Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-22T14:57:41.250Z Has data issue: false hasContentIssue false
Series:   Elements in Evolutionary Economics

Evolutionary Games and the Replicator Dynamics

Published online by Cambridge University Press:  13 May 2024

Saul Mendoza-Palacios  and
Onésimo Hernández-Lerma
Saul Mendoza-Palacios
Affiliation:
Centro de Investigación y Docencia Económicas
Onésimo Hernández-Lerma
Affiliation:
Centro de Investigacion y de Estudios Avanzados del IPN

Summary

This Element introduces the replicator dynamics for symmetric and asymmetric games where the strategy sets are metric spaces. Under this hypothesis the replicator dynamics evolves in a Banach space of finite signed measures. The authors provide a general framework to study the stability of the replicator dynamics for evolutionary games in this Banach space. This allows them to establish a relation between Nash equilibria and the stability of the replicator for normal a form games applicable to oligopoly models, theory of international trade, public good models, the tragedy of commons, and War of attrition game among others. They also provide conditions to approximate the replicator dynamics on a space of measures by means of a finite-dimensional dynamical system and a sequence of measure-valued Markov processes.
Get access

Keywords

games with continuous strategy spacesasymmetric evolutionary gamesevolutionary gamespopulation gamesreplicator dynamics
Type
Element
Information
Series: Elements in Evolutionary Economics
Online ISBN: 9781009472319
Publisher: Cambridge University Press
Print publication: 06 June 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Accinelli, E., and Sánchez Carrera, E. J.. The evolutionary game of poverty traps. The Manchester School, 80(4):381–400, 2012.CrossRefGoogle Scholar
Almudi, I., and Fatas-Villafranca, F.. Coevolution in economic systems. Cambridge University Press, Cambridge, 2021.CrossRefGoogle Scholar
Almudi, I., Fatas-Villafranca, F., Fernandez-Marquez, C. M., Potts, J., and Vazquez, F. J.. Absorptive capacity in a two-sector neo-Schumpeterian model: A new role for innovation policy. Industrial and Corporate Change, 29(2):507–531, 2020a.CrossRefGoogle Scholar
Almudi, I., Fatas-Villafranca, F., and Izquierdo, L. R.. Innovation, catch-up, and leadership in science-based industries. Industrial and Corporate Change, 21(2):345–375, 2012.CrossRefGoogle Scholar
Almudi, I., Fatas-Villafranca, F., Jarne, G., and Sanchez-Choliz, J.. An evolutionary growth model with banking activity. Metroeconomica, 71(2):392–430, 2020b.CrossRefGoogle Scholar
Almudi, I., Fatas-Villafranca, F., Palacio, J., and Sanchez-Choliz, J.. Pricing routines and industrial dynamics. Journal of Evolutionary Economics, 30(3):705–739, 2020c.CrossRefGoogle Scholar
Amir, R., Evstigneev, I. V., Hens, T., and Xu, L.. Evolutionary finance and dynamic games. Mathematics and Financial Economics, 5(3):161–184, 2011.CrossRefGoogle Scholar
Amir, R., Evstigneev, I. V., and Schenk-Hoppé, K. R.. Asset market games of survival: A synthesis of evolutionary and dynamic games. Annals of Finance, 9(2):121–144, 2013.CrossRefGoogle Scholar
Araujo, R. A. An evolutionary game theory approach to combat money laundering. Journal of Money Laundering Control, 13(1):70, 2010.Google Scholar
Araujo, R. A., and de Souza, N. A.. An evolutionary game theory approach to the dynamics of the labour market: A formal and informal perspective. Structural Change and Economic Dynamics, 21(2):101–110, 2010.CrossRefGoogle Scholar
Axelrod, R., and Hamilton, W. D.. The evolution of cooperation. Science, 211(4489):1390–1396, 1981.CrossRefGoogle ScholarPubMed
Balkenborg, D., and Schlag, K. H.. On the evolutionary selection of sets of Nash equilibria. Journal of Economic Theory, 133(1):295–315, 2007.CrossRefGoogle Scholar
