Random Matching in Adaptive Dynamics

This paper studies the effect of randomness in per-period matching on the long-run outcome of non-equilibrium adaptive processes. If there are many matchings between each strategy revision, the randomness due to matching will be small; our question is when a very small noise due to matching has a negligible effect. We study two different senses of this idea, and provide sufficient conditions for each. The less demanding sense corresponds to sending the matching noise to zero while holding fixed all other aspects of the adaptive process. The second sense in which matching noise can be negligible is that it does not alter the limit distribution obtained as the limit of the invariant distributions as an exogenous "mutation rate" goes to zero.

[1]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[2]  Fernando Vega-Redondo,et al.  Evolution, Games, and Economic Behaviour , 1996 .

[3]  R. Rob,et al.  Learning, Mutation, and Long Run Equilibria in Games , 1993 .

[4]  Drew Fudenberg,et al.  Monotone Imitation Dynamics in Large Populations , 2007, J. Econ. Theory.

[5]  H. Young,et al.  The Evolution of Conventions , 1993 .

[6]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[7]  Glenn Ellison.  Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution , 2000 .

[8]  Ken Binmore,et al.  Muddling Through: Noisy Equilibrium Selection☆ , 1997 .

[9]  C. D. Meyer,et al.  Comparison of perturbation bounds for the stationary distribution of a Markov chain , 2001 .

[10]  Marius Iosifescu,et al.  Finite Markov Processes and Their Applications , 1981 .

[11]  Drew Fudenberg,et al.  Imitation Processes with Small Mutations , 2004, J. Econ. Theory.

[12]  P. Billingsley,et al.  Probability and Measure , 1980 .

[13]  Glenn Ellison,et al.  Learning from Personal Experience: One Rational Guy and the Justification of Myopia , 1997 .

[14]  Drew Fudenberg,et al.  Evolutionary game dynamics in finite populations with strong selection and weak mutation. , 2006, Theoretical population biology.

[15]  J. Yorke,et al.  Basins of Attraction , 1996, Science.

[16]  Barton L. Lipman,et al.  Evolution with State-Dependent Mutations , 1996 .

[17]  Peter Secretan.  Learning , 1965, Mental Health.

[18]  F. Vega-Redondo,et al.  Efficient Equilibrium Selection in Evolutionary Games with Random Matching , 1996 .

[19]  M. Nowak,et al.  Evolutionary game dynamics in a Wright-Fisher process , 2006, Journal of mathematical biology.