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. BehrstockM. 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. BoerlijstM. Nowak and K. Sigmund, Equal pay for all prisoners, Amer. Math. Monthly, 104 (1997), 303-305. doi: 10.2307/2974578.

[10]

C. HilbeM. 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.

show all references

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. BehrstockM. 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. BoerlijstM. Nowak and K. Sigmund, Equal pay for all prisoners, Amer. Math. Monthly, 104 (1997), 303-305. doi: 10.2307/2974578.

[10]

C. HilbeM. 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.

Figure 1.  Competing Simple Smale Plans
Figure 2.  Example 3.10
Figure 3.  Example 3.12
Figure 4.  Theorem 4.10
[1]

Kashi Behrstock, Michel Benaïm, Morris W. Hirsch. Smale strategies for network prisoner's dilemma games. Journal of Dynamics & Games, 2015, 2 (2) : 141-155. doi: 10.3934/jdg.2015.2.141

[2]

Bin Yu. Behavior $0$ nonsingular Morse Smale flows on $S^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 509-540. doi: 10.3934/dcds.2016.36.509

[3]

François Béguin. Smale diffeomorphisms of surfaces: a classification algorithm. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 261-310. doi: 10.3934/dcds.2004.11.261

[4]

Sharon M. Cameron, Ariel Cintrón-Arias. Prisoner's Dilemma on real social networks: Revisited. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1381-1398. doi: 10.3934/mbe.2013.10.1381

[5]

Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17

[6]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829

[7]

Radosław Czaja, Waldyr M. Oliva, Carlos Rocha. On a definition of Morse-Smale evolution processes. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3601-3623. doi: 10.3934/dcds.2017155

[8]

Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang, Xiongtao Zhang. Emergent dynamics in the interactions of Cucker-Smale ensembles. Kinetic & Related Models, 2017, 10 (3) : 689-723. doi: 10.3934/krm.2017028

[9]

Bruno Buonomo. A simple analysis of vaccination strategies for rubella. Mathematical Biosciences & Engineering, 2011, 8 (3) : 677-687. doi: 10.3934/mbe.2011.8.677

[10]

Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447

[11]

Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073

[12]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987

[13]

Elismar R. Oliveira. Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 551-569. doi: 10.3934/dcds.2008.21.551

[14]

Hyeong-Ohk Bae, Young-Pil Choi, Seung-Yeal Ha, Moon-Jin Kang. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4419-4458. doi: 10.3934/dcds.2014.34.4419

[15]

Chun-Hsien Li, Suh-Yuh Yang. A new discrete Cucker-Smale flocking model under hierarchical leadership. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2587-2599. doi: 10.3934/dcdsb.2016062

[16]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1

[17]

Ewa Girejko, Luís Machado, Agnieszka B. Malinowska, Natália Martins. On consensus in the Cucker–Smale type model on isolated time scales. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 77-89. doi: 10.3934/dcdss.2018005

[18]

Young-Pil Choi, Jan Haskovec. Cucker-Smale model with normalized communication weights and time delay. Kinetic & Related Models, 2017, 10 (4) : 1011-1033. doi: 10.3934/krm.2017040

[19]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[20]

Laure Pédèches. Asymptotic properties of various stochastic cucker-smale dynamics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2731-2762. doi: 10.3934/dcds.2018115

 Impact Factor: 

Metrics

  • PDF downloads (9)
  • HTML views (10)
  • Cited by (1)

Other articles
by authors

[Back to Top]