Barelli, P., and Meneghel, I.. A note on the equilibrium existence problem in discontinuous games. Econometrica, 81(2):813–824, 2013.Google Scholar
Bartle, R. G. The elements of integration and Lebesgue measure. John Wiley & Sons, New York, 1995.CrossRefGoogle Scholar
Bauch, C. T. Imitation dynamics predict vaccinating behaviour. Proceedings of the Royal Society of London B: Biological Sciences, 272(1573):1669–1675, 2005.Google ScholarPubMed
Benaim, M., and Weibull, J. W.. Deterministic approximation of stochastic evolution in games. Econometrica, 71(3):873–903, 2003.CrossRefGoogle Scholar
Billingsley, P. Convergence of probability measures. John Wiley & Sons, New York, 2013.Google Scholar
Bischi, G. I., Lamantia, F., and Radi, D.. An evolutionary Cournot model with limited market knowledge. Journal of Economic Behavior & Organization, 116(C):219–238, 2015.CrossRefGoogle Scholar
Bishop, D., and Cannings, C.. A generalized war of attrition. Journal of Theoretical Biology, 70(1):85–124, 1978.CrossRefGoogle ScholarPubMed
Bobrowski, A. Functional analysis for probability and stochastic processes: An introduction. Cambridge University Press, Cambridge, 2005.CrossRefGoogle Scholar
Bogachev, V. I. Measure theory, volume 2. Springer, Berlin, 2007.CrossRefGoogle Scholar
Bomze, I. M. Dynamical aspects of evolutionary stability. Monatshefte für Mathematik, 110(3–4):189–206, 1990.CrossRefGoogle Scholar
Bomze, I. M. Cross entropy minimization in uninvadable states of complex populations. Journal of Mathematical Biology, 30(1):73–87, 1991.CrossRefGoogle Scholar
Bomze, I. M., and Pötscher, B. M.. Game theoretical foundations of evolutionary stability. Lecture Notes in Economics and Mathematical Systems, 324, Springer, Berlin, 1989.CrossRefGoogle Scholar
Böttcher, B., Schilling, R., and Wang, J.. Lévy matters III: Lévy-type processes: Construction, approximation and sample path properties. Lecture Notes in Mathematics, 2099. Springer, Cham, 2013.Google Scholar
Carbonell-Nicolau, O. On the existence of pure-strategy perfect equilibrium in discontinuous games. Games and Economic Behavior, 71(1):23–48, 2011.CrossRefGoogle Scholar
Carmona, G. An existence result for discontinuous games. Journal of Economic Theory, 144(3):1333–1340, 2009.CrossRefGoogle Scholar
Carmona, G., and Podczeck, K.. Existence of Nash equilibrium in games with a measure space of players and discontinuous payoff functions. Journal of Economic Theory, 152(1):130–178, 2014.CrossRefGoogle Scholar
Cheung, M.-W. Pairwise comparison dynamics for games with continuous strategy space. Journal of Economic Theory, 153(1):344–375, 2014.CrossRefGoogle Scholar
Cheung, M.-W. Imitative dynamics for games with continuous strategy space. Games and Economic Behavior, 99(C):206–223, 2016.CrossRefGoogle Scholar
Cleveland, J., and Ackleh, A. S.. Evolutionary game theory on measure spaces: Well-posedness. Nonlinear Analysis: Real World Applications, 14(1):785–797, 2013.Google Scholar
Corradi, V. and Sarin, R.. Continuous approximations of stochastic evolutionary game dynamics. Journal of Economic Theory, 94(2):163–191, 2000. Corrigendum: Journal of Economic Theory, 140(1) (2008).CrossRefGoogle Scholar
Cressman, R. Local stability of smooth selection dynamics for normal form games. Mathematical Social Sciences, 34(1):1–19, 1997.CrossRefGoogle Scholar
Cressman, R. Evolutionary dynamics and extensive form games, volume 5. Massachusetts Institute of Technology Press, London, 2003.CrossRefGoogle Scholar
Cressman, R. Stability of the replicator equation with continuous strategy space. Mathematical Social Sciences, 50(2):127–147, 2005.CrossRefGoogle Scholar
Cressman, R. Continuously stable strategies, neighborhood superiority and two-player games with continuous strategy space. International Journal of Game Theory, 38(2):221–247, 2009.CrossRefGoogle Scholar
Cressman, R., and Hofbauer, J.. Measure dynamics on a one-dimensional continuous trait space: Theoretical foundations for adaptive dynamics. Theoretical Population Biology, 67(1):47–59, 2005.CrossRefGoogle ScholarPubMed
