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In and out of equilibrium I: Evolution of strategies in repeated games with discounting

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  • García, Julián
  • van Veelen, Matthijs

Abstract

In the repeated prisoner's dilemma there is no strategy that is evolutionarily stable, and a profusion of neutrally stable ones. But how stable is neutrally stable? We show that in repeated games with large enough continuation probabilities, where the stage game is characterized by a conflict between individual and collective interests, there is always a neutral mutant that can drift into a population that is playing an equilibrium, and create a selective advantage for a second mutant. The existence of stepping stone paths out of any equilibrium determines the dynamics in finite populations playing the repeated prisoner's dilemma.

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  • García, Julián & van Veelen, Matthijs, 2016. "In and out of equilibrium I: Evolution of strategies in repeated games with discounting," Journal of Economic Theory, Elsevier, vol. 161(C), pages 161-189.
    • Handle: RePEc:eee:jetheo:v:161:y:2016:i:c:p:161-189
    DOI: 10.1016/j.jet.2015.11.007
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    1. Bomze Immanuel M. & Weibull Jorgen W., 1995. "Does Neutral Stability Imply Lyapunov Stability?," Games and Economic Behavior, Elsevier, vol. 11(2), pages 173-192, November.
    2. Glenn Ellison, 2000. "Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(1), pages 17-45.
    3. Bendor, Jonathan & Swistak, Piotr, 1997. "The Evolutionary Stability of Cooperation," American Political Science Review, Cambridge University Press, vol. 91(2), pages 290-307, June.
    4. van Damme, Eric, 1989. "Renegotiation-proof equilibria in repeated prisoners' dilemma," Journal of Economic Theory, Elsevier, vol. 47(1), pages 206-217, February.
    5. Farrell, Joseph & Maskin, Eric, 1989. "Renegotiation-proof equilibrium: Reply," Journal of Economic Theory, Elsevier, vol. 49(2), pages 376-378, December.
    6. Sandholm, William H. & Staudigl, Mathias, 2016. "Large deviations and stochastic stability in the small noise double limit," Theoretical Economics, Econometric Society, vol. 11(1), January.
    7. van Veelen, Matthijs & García, Julián, 2019. "In and out of equilibrium II: Evolution in repeated games with discounting and complexity costs," Games and Economic Behavior, Elsevier, vol. 115(C), pages 113-130.
    8. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
    9. Drew Fudenberg & Eric Maskin, 2008. "The Folk Theorem In Repeated Games With Discounting Or With Incomplete Information," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 11, pages 209-230, World Scientific Publishing Co. Pte. Ltd..
    10. Selten, Reinhard, 1983. "Evolutionary stability in extensive two-person games," Mathematical Social Sciences, Elsevier, vol. 5(3), pages 269-363, September.
    11. Matthijs van Veelen, 2010. "But Some Neutrally Stable Strategies are More Neutrally Stable than Others," Tinbergen Institute Discussion Papers 10-033/1, Tinbergen Institute.
    12. Kandori, Michihiro & Mailath, George J & Rob, Rafael, 1993. "Learning, Mutation, and Long Run Equilibria in Games," Econometrica, Econometric Society, vol. 61(1), pages 29-56, January.
    13. Bergin, James & Lipman, Barton L, 1996. "Evolution with State-Dependent Mutations," Econometrica, Econometric Society, vol. 64(4), pages 943-956, July.
    14. Linster, Bruce G, 1994. "Stochastic Evolutionary Dynamics in the Repeated Prisoners' Dilemma," Economic Inquiry, Western Economic Association International, vol. 32(2), pages 342-357, April.
    15. Samuelson, Larry & Swinkels, Jeroen M., 2003. "Evolutionary stability and lexicographic preferences," Games and Economic Behavior, Elsevier, vol. 44(2), pages 332-342, August.
    16. William H. Sandholm, 2001. "Almost global convergence to p-dominant equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(1), pages 107-116.
    17. Cressman, R., 1996. "Evolutionary Stability in the Finitely Repeated Prisoner 's Dilemma Game," Journal of Economic Theory, Elsevier, vol. 68(1), pages 234-248, January.
    18. Karl H. Schlag & Dieter Balkenborg, 2001. "Evolutionarily stable sets," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(4), pages 571-595.
    19. Robson, A.J., 1989. "Efficiency In Evolutionary Games: Darwin, Nash And Secret Handshake," Papers 89-22, Michigan - Center for Research on Economic & Social Theory.
    20. Jack Hirshleifer & Juan Carlos Martinez Coll, 1988. "What Strategies Can Support the Evolutionary Emergence of Cooperation?," Journal of Conflict Resolution, Peace Science Society (International), vol. 32(2), pages 367-398, June.
    21. Matthijs Veelen & Peter Spreij, 2009. "Evolution in games with a continuous action space," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(3), pages 355-376, June.
    22. Miller, John H., 1996. "The coevolution of automata in the repeated Prisoner's Dilemma," Journal of Economic Behavior & Organization, Elsevier, vol. 29(1), pages 87-112, January.
    23. García, Julián & van Veelen, Matthijs, 2016. "In and out of equilibrium I: Evolution of strategies in repeated games with discounting," Journal of Economic Theory, Elsevier, vol. 161(C), pages 161-189.
    24. Cooper, David J., 1996. "Supergames Played by Finite Automata with Finite Costs of Complexity in an Evolutionary Setting," Journal of Economic Theory, Elsevier, vol. 68(1), pages 266-275, January.
    25. , & , H. & ,, 2015. "Sampling best response dynamics and deterministic equilibrium selection," Theoretical Economics, Econometric Society, vol. 10(1), January.
    26. Kim, Yong-Gwan, 1994. "Evolutionarily stable strategies in the repeated prisoner's dilemma," Mathematical Social Sciences, Elsevier, vol. 28(3), pages 167-197, December.
    27. Peyton Young, H. & Foster, Dean, 1991. "Cooperation in the long-run," Games and Economic Behavior, Elsevier, vol. 3(1), pages 145-156, February.
    28. Ho, Teck-Hua, 1996. "Finite automata play repeated prisoner's dilemma with information processing costs," Journal of Economic Dynamics and Control, Elsevier, vol. 20(1-3), pages 173-207.
    29. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, vol. 56(2), pages 397-410, March.
    30. Binmore, Kenneth G. & Samuelson, Larry, 1992. "Evolutionary stability in repeated games played by finite automata," Journal of Economic Theory, Elsevier, vol. 57(2), pages 278-305, August.
    31. Abreu, Dilip & Rubinstein, Ariel, 1988. "The Structure of Nash Equilibrium in Repeated Games with Finite Automata," Econometrica, Econometric Society, vol. 56(6), pages 1259-1281, November.
    32. Rubinstein, Ariel, 1979. "Equilibrium in supergames with the overtaking criterion," Journal of Economic Theory, Elsevier, vol. 21(1), pages 1-9, August.
    33. Jonathan Bendor & Piotr Swistak, 1998. "Evolutionary Equilibria: Characterization Theorems and Their Implications," Theory and Decision, Springer, vol. 45(2), pages 99-159, October.
    34. Rubinstein, Ariel, 1986. "Finite automata play the repeated prisoner's dilemma," Journal of Economic Theory, Elsevier, vol. 39(1), pages 83-96, June.
    35. Pedro Dal BÛ, 2002. "Cooperation Under the Shadow of the Future: Experimental Evidence from Infinitely Repeated Games," Working Papers 2002-20, Brown University, Department of Economics.
    36. Fudenberg, Drew & Maskin, Eric, 1990. "Evolution and Cooperation in Noisy Repeated Games," American Economic Review, American Economic Association, vol. 80(2), pages 274-279, May.
    37. van Veelen, Matthijs, 2012. "Robustness against indirect invasions," Games and Economic Behavior, Elsevier, vol. 74(1), pages 382-393.
    38. Volij, Oscar, 2002. "In Defense of DEFECT," Games and Economic Behavior, Elsevier, vol. 39(2), pages 309-321, May.
    39. Binmore, Ken & Samuelson, Larry, 1997. "Muddling Through: Noisy Equilibrium Selection," Journal of Economic Theory, Elsevier, vol. 74(2), pages 235-265, June.
    40. Pedro Dal Bó, 2005. "Cooperation under the Shadow of the Future: Experimental Evidence from Infinitely Repeated Games," American Economic Review, American Economic Association, vol. 95(5), pages 1591-1604, December.
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    Cited by:

