{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T08:59:39Z","timestamp":1769072379363,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2011,9,14]],"date-time":"2011-09-14T00:00:00Z","timestamp":1315958400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>The replicator-mutator dynamics is a set of differential equations frequently used in biological and socioeconomic contexts to model evolutionary processes subject to mutation, error or experimentation. The replicator-mutator dynamics generalizes the widely used replicator dynamics, which appears in this framework as the extreme case where replication is perfectly precise. This paper studies the influence of strictly dominated strategies on the location of the rest points of the replicator-mutator dynamics, at the limit where the mutation terms become arbitrarily small. It can be proved that such limit rest points for small mutation are Nash equilibria, so strictly dominated strategies do not occur at limit stationary points. However, we show through a simple case how strictly dominated strategies can have an influence on the location of the limit rest points for small mutation. Consequently, the characterization of the limit rest points of the replicator-mutator dynamics cannot in general proceed safely by readily eliminating strictly dominated strategies.<\/jats:p>","DOI":"10.3390\/g2030355","type":"journal-article","created":{"date-parts":[[2011,9,15]],"date-time":"2011-09-15T16:44:05Z","timestamp":1316105045000},"page":"355-364","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Strictly Dominated Strategies in the Replicator-Mutator Dynamics"],"prefix":"10.3390","volume":"2","author":[{"given":"Segismundo S.","family":"Izquierdo","sequence":"first","affiliation":[{"name":"Department of Industrial Organization, Universidad de Valladolid, Valladolid, 47011 Spain"},{"name":"Social Systems Engineering Centre (InSiSoc), Valladolid, 47011 Spain"}]},{"given":"Luis R.","family":"Izquierdo","sequence":"additional","affiliation":[{"name":"Social Systems Engineering Centre (InSiSoc), Valladolid, 47011 Spain"},{"name":"Department of Civil Engineering, Universidad de Burgos, Burgos, 09001 Spain"}]}],"member":"1968","published-online":{"date-parts":[[2011,9,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/BF00276557","article-title":"The selection mutation equation","volume":"23","author":"Hofbauer","year":"1985","journal-title":"J. Math. Biol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/S0022-5193(02)93112-7","article-title":"Unifying evolutionary dynamics","volume":"219","author":"Page","year":"2002","journal-title":"J. Theor. Biol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/BF00948894","article-title":"Mutation in autocatalytic reaction networks\u2014An analysis based on perturbation-theory","volume":"30","author":"Stadler","year":"1992","journal-title":"J. Math. Biol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"137:1","DOI":"10.1186\/1471-2148-10-137","article-title":"Viral quasispecies profiles as the result of the interplay of competition cooperation","volume":"10","author":"Arbiza","year":"2010","journal-title":"BMC Evol. Biol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"10797","DOI":"10.1073\/pnas.0502589102","article-title":"Evolutionary cycles of cooperation and defection","volume":"102","author":"Imhof","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.jtbi.2004.05.004","article-title":"Replicator-mutator equation, universality property and population dynamics of learning","volume":"230","author":"Komarova","year":"2004","journal-title":"J. Theor. Biol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1126\/science.291.5501.114","article-title":"Evolution of universal grammar","volume":"291","author":"Nowak","year":"2001","journal-title":"Science"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"18517691:1","DOI":"10.1103\/PhysRevE.77.046108","article-title":"Coherence thresholds in models of language change and evolution: The effects of noise, dynamics, and network of interactions","volume":"77","author":"Tavares","year":"2008","journal-title":"Phys. Rev. E."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1016\/j.jtbi.2008.07.010","article-title":"Feasibility of communication in binary signaling games","volume":"254","author":"Hofbauer","year":"2008","journal-title":"J. Theor. Biol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s11229-009-9477-0","article-title":"Evolutionary dynamics of Lewis signaling games: Signaling systems vs. partial pooling","volume":"172","author":"Huttegger","year":"2010","journal-title":"Synthese"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s11538-009-9450-6","article-title":"A rapid-mutation approximation for cell population dynamics","volume":"72","author":"Sachs","year":"2010","journal-title":"B. Math. Biol."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1006\/game.1995.1047","article-title":"Stability by mutation in evolutionary games","volume":"11","author":"Bomze","year":"1995","journal-title":"Game. Econ. Behav."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/3213231","article-title":"A simple model for the balance between selection and mutation","volume":"15","author":"Kingman","year":"1978","journal-title":"J. Appl. Prob."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Akin, E. (1979). The Geometry of Population Genetics, Springer.","DOI":"10.1007\/978-3-642-93128-4"},{"key":"ref_15","unstructured":"Samuelson, L. (1997). Evolutionary Games and Equilibrium Selection, IT Press."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Hofbauer, J., and Sigmund, K. (1998). Evolutionary Games and Population Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9781139173179"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Vega-Redondo, F. (2003). Economics and the Theory of Games, Cambridge University Press.","DOI":"10.1017\/CBO9780511753954"},{"key":"ref_18","unstructured":"Weibull, J.W. (1995). Evolutionary Game Theory, MIT Press."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1111\/1467-937X.00091","article-title":"Evolutionary drift and equilibrium selection","volume":"66","author":"Binmore","year":"1999","journal-title":"Rev. Econ. Stud."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/0022-0531(92)90041-F","article-title":"Evolutionary stability in asymmetric games","volume":"57","author":"Samuelson","year":"1992","journal-title":"J. Econ. Theor."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0025-5564(80)90039-5","article-title":"Domination or equilibrium","volume":"50","author":"Akin","year":"1980","journal-title":"Math. Biosci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1006\/jeth.1996.0133","article-title":"Evolutionary selection against dominated strategies","volume":"71","author":"Hofbauer","year":"1996","journal-title":"J. Econ. Theor."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Hofbauer, J., and Sandholm, W.H. (2011). Survival of dominated strategies under evolutionary dynamics. Theor. Econ., in press.","DOI":"10.3982\/TE771"},{"key":"ref_24","unstructured":"Hirsch, M.W., and Smale, S. (1974). Differential Equations, Dynamical Systems and Linear Algebra, Academic Press."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Clement, P., and Lumer, G. (1994). Evolution Equations, Control Theory, and Biomathematics, Marcel Dekker, Inc.","DOI":"10.2307\/2532823"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/0025-5564(94)90055-8","article-title":"The influence of mutation on autocatalytic reaction networks","volume":"122","author":"Stadler","year":"1994","journal-title":"Math. Biosci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0092-8240(94)00022-5","article-title":"Dynamics of small autocatalytic reaction networks 2. Replication, mutation and catalysis","volume":"57","author":"Stadler","year":"1995","journal-title":"B. Math. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.jtbi.2011.02.024","article-title":"Sex and evolutionary stability","volume":"278","author":"Binmore","year":"2011","journal-title":"J. Theor. Biol."}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/2\/3\/355\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:57:22Z","timestamp":1760219842000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/2\/3\/355"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,9,14]]},"references-count":28,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2011,9]]}},"alternative-id":["g2030355"],"URL":"https:\/\/doi.org\/10.3390\/g2030355","relation":{},"ISSN":["2073-4336"],"issn-type":[{"value":"2073-4336","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,9,14]]}}}