Time averages, recurrence and transience in the stochastic replicator dynamics

Annals of Applied Probability

Volumn 19, Issue 4, 2009, Pages 1347-1368

  Hofbauer, Josef ^a,b   Imhof, Lorens A ^a,b  

^a UNIVERSITY OF VIENNA   (Austria)

^b UNIVERSITY OF BONN   (Germany)

Author keywords

Averaging principle; Dirichlet distribution; Exclusion principle; Invariant distribution; Lyapunov function; Nash equilibrium; Stochastic asymptotic stability; Stochastic differential equation

Indexed keywords

EID: 70049094993     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/08-AAP577     Document Type: Article

References (36)

1. AMIR, M. and BERNINGHAUS, S. (1998). Scale functions in equilibrium selection games. J. Evol. Econom. 8 1-13. (Pubitemid 128353591)
2. ARNOLD, L. (1974). Stochastic Differential Equations: Theory and Applications. Wiley, New York. MR0443083
3. BEGGS, A. (2002). Stochastic evolution with slow learning. Econom. Theory 19 379-405. MR1889582
4. BENAÏM, M., HOFBAUER, J. and SANDHOLM, W. H. (2008). Robust permanence and impermanence for stochastic replicator dynamics. J. Biol. Dyn. 2 180-195. MR2427526
5. BENAÏM, M., SCHREIBER, S. J. and TARRÉS, P. (2004). Generalized urn models of evolutionary processes. Ann. Appl. Probab. 14 1455-1478. MR2071430
6. BHATTACHARYA, R. N. (1978). Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab. 6 541-553. MR0494525
7. BHATTACHARYA, R. N. andWAYMIRE, E. C. (1990). Stochastic Processes with Applications. Wiley, New York. MR1054645
8. CABRALES, A. (2000). Stochastic replicator dynamics. Internat. Econom. Rev. 41 451-481. MR1760461
9. CORRADI, V. and SARIN, R. (2000). Continuous approximations of stochastic evolutionary game dynamics. J. Econom. Theory 94 163-191. MR1791297
10. FOSTER, D. and YOUNG, P. (1990). Stochastic evolutionary game dynamics. Theoret. Population Biol. 38 219-232. MR1075000
11. FRIEDMAN, A. (1975). Stochastic Differential Equations and Applications. Vol. 1. Probability and Mathematical Statistics 28. Academic Press, New York. MR0494490
12. FUDENBERG, D. and HARRIS, C. (1992). Evolutionary dynamics with aggregate shocks. J. Econom. Theory 57 420-441. MR1180005
13. FUDENBERG, D. and LEVINE, D. K. (1995). Consistency and cautious fictitious play. J. Econom. Dynam. Control 19 1065-1089. MR1346026
14. GICHMAN, I. I. and SKOROCHOD, A. W. (1971). Stochastische Differentialgleichungen. Akademie Verlag, Berlin. MR0346905
15. HART, S. (2005). Adaptive heuristics. Econometrica 73 1401-1430. MR2156669
16. HAS'MINSKIǏ , R. Z. (1980). Stochastic Stability of Differential Equations. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis 7. Sijthoff and Noordhoff, Germantown, MD. MR600653
17. HOFBAUER, J. (1981). A general cooperation theorem for hypercycles. Monatsh. Math. 91 233-240. MR619966
18. HOFBAUER, J. (2005). Time averages of the replicator dynamics and correlated equilibria. Preprint.
19. HOFBAUER, J. and SIGMUND, K. (1998). Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge. MR1635735
20. HOFBAUER, J. and SIGMUND, K. (2003). Evolutionary game dynamics. Bull. Amer. Math. Soc. (N.S.) 40 479-519 (electronic). MR1997349
21. IMHOF, L. A. (2005a). The long-run behavior of the stochastic replicator dynamics. Ann. Appl. Probab. 15 1019-1045. MR2114999
22. IMHOF, L. A. (2005b). Evolutionary game dynamics under stochastic influences. Jahresber. Deutsch. Math.-Verein. 107 197-213. MR2195562
23. IMHOF, L. A. (2008). Multiple-trial conflicts and stochastic evolutionary game dynamics. Adv. in Appl. Probab. 40 1174-1197.
24. KARATZAS, I. and SHREVE, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York. MR1121940
25. KHAS'MINSKIǏ, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Theory Probab. Appl. 5 179-196. MR0133871
26. KHASMINSKII, R. and POTSEPUN, N. (2006). On the replicator dynamics behavior under Stratonovich type random perturbations. Stoch. Dyn. 6 197-211. MR2239089
27. NOWAK, M. A. and SIGMUND, K. (2004). Evolutionary dynamics of biological games. Science 303 793-799.
28. PINSKY, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics 45. Cambridge Univ. Press, Cambridge. MR1326606
29. SAITO, M. (1997). A note on ergodic distributions in two-agent economies. J. Math. Econom. 27 133-141. MR1431151
30. SANDHOLM, W. H. (2007). Evolutionary game theory. Preprint.
31. SCHREIBER, S. J. (2001). Urn models, replicator processes, and random genetic drift. SIAM J. Appl. Math. 61 2148-2167 (electronic). MR1856886
32. SCHUSTER, P., SIGMUND, K., HOFBAUER, J. andWOLFF, R. (1981). Self-regulation of behaviour in animal societies. I. Symmetric contests. Biol. Cybernet. 40 1-8. MR609925
33. TAYLOR, P. D. and JONKER, L. B. (1978). Evolutionarily stable strategies and game dynamics. Math. Biosci. 40 145-156. MR0489983
34. TURELLI, M. (1977). Random environments and stochastic calculus. Theoret. Population Biology 12 140-178. MR0465290
35. WEIBULL, J. W. (1995). Evolutionary Game Theory. MIT Press, Cambridge, MA. MR1347921
36. YOUNG, H. P. (2004). Strategic Learning and Its Limits. Oxford Univ. Press, Oxford.