    1. Heller, Yuval, 2015. "Three steps ahead," Theoretical Economics, Econometric Society, vol. 10(1), January.
    2. van Veelen, Matthijs & García, Julián, 2019. "In and out of equilibrium II: Evolution in repeated games with discounting and complexity costs," Games and Economic Behavior, Elsevier, vol. 115(C), pages 113-130.
    3. Shun Kurokawa & Joe Yuichiro Wakano & Yasuo Ihara, 2018. "Evolution of Groupwise Cooperation: Generosity, Paradoxical Behavior, and Non-Linear Payoff Functions," Games, MDPI, vol. 9(4), pages 1-24, December.
    4. Matthijs van Veelen, 2020. "The evolution of morality," Tinbergen Institute Discussion Papers 20-063/I, Tinbergen Institute.
    5. Saral, Ali Seyhun, 2020. "Evolution of Conditional Cooperation in Prisoner's Dilemma," OSF Preprints wcpkz, Center for Open Science.
    6. Matthijs van Veelen & Benjamin Allen & Moshe Hoffman & Burton Simon & Carl Veller, 2016. "Inclusive Fitness," Tinbergen Institute Discussion Papers 16-055/I, Tinbergen Institute.
    7. Peter S. Park & Martin A. Nowak & Christian Hilbe, 2022. "Cooperation in alternating interactions with memory constraints," Nature Communications, Nature, vol. 13(1), pages 1-11, December.
    8. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    9. García, Julián & van Veelen, Matthijs, 2016. "In and out of equilibrium I: Evolution of strategies in repeated games with discounting," Journal of Economic Theory, Elsevier, vol. 161(C), pages 161-189.
    10. Izquierdo, Luis R. & Izquierdo, Segismundo S. & Sandholm, William H., 2019. "An introduction to ABED: Agent-based simulation of evolutionary game dynamics," Games and Economic Behavior, Elsevier, vol. 118(C), pages 434-462.
    11. Maria Kleshnina & Christian Hilbe & Štěpán Šimsa & Krishnendu Chatterjee & Martin A. Nowak, 2023. "The effect of environmental information on evolution of cooperation in stochastic games," Nature Communications, Nature, vol. 14(1), pages 1-11, December.
    12. Benjamin M Zagorsky & Johannes G Reiter & Krishnendu Chatterjee & Martin A Nowak, 2013. "Forgiver Triumphs in Alternating Prisoner's Dilemma," PLOS ONE, Public Library of Science, vol. 8(12), pages 1-8, December.

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    More about this item

    Keywords

    Repeated games; Evolutionary game theory;

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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