Cressman, R., Hofbauer, J., and Riedel, F.. Stability of the replicator equation for a single species with a multi-dimensional continuous trait space. Journal of Theoretical Biology, 239(2):273–288, 2006.CrossRefGoogle ScholarPubMed
Cressman, R., Morrison, W. G., and Wen, J.-F.. On the evolutionary dynamics of crime. Canadian Journal of Economics, 31(5):1101–1117, 1998.CrossRefGoogle Scholar
Dasgupta, P., and Maskin, E.. The existence of equilibrium in discontinuous economic games, I: Theory. The Review of Economic Studies, 53(1):1–26, 1986a.CrossRefGoogle Scholar
Dasgupta, P., and Maskin, E.. The existence of equilibrium in discontinuous economic games, II: Applications. The Review of Economic Studies, 53(1):27–41, 1986b.CrossRefGoogle Scholar
Dosi, G., and Nelson, R. R.. An introduction to evolutionary theories in economics. Journal of Evolutionary Economics, 4(3):153–172, 1994.CrossRefGoogle Scholar
Dosi, G., Fagiolo, G., and Roventini, A.. Schumpeter meeting Keynes: A policy-friendly model of endogenous growth and business cycles. Journal of Economic Dynamics and Control, 34(9):1748–1767, 2010.CrossRefGoogle Scholar
Dynkin, E. B. An introduction to branching measure-valued processes. American Mathematical Society, Providence (Rhode Island), 1994.Google Scholar
Eshel, I. and Sansone, E.. Evolutionary and dynamic stability in continuous population games. Journal of Mathematical Biology, 46(5):445–459, 2003.CrossRefGoogle ScholarPubMed
Ethier, S. N., and Kurtz, T. G.. Markov processes: Characterization and convergence, volume 282. John Wiley & Sons, New York, 1984.Google Scholar
Fudenberg, D., and Tirole, J.. Game theory, 1991. Massachusetts Institute of Technology Press, Cambridge, 1991.Google Scholar
Fujita, M., Krugman, P. R., and Venables, A. J.. The spatial economy: Cities, regions, and international trade. Massachusetts Institute of Technology Press, London, 2001.Google Scholar
Gatenby, R. A., and Gillies, R. J.. A microenvironmental model of carcinogenesis. Nature Reviews Cancer, 8(1):56–61, 2008.CrossRefGoogle ScholarPubMed
Gibbons, R. Game theory for applied economists. Princeton University Press, Princeton, 1992.Google Scholar
Givens, C. R., and Shortt, R. M.. A class of Wasserstein metrics for probability distributions. The Michigan Mathematical Journal, 31(2):231–240, 1984.CrossRefGoogle Scholar
Glicksberg, I. L. A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proceedings of the American Mathematical Society, 3(1):170–174, 1952.Google Scholar
Goodwin, R. M. A growth cycle. In Feinstein, C., editor, Socialism Capitalism and Economic Growth, pages 165–170. Cambridge University Press, Cambridge, 1967.Google Scholar
Harsanyi, J. C. Oddness of the number of equilibrium points: A new proof. International Journal of Game Theory, 2(1):235–250, 1973.CrossRefGoogle Scholar
Heidergott, B. and Leahu, H.. Weak differentiability of product measures. Mathematics of Operations Research, 35(1):27–51, 2010.CrossRefGoogle Scholar
Heifetz, A., Shannon, C., and Spiegel, Y.. The dynamic evolution of preferences. Economic Theory, 32(2):251–286, 2007.CrossRefGoogle Scholar
Hewitt, E., and Savage, L. J.. Symmetric measures on Cartesian products. Transactions of the American Mathematical Society, 80(2):470–501, 1955.CrossRefGoogle Scholar
Hofbauer, J., and Sigmund, K.. Evolutionary games and population dynamics. Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
Hofbauer, J., and Sigmund, K.. Evolutionary game dynamics. Bulletin of the American Mathematical Society, 40(4):479–519, 2003.CrossRefGoogle Scholar
Hofbauer, J., and Weibull, J. W.. Evolutionary selection against dominated strategies. Journal of Economic Theory, 71(2):558–573, 1996.CrossRefGoogle Scholar
Hofbauer, J., Oechssler, J., and Riedel, F.. Brown–von Neumann–Nash dynamics: The continuous strategy case. Games and Economic Behavior, 65(2):406–429, 2009.CrossRefGoogle Scholar
Hummert, S., Bohl, K., Basanta, D. et al. Evolutionary game theory: Cells as players. Molecular BioSystems, 10(12):3044–3065, 2014.CrossRefGoogle ScholarPubMed
Kallenberg, O. Foundations of modern probability. Springer, New York, 2002.CrossRefGoogle Scholar
Katsikas, S., Kolokoltsov, V., and Yang, W.. Evolutionary inspection and corruption games. Games, 7(4):31, 2016.CrossRefGoogle Scholar
Kolokoltsov, V. N. Kinetic equations for the pure jump models of k-nary interacting particle systems. Markov Process and Related Fields, 12(1):95–138, 2006.Google Scholar
Kolokoltsov, V. N. Nonlinear Markov processes and kinetic equations, volume 182. Cambridge University Press, Cambridge, 2010.CrossRefGoogle Scholar
Kolokoltsov, V. N. and Malafeyev, O. A.. Understanding game theory: Introduction to the analysis of many agent systems with competition and cooperation. World Scientific, Singapore, 2010.CrossRefGoogle Scholar
Kou, P., Han, Y., and Li, Y.. An evolutionary analysis of corruption in the process of collecting environmental tax in China. Environmental Science and Pollution Research, 28(39):54852–54862, 2021.CrossRefGoogle ScholarPubMed
Kravvaritis, D., and Papanicolaou, V. G.. Singular equilibrium solutions for a replicator dynamics model. Electronic Journal of Differential Equations, 2011(87):1–8, 2011.Google Scholar
Kravvaritis, D., Papanicolaou, V. G., and Yannacopoulos, A. N.. Similarity solutions for a replicator dynamics equation. Indiana University Mathematics Journal, 57(4):1929–1945, 2008.CrossRefGoogle Scholar
Kravvaritis, D., Papanicolaou, V., Xepapadeas, A., and Yannacopoulos, A.. On a class of operator equations arising in infinite dimensional replicator dynamics. Nonlinear Analysis: Real World Applications, 11 (4): 2537–2556, 2010.Google Scholar
Kravvaritis, D., Papanicolaou, V., Xepapadeas, T., and Yannacopoulos, A.. A class of infinite dimensional replicator dynamics. In Dynamics, games and science, volume I, pages 529–532. Springer, 2011.CrossRefGoogle Scholar
Kurtz, T. G. Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability, 7(1):49–58, 1970.CrossRefGoogle Scholar
Kurtz, T. G. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. Journal of Applied Probability, 8(2):344–356, 1971.CrossRefGoogle Scholar
Lahkar, R., and Riedel, F.. The logit dynamic for games with continuous strategy sets. Games and Economic Behavior, 2015.CrossRefGoogle Scholar
Lahkar, R., Mukherjee, S., and Roy, S.. Generalized perturbed best response dynamics with a continuum of strategies. Journal of Economic Theory, 200: 105398, 2022.CrossRefGoogle Scholar
Lamantia, F. Evolutionary modelling in environmental economics. Journal of Difference Equations and Applications, 23(7):1255–1285, 2017.CrossRefGoogle Scholar
Lang, S. Differential and Riemannian manifolds. Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Lewontin, R. C. Evolution and the theory of games. Journal of Theoretical Biology, 1(3):382–403, 1961.CrossRefGoogle ScholarPubMed
Li, Z. Measure-valued branching Markov processes. Springer, Berlin, 2010.Google Scholar
Mas-Colell, A., Whinston, M. D., and Green, J. R.. Microeconomic theory, volume 1. Oxford University Press, New York, 1995.Google Scholar
Maynard Smith, J. On evolution. Edinburgh University Press, Edinburgh, 1972.Google Scholar
Maynard Smith, J. The theory of games and the evolution of animal conflicts. Journal of Theoretical Biology, 47(1):209–221, 1974.CrossRefGoogle Scholar
Maynard Smith, J. Evolution and the theory of games. Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
Maynard Smith, J., and Parker, G. A.. The logic of asymmetric contests. Animal Behaviour, 24(1):159–175, 1976.CrossRefGoogle Scholar
Maynard Smith, J., and Price, G. R.. The logic of animal conflict. Nature, 246(5427):15–18, 1973.CrossRefGoogle Scholar
