10","author":"RD McKelvey","year":"1995","unstructured":"McKelvey, R.D., Palfrey, T.R.: Quantal response equilibria for normal form games. Games Econ. Behav. 10, 6\u201338 (1995)","journal-title":"Games Econ. Behav."},{"key":"2433_CR34","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s00199-009-0440-6","volume":"42","author":"PB Miltersen","year":"2010","unstructured":"Miltersen, P.B., S\u00f8rensen, T.B.: Computing a quasi-perfect equilibrium of a two-player game. Econ. Theory 42, 175\u2013192 (2010)","journal-title":"Econ. Theory"},{"key":"2433_CR35","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF01753236","volume":"7","author":"RB Myerson","year":"1978","unstructured":"Myerson, R.B.: Refinements of the Nash equilibrium concept. Int. J. Game Theory 7, 73\u201380 (1978)","journal-title":"Int. J. Game Theory"},{"key":"2433_CR36","volume-title":"Game Theory: Analysis of Conflict","author":"RB Myerson","year":"1991","unstructured":"Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)"},{"key":"2433_CR37","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1257\/jel.37.3.1067","volume":"37","author":"RB Myerson","year":"1999","unstructured":"Myerson, R.B.: Nash equilibrium and the history of economic theory. J. Econ. Lit. 37, 1067\u20131082 (1999)","journal-title":"J. Econ. Lit."},{"key":"2433_CR38","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1073\/pnas.36.1.48","volume":"36","author":"Nash JrJF","year":"1950","unstructured":"JrJF, Nash: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36, 48\u201349 (1950)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"2433_CR39","first-page":"289","volume":"54","author":"Nash JrJF","year":"1951","unstructured":"JrJF, Nash: Noncooperative games. Ann. Math. 54, 289\u2013295 (1951)","journal-title":"Ann. Math."},{"key":"2433_CR40","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/978-3-642-11339-0_4","volume-title":"Nonlinear Optimization","author":"I P\u00f3lik","year":"2010","unstructured":"P\u00f3lik, I., Terlaky, T.: Interior point methods for nonlinear optimization. In: Di Pillo, G., Schoen, F. (eds.) Nonlinear Optimization, pp. 215\u2013276. Springer, Berlin (2010)"},{"key":"2433_CR41","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1023\/A:1014861331301","volume":"113","author":"L Qi","year":"2002","unstructured":"Qi, L., Sun, D.: Smoothing functions and smoothing Newton method for complementarity and variational inequality problems. J. Optim. Theory Appl. 113, 121\u2013147 (2002)","journal-title":"J. Optim. Theory Appl."},{"key":"2433_CR42","volume-title":"Variational Analysis","author":"RT Rockaffelar","year":"2009","unstructured":"Rockaffelar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (2009)"},{"key":"2433_CR43","doi-asserted-by":"crossref","first-page":"1328","DOI":"10.1137\/0115116","volume":"15","author":"HE Scarf","year":"1967","unstructured":"Scarf, H.E.: The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math. 15, 1328\u20131343 (1967)","journal-title":"SIAM J. Appl. Math."},{"key":"2433_CR44","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF01766400","volume":"4","author":"R Selten","year":"1975","unstructured":"Selten, R.: Reexamination of the perfectness concept for equilibrium points in extensive games. Int. J. Game Theory 4, 25\u201355 (1975)","journal-title":"Int. J. Game Theory"},{"key":"2433_CR45","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1111\/1468-0262.00300","volume":"70","author":"B von Stengel","year":"2002","unstructured":"von Stengel, B., van den Elzen, A., Talman, D.: Computing normal form perfect equilibria for extensive two-person games. Econometrica 70, 693\u2013715 (2002)","journal-title":"Econometrica"},{"key":"2433_CR46","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-50327-6","volume-title":"The Computation of Fixed Points and Applications","author":"MJ Todd","year":"1976","unstructured":"Todd, M.J.: The Computation of Fixed Points and Applications. Springer, Berlin (1976)"},{"key":"2433_CR47","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.geb.2004.04.003","volume":"51","author":"TL Turocy","year":"2005","unstructured":"Turocy, T.L.: A dynamic homotopy interpretation of the logistic quantal response equilibrium correspondence. Games Econ. Behav. 51, 243\u2013263 (2005)","journal-title":"Games Econ. Behav."},{"key":"2433_CR48","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/BF01240056","volume":"22","author":"Y Yamamoto","year":"1993","unstructured":"Yamamoto, Y.: A path-following procedure to find a proper equilibrium of finite games. Int. J. Game Theory 22, 249\u2013259 (1993)","journal-title":"Int. J. Game Theory"},{"key":"2433_CR49","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1007\/s10957-016-0939-5","volume":"175","author":"Y Zhao","year":"2017","unstructured":"Zhao, Y., Zhang, J., Yang, X., Lin, G.H.: Expected residual minimization formulation for a class of stochastic vector variational inequalities. J. Optim. Theory Appl. 175, 545\u2013566 (2017)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02433-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02433-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02433-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,24]],"date-time":"2024-06-24T07:18:41Z","timestamp":1719213521000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02433-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,3]]},"references-count":49,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,6]]}},"alternative-id":["2433"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02433-2","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2024,5,3]]},"assertion":[{"value":"13 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 March 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 May 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}