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 and 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, University of California at Berkeley, 2001.

[2]

R. Axelrod, The Evolution of Cooperation, Basic Book, Inc Publishers, New York, 1984.

[3]

M. Benaïm, A dynamical system approach to stochastic approximation, SIAM Journal on Optimization and Control, 34 (1996), 437-472. doi: 10.1137/S0363012993253534.

[4]

M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités XXXIII, Lecture Notes in Math, 1709 (1999), 1-68. 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), 141-176. doi: 10.1007/BF02218617.

[6]

M. Benaïm and M. Hirsch, Stochastic Adaptive Behavior for Prisoner's Dilemma, Unpublished manuscript, University of California at Berkeley, 1996.

[7]

M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, SIAM Journal on Optimization and Control, 44 (2005), 328-348. doi: 10.1137/S0363012904439301.

[8]

M. Berger, Géométrie, Vol 3 : Convexes et Polytopes, Polyèdres Réguliers, Aires et Volumes, Fernand-Nathan, Paris, 1977.

[9]

D. Blackwell, An analog of the minmax theorem for vector payoffs, Pacific Journal of Mathematics, 6 (1956), 1-8. 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), 624-640. doi: 10.1287/moor.1100.0455.

[11]

G. Hardin, The tragedy of the commons, Journal of Natural Resources Policy Research, 1 (2009), 243-253. doi: 10.1080/19390450903037302.

[12]

V. Perchet, Approachability, regret and calibration: Implications and equivalences, Journal of Dynamics and Games, 1 (2014), 181-254. 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), 1617-1634. doi: 10.2307/1911925.

show all references

References:
[1]

K. Abhyankar, Smale Strategies for Prisoner's Dilemma Type Games, Doctoral dissertation, University of California at Berkeley, 2001.

[2]

R. Axelrod, The Evolution of Cooperation, Basic Book, Inc Publishers, New York, 1984.

[3]

M. Benaïm, A dynamical system approach to stochastic approximation, SIAM Journal on Optimization and Control, 34 (1996), 437-472. doi: 10.1137/S0363012993253534.

[4]

M. Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités XXXIII, Lecture Notes in Math, 1709 (1999), 1-68. 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), 141-176. doi: 10.1007/BF02218617.

[6]

M. Benaïm and M. Hirsch, Stochastic Adaptive Behavior for Prisoner's Dilemma, Unpublished manuscript, University of California at Berkeley, 1996.

[7]

M. Benaïm, J. Hofbauer and S. Sorin, Stochastic approximations and differential inclusions, SIAM Journal on Optimization and Control, 44 (2005), 328-348. doi: 10.1137/S0363012904439301.

[8]

M. Berger, Géométrie, Vol 3 : Convexes et Polytopes, Polyèdres Réguliers, Aires et Volumes, Fernand-Nathan, Paris, 1977.

[9]

D. Blackwell, An analog of the minmax theorem for vector payoffs, Pacific Journal of Mathematics, 6 (1956), 1-8. 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), 624-640. doi: 10.1287/moor.1100.0455.

[11]

G. Hardin, The tragedy of the commons, Journal of Natural Resources Policy Research, 1 (2009), 243-253. doi: 10.1080/19390450903037302.

[12]

V. Perchet, Approachability, regret and calibration: Implications and equivalences, Journal of Dynamics and Games, 1 (2014), 181-254. 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), 1617-1634. doi: 10.2307/1911925.

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