McLennan, A., Monteiro, P. K., and Tourky, R.. Games with discontinuous payoffs: A strengthening of Reny’s existence theorem. Econometrica, 79(5):1643–1664, 2011.Google Scholar
Mendoza-Palacios, S., and Hernández-Lerma, O.. Evolutionary dynamics on measurable strategy spaces: Asymmetric games. Journal of Differential Equations, 259(11):5709–5733, 2015.CrossRefGoogle Scholar
Mendoza-Palacios, S., and Hernández-Lerma, O.. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics and Games, 4(4):319–333, 2017.CrossRefGoogle Scholar
Mendoza-Palacios, S., and Hernández-Lerma, O.. A survey on the replicator dynamics for games with strategies in metric spaces. Pure and Applied Functional Analysis, 4(3):603–628, 2019.Google Scholar
Mendoza-Palacios, S., and Hernández-Lerma, O.. The replicator dynamics for games in metric spaces: Finite approximations. In Ramsey, D. M. and Renault, R. J, editors, Advances in Dynamic Games: Annals of the International Society of Dynamic Games, vol 17, pages 163–186. Birkhäuser, Cham, 2020.Google Scholar
Mendoza-Palacios, S., and Mercado, A.. A note on the big push as industrialization process. In Mendoza-Palacios, S. and Mercado, A., editors, Games and evolutionary dynamics: Selected theoretical and applied developments, pages 153–169. El Colegio de México, 2021.Google Scholar
Mendoza-Palacios, S., Berasaluce, J., and Mercado, A.. On industrialization, human resources training, and policy coordination. Journal of Industry, Competition and Trade, 22(2):179–206, 2022.CrossRefGoogle Scholar
Myerson, R. B. Game theory. Harvard University Press, London, 1997.Google Scholar
Narang, A., and Shaiju, A.. Evolutionary stability of polymorphic profiles in asymmetric games. Dynamic Games and Applications, 9(4):1126–1142, 2019.CrossRefGoogle Scholar
Narang, A., and Shaiju, A.. Globally strong uninvadable sets of profiles in asymmetric games. International Game Theory Review, 22(01):1950014, 2020.CrossRefGoogle Scholar
Narang, A., and Shaiju, A.. Neighborhood strong superiority and evolutionary stability of polymorphic profiles in asymmetric games. Journal of Dynamics and Games, 9(3):253–266, 2022.CrossRefGoogle Scholar
Norman, T. W. Dynamically stable sets in infinite strategy spaces. Games and Economic Behavior, 62(2):610–627, 2008.CrossRefGoogle Scholar
Norman, T. W. Equilibrium selection and the dynamic evolution of preferences. Games and Economic Behavior, 74(1):311–320, 2012.CrossRefGoogle Scholar
Oechssler, J., and Riedel, F.. Evolutionary dynamics on infinite strategy spaces. Economic Theory, 17(1):141–162, 2001.CrossRefGoogle Scholar
Oechssler, J., and Riedel, F.. On the dynamic foundation of evolutionary stability in continuous models. Journal of Economic Theory, 107(2):223–252, 2002.CrossRefGoogle Scholar
Osborne, M. J., and Rubinstein, A.. A course in game theory. Massachusetts Institute of Technology Press, London, 1994.Google Scholar
Papanicolaou, V. G., and Smyrlis, G.. Similarity solutions for a multi-dimensional replicator dynamics equation. Nonlinear Analysis: Theory, Methods & Applications, 71(7):3185–3196, 2009.CrossRefGoogle Scholar
Parthasarathy, K. R. Probability measures on metric spaces. Academic Press, New York, 1967.CrossRefGoogle Scholar
Pedersen, G. K. Analysis now. Springer, New York, 2012.Google Scholar
Prokopovych, P., and Yannelis, N. C.. On the existence of mixed strategy Nash equilibria. Journal of Mathematical Economics, 52:87–97, 2014.CrossRefGoogle Scholar
Quinteros, M. J., and Villena, M. J.. On the dynamics and stability of the crime and punishment game. Complexity, 2022(3):1–14, 2022.CrossRefGoogle Scholar
Rabanal, J. P. On the evolution of continuous types under replicator and gradient dynamics: Two examples. Dynamic Games and Applications, pages 7(1), No 6, 76–92, 2017.CrossRefGoogle Scholar
Reiss, R.-D. Approximate distributions of order statistics. Springer, New York, 1989.CrossRefGoogle Scholar
