# American Institute of Mathematical Sciences

April  2015, 2(2): 141-155. doi: 10.3934/jdg.2015.2.141

## Smale strategies for network prisoner's dilemma games

 1 Kashi Abhyankar Behrstock works in the nancial industry in New York, United States 2 Institut de Mathématiques, Université de Neuchâtel, Switzerland 3 Mathematics Department, University Wisconsin at Madison and University of California at Berkeley, United States

Received  March 2015 Revised  October 2015 Published  December 2015

Smale's approach [13] to the classical two-players repeated Prisoner's Dilemma game is revisited here for $N$-players and Network games in the framework of Blackwell's approachability, stochastic approximations and differential inclusions.
Citation: 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
##### References:
 [1] K. Abhyankar, Smale Strategies for Prisoner's Dilemma Type Games,, Doctoral dissertation, (2001). [2] R. Axelrod, The Evolution of Cooperation,, Basic Book, (1984). [3] M. Benaïm, A dynamical system approach to stochastic approximation,, SIAM Journal on Optimization and Control, 34 (1996), 437. doi: 10.1137/S0363012993253534. [4] M. Benaïm, Dynamics of stochastic approximation algorithms,, Séminaire de Probabilités XXXIII, 1709 (1999), 1. doi: 10.1007/BFb0096509. [5] M. Benaïm and M. W. Hirsch, Asymptotic pseudotrajectories and chain recurrent flows, with applications,, J. Dynam. Differential Equations, 8 (1996), 141. doi: 10.1007/BF02218617. [6] M. Benaïm and M. Hirsch, Stochastic Adaptive Behavior for Prisoner's Dilemma,, Unpublished manuscript, (1996). [7] M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions,, SIAM Journal on Optimization and Control, 44 (2005), 328. doi: 10.1137/S0363012904439301. [8] M. Berger, Géométrie, Vol 3 : Convexes et Polytopes, Polyèdres Réguliers, Aires et Volumes,, Fernand-Nathan, (1977). [9] D. Blackwell, An analog of the minmax theorem for vector payoffs,, Pacific Journal of Mathematics, 6 (1956), 1. doi: 10.2140/pjm.1956.6.1. [10] M. Faure and G. Roth, Stochastic approximations of set-valued dynamical systems: Convergence with positive probability to an attractor,, Mathematics of Operation Research, 35 (2010), 624. doi: 10.1287/moor.1100.0455. [11] G. Hardin, The tragedy of the commons,, Journal of Natural Resources Policy Research, (2009), 243. doi: 10.1080/19390450903037302. [12] V. Perchet, Approachability, regret and calibration: Implications and equivalences,, Journal of Dynamics and Games, 1 (2014), 181. doi: 10.3934/jdg.2014.1.181. [13] S. Smale, The prisoner's dilemma and dynamical systems associated to non-cooperative games,, Econometrica, 48 (1980), 1617. doi: 10.2307/1911925.

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##### References:
 [1] K. Abhyankar, Smale Strategies for Prisoner's Dilemma Type Games,, Doctoral dissertation, (2001). [2] R. Axelrod, The Evolution of Cooperation,, Basic Book, (1984). [3] M. Benaïm, A dynamical system approach to stochastic approximation,, SIAM Journal on Optimization and Control, 34 (1996), 437. doi: 10.1137/S0363012993253534. [4] M. Benaïm, Dynamics of stochastic approximation algorithms,, Séminaire de Probabilités XXXIII, 1709 (1999), 1. doi: 10.1007/BFb0096509. [5] M. Benaïm and M. W. Hirsch, Asymptotic pseudotrajectories and chain recurrent flows, with applications,, J. Dynam. Differential Equations, 8 (1996), 141. doi: 10.1007/BF02218617. [6] M. Benaïm and M. Hirsch, Stochastic Adaptive Behavior for Prisoner's Dilemma,, Unpublished manuscript, (1996). [7] M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions,, SIAM Journal on Optimization and Control, 44 (2005), 328. doi: 10.1137/S0363012904439301. [8] M. Berger, Géométrie, Vol 3 : Convexes et Polytopes, Polyèdres Réguliers, Aires et Volumes,, Fernand-Nathan, (1977). [9] D. Blackwell, An analog of the minmax theorem for vector payoffs,, Pacific Journal of Mathematics, 6 (1956), 1. doi: 10.2140/pjm.1956.6.1. [10] M. Faure and G. Roth, Stochastic approximations of set-valued dynamical systems: Convergence with positive probability to an attractor,, Mathematics of Operation Research, 35 (2010), 624. doi: 10.1287/moor.1100.0455. [11] G. Hardin, The tragedy of the commons,, Journal of Natural Resources Policy Research, (2009), 243. doi: 10.1080/19390450903037302. [12] V. Perchet, Approachability, regret and calibration: Implications and equivalences,, Journal of Dynamics and Games, 1 (2014), 181. doi: 10.3934/jdg.2014.1.181. [13] S. Smale, The prisoner's dilemma and dynamical systems associated to non-cooperative games,, Econometrica, 48 (1980), 1617. doi: 10.2307/1911925.
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