{"title":"有限种群进化对策中固定概率的Bernstein多项式逼近","authors":"Jiyeon Park, Paul K. Newton","doi":"10.1007/s13235-023-00509-8","DOIUrl":null,"url":null,"abstract":"We use the Bernstein polynomials of degree d as the basis for constructing a uniform approximation to the rate of evolution (related to the fixation probability) of a species in a two-component finite-population, well-mixed, frequency-dependent evolutionary game setting. The approximation is valid over the full range $$0 \\le w \\le 1$$ , where w is the selection pressure parameter, and converges uniformly to the exact solution as $$d \\rightarrow \\infty $$ . We compare it to a widely used non-uniform approximation formula in the weak-selection limit ( $$w \\sim 0$$ ) as well as numerically computed values of the exact solution. Because of a boundary layer that occurs in the weak-selection limit, the Bernstein polynomial method is more efficient at approximating the rate of evolution in the strong selection region ( $$w \\sim 1$$ ) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.","PeriodicalId":48933,"journal":{"name":"Dynamic Games and Applications","volume":"181 1","pages":"0"},"PeriodicalIF":1.8000,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games\",\"authors\":\"Jiyeon Park, Paul K. Newton\",\"doi\":\"10.1007/s13235-023-00509-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We use the Bernstein polynomials of degree d as the basis for constructing a uniform approximation to the rate of evolution (related to the fixation probability) of a species in a two-component finite-population, well-mixed, frequency-dependent evolutionary game setting. The approximation is valid over the full range $$0 \\\\le w \\\\le 1$$ , where w is the selection pressure parameter, and converges uniformly to the exact solution as $$d \\\\rightarrow \\\\infty $$ . We compare it to a widely used non-uniform approximation formula in the weak-selection limit ( $$w \\\\sim 0$$ ) as well as numerically computed values of the exact solution. Because of a boundary layer that occurs in the weak-selection limit, the Bernstein polynomial method is more efficient at approximating the rate of evolution in the strong selection region ( $$w \\\\sim 1$$ ) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.\",\"PeriodicalId\":48933,\"journal\":{\"name\":\"Dynamic Games and Applications\",\"volume\":\"181 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2023-06-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Dynamic Games and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s13235-023-00509-8\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Dynamic Games and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s13235-023-00509-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games
We use the Bernstein polynomials of degree d as the basis for constructing a uniform approximation to the rate of evolution (related to the fixation probability) of a species in a two-component finite-population, well-mixed, frequency-dependent evolutionary game setting. The approximation is valid over the full range $$0 \le w \le 1$$ , where w is the selection pressure parameter, and converges uniformly to the exact solution as $$d \rightarrow \infty $$ . We compare it to a widely used non-uniform approximation formula in the weak-selection limit ( $$w \sim 0$$ ) as well as numerically computed values of the exact solution. Because of a boundary layer that occurs in the weak-selection limit, the Bernstein polynomial method is more efficient at approximating the rate of evolution in the strong selection region ( $$w \sim 1$$ ) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.
期刊介绍:
Dynamic Games and Applications is devoted to the development of all classes of dynamic games, namely, differential games, discrete-time dynamic games, evolutionary games, repeated and stochastic games, and their applications in all fields