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Imitation Processes with Small Mutations

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  • Fudenberg, Drew
  • Imhof, Lorens

Abstract

This note characterizes the impact of adding rare stochastic mutations to an “imitation dynamic,†meaning a process with the properties that absent strategies remain absent, and non-homogeneous states are transient. The resulting system will spend almost all of its time at the absorbing states of the no-mutation process. The work of Freidlin and Wentzell [Random Perturbations of Dynamical Systems, Springer, New York, 1984] and its extensions provide a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply. This note provides a simpler and more intuitive algorithm. Loosely speaking, in a process with K strategies, it is sufficient to find the invariant distribution of a K×K Markov matrix on the K homogeneous states, where the probability of a transit from “all play i†to “all play j†is the probability of a transition from the state “all agents but 1 play i, 1 plays j†to the state “all play jâ€.

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Bibliographic Info

Paper provided by Harvard University Department of Economics in its series Scholarly Articles with number 3190369.

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Date of creation: 2006
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Publication status: Published in Journal of Economic Theory
Handle: RePEc:hrv:faseco:3190369

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References

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  1. BERGIN, James & LIPMAN, Bart, 1994. "Evolution with State-Dependent Mutations," CORE Discussion Papers 1994055, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  2. Ellison, Glenn, 2000. "Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution," Review of Economic Studies, Wiley Blackwell, vol. 67(1), pages 17-45, January.
  3. Kandori Michihiro & Rob Rafael, 1995. "Evolution of Equilibria in the Long Run: A General Theory and Applications," Journal of Economic Theory, Elsevier, vol. 65(2), pages 383-414, April.
  4. Schlag, Karl H., 1998. "Why Imitate, and If So, How?, : A Boundedly Rational Approach to Multi-armed Bandits," Journal of Economic Theory, Elsevier, vol. 78(1), pages 130-156, January.
  5. A. Banerjee & Drew Fudenberg, 2010. "Word-of-Mouth Communication and Social Learning," Levine's Working Paper Archive 425, David K. Levine.
  6. G. Ellison & D. Fudenberg, 2010. "Rules of Thumb for Social Learning," Levine's Working Paper Archive 435, David K. Levine.
  7. Michihiro, Kandori & Rob, Rafael, 1998. "Bandwagon Effects and Long Run Technology Choice," Games and Economic Behavior, Elsevier, vol. 22(1), pages 30-60, January.
  8. Noeldecke,Georg & Samuelson,Larry, . "An evolutionary analysis of backward and forward induction," Discussion Paper Serie B 228, University of Bonn, Germany.
  9. Karandikar, Rajeeva & Mookherjee, Dilip & Ray, Debraj & Vega-Redondo, Fernando, 1998. "Evolving Aspirations and Cooperation," Journal of Economic Theory, Elsevier, vol. 80(2), pages 292-331, June.
  10. M. Kandori & G. Mailath & R. Rob, 1999. "Learning, Mutation and Long Run Equilibria in Games," Levine's Working Paper Archive 500, David K. Levine.
  11. Drew Fudenberg & David K. Levine, 1996. "The Theory of Learning in Games," Levine's Working Paper Archive 624, David K. Levine.
  12. Sandholm, William H., 1998. "Simple and clever decision rules for a model of evolution," Economics Letters, Elsevier, vol. 61(2), pages 165-170, November.
  13. Glen Ellison, 2010. "Learning, Local Interaction, and Coordination," Levine's Working Paper Archive 391, David K. Levine.
  14. K. Schlag, 2010. "Why Imitate, and if so, How? Exploring a Model of Social Evolution," Levine's Working Paper Archive 454, David K. Levine.
  15. P. Young, 1999. "The Evolution of Conventions," Levine's Working Paper Archive 485, David K. Levine.
  16. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
  17. D. Foster & P. Young, 2010. "Stochastic Evolutionary Game Dynamics," Levine's Working Paper Archive 493, David K. Levine.
  18. Nowak, Martin & Sasaki, Akira & Fudenberg, Drew & Taylor, Christine, 2004. "Emergence of Cooperation and Evolutionary Stability in Finite Populations," Scholarly Articles 3196331, Harvard University Department of Economics.
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Citations

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Cited by:
  1. Glenn Ellison & Drew Fudenberg & Lorens A. Imhof, 2007. "Random matching in adaptive dynamics," Levine's Bibliography 321307000000000815, UCLA Department of Economics.
  2. Matthey, Astrid, 2010. "Imitation with intention and memory: An experiment," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 39(5), pages 585-594, October.
  3. van Veelen, Matthijs, 2012. "Robustness against indirect invasions," Games and Economic Behavior, Elsevier, vol. 74(1), pages 382-393.
  4. repec:dgr:uvatin:2010033 is not listed on IDEAS
  5. Kevin Hasker, 2014. "The Emergent Seed: A Representation Theorem for Models of Stochastic Evolution and two formulas for Waiting Time," Levine's Working Paper Archive 786969000000000954, David K. Levine.
  6. Sandholm, William H., 2012. "Stochastic imitative game dynamics with committed agents," Journal of Economic Theory, Elsevier, vol. 147(5), pages 2056-2071.
  7. Fudenberg, Drew & Imhof, Lorens A., 2008. "Monotone imitation dynamics in large populations," Journal of Economic Theory, Elsevier, vol. 140(1), pages 229-245, May.
  8. Sekiguchi, Takuya, 2013. "General conditions for strategy abundance through a self-referential mechanism under weak selection," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(13), pages 2886-2892.
  9. Darong DAI & Kunrong SHEN, 2014. "Stochastic evolutionary cartel formation," Theoretical and Applied Economics, Asociatia Generala a Economistilor din Romania - AGER, vol. 0(1(590)), pages 7-26, January.
  10. Cui, Zhiwei & Zhai, Jian, 2010. "Escape dynamics and equilibria selection by iterative cycle decomposition," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1015-1029, November.
  11. Bin Wu & Arne Traulsen & Chaitanya S. Gokhale, 2013. "Dynamic Properties of Evolutionary Multi-player Games in Finite Populations," Games, MDPI, Open Access Journal, vol. 4(2), pages 182-199, May.
  12. Manapat, Michael L. & Nowak, Martin A. & Rand, David G., 2013. "Information, irrationality, and the evolution of trust," Journal of Economic Behavior & Organization, Elsevier, vol. 90(S), pages S57-S75.

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