October  2018, 5(4): 283-309. doi: 10.3934/jdg.2018018

Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received  April 2017 Revised  October 2017 Published  August 2018

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

Citation: Andrew Vlasic. Long-run analysis of the stochastic replicator dynamics in the presence of random jumps. Journal of Dynamics and Games, 2018, 5 (4) : 283-309. doi: 10.3934/jdg.2018018
References:
[1]

M. Abundo, On first-passage times problem for one-dimensional jump-diffusion processes, Probability and Mathematical Statistics, 20 (2000), 399-423. 

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge, 2004.

[3]

N. BalabanJ. MerrinR. ChaitL. Kowalik and S. Leibler, Bacterial persistence as a phenotypic switch, Science, 305 (2004), 1622-1625.  doi: 10.1126/science.1099390.

[4]

R. BartoszynskiW. J. BűhlerW. Chan and D. K. Pearl, Population processes under the influence of disasters occurring independently of population size, Journal of Mathematical Biology, 27 (1989), 167-178.  doi: 10.1007/BF00276101.

[5]

M. BenaïmJ. Hofbauer and W. Sandholm, Robust permanence and impermanence for the stochastic replicator dynamic, Journal of Biological Dynamics, 2 (2008), 180-195. 

[6]

J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996.

[7]

R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Springer-Science+Business Media, B. V., London, 1957. doi: 10.1007/978-94-011-2106-4.

[8]

S. Bruan and W. Flűckiger, Increased population of the aphid aphis pomi at a motorway. part 2-the effect of drought and deicing salt, Environmental Pollution Series A, Ecological and Biological, 36 (1984), 261-270.  doi: 10.1016/0143-1471(84)90007-2.

[9]

A. Cabrales, Stochastic replicator dynamics, International Economic Review, 41 (2000), 451-481.  doi: 10.1111/1468-2354.00071.

[10]

D. DownS. P. Meyn and R. L. Tweedie, Exponential and uniform ergodicity of markov processes, Annals of Applied Probability, 23 (1995), 1671-1691.  doi: 10.1214/aop/1176987798.

[11]

E. Dynkin, Markov Processes, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.

[12]

D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, Journal of Economic Theory, 57 (1992), 420-441.  doi: 10.1016/0022-0531(92)90044-I.

[13]

I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York, 1972.

[14]

F. Hanson and H. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187.  doi: 10.1007/s002850050096.

[15]

R. Z. Has'minski${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Rockville, Maryland, USA, 2004.

[16]

J. Hofbauer and L. Imhof, Time averages, recurrence and transience in the stochastic replicator dynamic, Annals of Applied Probability, 19 (2009), 1347-1368.  doi: 10.1214/08-AAP577.

[17]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[18]

L. Imhof, The long-run behavior of the stochastic replicator dynamics, Annals of Applied Probability, 15 (2005), 1019-1045.  doi: 10.1214/105051604000000837.

[19]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991.

[20]

R. Khasminskii and N. Potsepun, On the replicator dynamics behavior under Stratonovich type random perturbations, Stochastic and Dynamics, 6 (2006), 197-211.  doi: 10.1142/S0219493706001712.

[21]

R. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Netherlands, 1980.

[22]

H. Kushner, Stochastic Stability and Control, Academic Press Inc., New York, 1967.

[23]

H. Masuda, Ergodicity and exponential $β$-mixing bounds for multidimensional diffusions with jumps, Stochastic Processes and their Applications, 25 (2007), 35-56. 

[24]

M. Menotti-Raymond and S. O'Brien, Dating the genetic bottleneck of the african cheetah, Proc. Natl. Acad. Sci., 90 (1993), 3172-3176.  doi: 10.1073/pnas.90.8.3172.

[25]

P. Mertikopoulos and Y. Viossat, Imitation dynamics with payoff shocks, International Journal of Game Theory, 45 (2016), 291-320.  doi: 10.1007/s00182-015-0505-7.

[26]

S. P. Meyn and R. L. Tweedie, Stability of markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Annals of Applied Probability, 25 (1993), 518-548. 

[27]

S. P. Meyn and R. L. Tweedie, A survey of Foster-Lyapunov techniques for general state space Markov processes, Proc. Workshop Stochastic Stability and Stochastic Stabilization.

[28]

N. RabalaisR. Turner and W. Wiseman, Gulf of mexico hypoxia, a.k.a, "the dead zone", Ann Rev Ecol Sys, 33 (2002), 235-263.  doi: 10.1146/annurev.ecolsys.33.010802.150513.

[29]

W. Sandholm and M. Staudigl, Large deviations and stochastic stability in the small noise double limit, Theoretical Economics, 11 (2016), 279-355. 

[30]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.

[31]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society, Moscow, 1989.

[32]

K. Taira, Diffusion processes and partial differential equations, North-Holland Mathematics Studies, 98 (1984), 197-210.  doi: 10.1016/S0304-0208(08)71499-3.

[33]

K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer-Verlag, Berlin-Heidelberg, 2004.

[34]

H. C. Tuckwell, On the first-exit time problem for temporally homogeneous markov processes, Journal of Applied Probability, 13 (1976), 39-48. 

[35]

A. Vlasic, Stochastic replicator dynamics subject to Markovian switching, Journal of Mathematical Analysis and Applications, 427 (2015), 235-247.  doi: 10.1016/j.jmaa.2015.02.016.

[36]

P. Young and D. Foster, Stochastic evolutionary game dynamics, Theoretical Population Biololgy, 38 (1990), 219-232. 

show all references

References:
[1]

M. Abundo, On first-passage times problem for one-dimensional jump-diffusion processes, Probability and Mathematical Statistics, 20 (2000), 399-423. 

[2]

D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Cambridge, 2004.

[3]

N. BalabanJ. MerrinR. ChaitL. Kowalik and S. Leibler, Bacterial persistence as a phenotypic switch, Science, 305 (2004), 1622-1625.  doi: 10.1126/science.1099390.

[4]

R. BartoszynskiW. J. BűhlerW. Chan and D. K. Pearl, Population processes under the influence of disasters occurring independently of population size, Journal of Mathematical Biology, 27 (1989), 167-178.  doi: 10.1007/BF00276101.

[5]

M. BenaïmJ. Hofbauer and W. