# American Institute of Mathematical Sciences

July 2017, 4(3): 217-253. doi: 10.3934/jdg.2017014

## Good strategies for the Iterated Prisoner's Dilemma: Smale vs. Markov

 Mathematics Department, The City College, 137 Street and Convent Avenue, New York City, NY 10031, USA

Received  October 2016 Revised  March 2017 Published  May 2017

In 1980 Steven Smale introduced a class of strategies for the Iterated Prisoner's Dilemma which used as data the running average of the previous payoff pairs. This approach is quite different from the Markov chain approach, common before and since, which used as data the outcome of the just previous play, the memory-one strategies. Our purpose here is to compare these two approaches focusing upon good strategies which, when used by a player, assure that the only way an opponent can obtain at least the cooperative payoff is to behave so that both players receive the cooperative payoff. In addition, we prove a version for the Smale approach of the so-called Folk Theorem concerning the existence of Nash equilibria in repeated play. We also consider the dynamics when certain simple Smale strategies are played against one another.

Citation: Ethan Akin. Good strategies for the Iterated Prisoner's Dilemma: Smale vs. Markov. Journal of Dynamics & Games, 2017, 4 (3) : 217-253. doi: 10.3934/jdg.2017014
##### References:
 [1] K. Abhyankar, Smale strategies for Prisoner's Dilemma type games, in Algebra, Arithmetic and Geometry with Applications; Papers from Shreeram S. Abhyankar's 70th Birthday Conference (eds. C. Christensen et al. ), Springer-Verlag, Berlin, (2004), 45-48. [2] E. Akin, The differential geometry of population genetics and evolutionary games, in Mathematical and Statistical Developments of Evolutionary Theory (ed. S. Lessard), Kluwer, Dordrecht, 229 (1990), 1-93. [3] E. Akin, The iterated prisoner's dilemma: Good strategies and their dynamics, Ergodic theory, De Gruyter, Berlin, (2016), 77{107. arXiv: 1211.0969v3. [4] E. Akin, What you gotta know to play good in the Iterated Prisoner's Dilemma, Games, 6 (2015), 175-190. doi: 10.3390/g6030175. [5] E. Akin, The Iterated Prisoner's Dilemma: Good strategies and their dynamics, in Ergodic Theory, Advances in Dynamical Systems (ed. I. Assani), De Gruyter, Berlin, (2016), 77{107. [6] R. Axelrod, The Evolution of Cooperation, Basic Books, New York, NY, 1984. [7] K. Behrstock, M. Benaim and M. Hirsch, Smale strategies for network Prisoner's Dilemma Games, Journal of Dynamics and Games, 2 (2015), 141-155. doi: 10.3934/jdg.2015.2.141. [8] M. Benaim and M. Hirsch, Stochastic adaptive behavior for Prisoner's Dilemma, unpublished manuscript. [9] M. Boerlijst, M. Nowak and K. Sigmund, Equal pay for all prisoners, Amer. Math. Monthly, 104 (1997), 303-305. doi: 10.2307/2974578. [10] C. Hilbe, M. Nowak and K. Sigmund, The evolution of extortion in Iterated Prisoner's Dilemma games, Proceedings of the National Academy of Sciences, 110 (2013), 6913-6918. doi: 10.1073/pnas.1214834110. [11] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. Press, Cambridge, UK, 1998. doi: 10.1017/CBO9781139173179. [12] G. Kendall, X. Yao and S. W. Chong (eds. ), The Iterated Prisoner's Dilemma, 20 Years On, Advances in Natural Computation vol. 4, World Scientific, Singapore, 2007. [13] J. Maynard Smith, Evolution and the Theory of Games, Cambridge Univ. Press, Cambridge, UK, 1982. doi: 10.1017/CBO9780511806292. [14] M. Nowak, Evolutionary Dynamics, Harvard Univ. Press, Cambridge, MA, 2006. [15] W. Press and F. Dyson, Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent, Birds and Frogs, (2015), 329-341. doi: 10.1142/9789814602877_0029. [16] K. Sigmund, Games of Life, Oxford Univ. Press, Oxford, UK, 1993. [17] K. Sigmund, The Calculus of Selfishness, Princeton Univ. Press, Princeton, NJ, 2010. doi: 10.1515/9781400832255. [18] S. Smale, The Prisoner's Dilemma and dynamical systems associated to non-cooperative games, Econometrica, 48 (1980), 1617-1634. doi: 10.2307/1911925. [19] A. Stewart and J. Plotkin, Extortion and cooperation in the Prisoner's Dilemma, Proceedings of the National Academy of Sciences, 109 (2012), 10134-10135. [20] A. Stewart and J. Plotkin, From extortion to generosity, evolution in the Iterated Prisoner's Dilemma, Proceedings of the National Academy of Sciences, 110 (2013), 15348-15353. doi: 10.1073/pnas.1306246110. [21] P. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics, Math. Biosciences, 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.

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##### References:
 [1] K. Abhyankar, Smale strategies for Prisoner's Dilemma type games, in Algebra, Arithmetic and Geometry with Applications; Papers from Shreeram S. Abhyankar's 70th Birthday Conference (eds. C. Christensen et al. ), Springer-Verlag, Berlin, (2004), 45-48. [2] E. Akin, The differential geometry of population genetics and evolutionary games, in Mathematical and Statistical Developments of Evolutionary Theory (ed. S. Lessard), Kluwer, Dordrecht, 229 (1990), 1-93. [3] E. Akin, The iterated prisoner's dilemma: Good strategies and their dynamics, Ergodic theory, De Gruyter, Berlin, (2016), 77{107. arXiv: 1211.0969v3. [4] E. Akin, What you gotta know to play good in the Iterated Prisoner's Dilemma, Games, 6 (2015), 175-190. doi: 10.3390/g6030175. [5] E. Akin, The Iterated Prisoner's Dilemma: Good strategies and their dynamics, in Ergodic Theory, Advances in Dynamical Systems (ed. I. Assani), De Gruyter, Berlin, (2016), 77{107. [6] R. Axelrod, The Evolution of Cooperation, Basic Books, New York, NY, 1984. [7] K. Behrstock, M. Benaim and M. Hirsch, Smale strategies for network Prisoner's Dilemma Games, Journal of Dynamics and Games, 2 (2015), 141-155. doi: 10.3934/jdg.2015.2.141. [8] M. Benaim and M. Hirsch, Stochastic adaptive behavior for Prisoner's Dilemma, unpublished manuscript. [9] M. Boerlijst, M. Nowak and K. Sigmund, Equal pay for all prisoners, Amer. Math. Monthly, 104 (1997), 303-305. doi: 10.2307/2974578. [10] C. Hilbe, M. Nowak and K. Sigmund, The evolution of extortion in Iterated Prisoner's Dilemma games, Proceedings of the National Academy of Sciences, 110 (2013), 6913-6918. doi: 10.1073/pnas.1214834110. [11] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. Press, Cambridge, UK, 1998. doi: 10.1017/CBO9781139173179. [12] G. Kendall, X. Yao and S. W. Chong (eds. ), The Iterated Prisoner's Dilemma, 20 Years On, Advances in Natural Computation vol. 4, World Scientific, Singapore, 2007. [13] J. Maynard Smith, Evolution and the Theory of Games, Cambridge Univ. Press, Cambridge, UK, 1982. doi: 10.1017/CBO9780511806292. [14] M. Nowak, Evolutionary Dynamics, Harvard Univ. Press, Cambridge, MA, 2006. [15] W. Press and F. Dyson, Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent, Birds and Frogs, (2015), 329-341. doi: 10.1142/9789814602877_0029. [16] K. Sigmund, Games of Life, Oxford Univ. Press, Oxford, UK, 1993. [17] K. Sigmund, The Calculus of Selfishness, Princeton Univ. Press, Princeton, NJ, 2010. doi: 10.1515/9781400832255. [18] S. Smale, The Prisoner's Dilemma and dynamical systems associated to non-cooperative games, Econometrica, 48 (1980), 1617-1634. doi: 10.2307/1911925. [19] A. Stewart and J. Plotkin, Extortion and cooperation in the Prisoner's Dilemma, Proceedings of the National Academy of Sciences, 109 (2012), 10134-10135. [20] A. Stewart and J. Plotkin, From extortion to generosity, evolution in the Iterated Prisoner's Dilemma, Proceedings of the National Academy of Sciences, 110 (2013), 15348-15353. doi: 10.1073/pnas.1306246110. [21] P. Taylor and L. Jonker, Evolutionarily stable strategies and game dynamics, Math. Biosciences, 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.
Competing Simple Smale Plans
Example 3.10
Example 3.12
Theorem 4.10
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