Reny, P. J. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica, 67(5):1029–1056, 1999.CrossRefGoogle Scholar
Ritzberger, K. The theory of normal form games from the differentiable viewpoint. International Journal of Game Theory, 23(3):207–236, 1994.CrossRefGoogle Scholar
Ritzberger, K., and Weibull, J. W.. Evolutionary selection in normal-form games. Econometrica, 63(6):1371–1399, 1995.CrossRefGoogle Scholar
Rosen, J. B. Existence and uniqueness of equilibrium points for concave -person games. Econometrica, 33(3):520–534, 1965.CrossRefGoogle Scholar
Royden, H. L. Real analysis. Macmillan, New York, 1988.Google Scholar
Ruijgrok, M. and Ruijgrok, T. W.. An effective replicator equation for games with a continuous strategy set. Dynamic Games and Applications, 5(2):157–179, 2015.CrossRefGoogle Scholar
Safarzyńska, K., and van den Bergh, J. C.. Evolutionary models in economics: A survey of methods and building blocks. Journal of Evolutionary Economics, 20(3):329–373, 2010.CrossRefGoogle Scholar
Samuelson, L., and Zhang, J.. Evolutionary stability in asymmetric games. Journal of Economic Theory, 57(2):363–391, 1992.CrossRefGoogle Scholar
Sandholm, W. H. Potential games with continuous player sets. Journal of Economic Theory, 97(1):81–108, 2001.CrossRefGoogle Scholar
Sandholm, W. H. Evolution and equilibrium under inexact information. Games and Economic Behavior, 44(2):343–378, 2003.CrossRefGoogle Scholar
Sandholm, W. H. Population games and evolutionary dynamics. Massachusetts Institute of Technology Press, 2010.Google Scholar
Selten, R. A note on evolutionarily stable strategies in asymmetric animal conflicts. Journal of Theoretical Biology, 84(1):93–101, 1980.CrossRefGoogle ScholarPubMed
Sethi, R. and Somanathan, E.. The evolution of social norms in common property resource use. The American Economic Review, pages 766–788, 1996.Google Scholar
Shahi, C. and Kant, S.. An evolutionary game-theoretic approach to the strategies of community members under joint forest management regime. Forest Policy and Economics, 9(7):763–775, 2007.CrossRefGoogle Scholar
Shiryaev, A. N. Probability. Springer-Verlag, New York, 1996.CrossRefGoogle Scholar
Simon, L. K. Games with discontinuous payoffs. The Review of Economic Studies, 54(4):569–597, 1987.CrossRefGoogle Scholar
Van Veelen, M., and Spreij, P.. Evolution in games with a continuous action space. Economic Theory, 39(3):355–376, 2009.CrossRefGoogle Scholar
Varian, H. R. A model of sales. The American Economic Review, 70(4):651–659, 1980.Google Scholar
Varian, H. R. Microeconomic analysis. W. W. Norton, New York, 1992.Google Scholar
Villani, C. Optimal transport: Old and new. Springer-Verlag, Berlin, 2008.Google Scholar
Vincent, T. L., and Gatenby, R. A.. Modeling cancer as an evolutionary game. International Game Theory Review, 7(03):331–346, 2005.CrossRefGoogle Scholar
Webb, J. N. Game theory: Decisions, interaction and evolution. Springer, London, 2007.Google Scholar
Weibull, J. W. Evolutionary game theory. Massachusetts Institute of Technology Press, London, 1997.Google Scholar
Winter, S. G., Kaniovski, Y. M., and Dosi, G.. A baseline model of industry evolution. Journal of Evolutionary Economics, 13(4):355–383, 2003.CrossRefGoogle Scholar
Wright, S. Evolution in Mendelian populations. Genetics, 16(2):97, 1931.CrossRefGoogle ScholarPubMed
Yang, F. and Yang, Q.. Model for the spread of SIS epidemic based on evolution game. Technology and Health Care, 23(Suppl 1):s71–s75, 2015.CrossRefGoogle ScholarPubMed

Save element to Kindle

To save this element to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Evolutionary Games and the Replicator Dynamics
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Evolutionary Games and the Replicator Dynamics
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Evolutionary Games and the Replicator Dynamics
Available formats
×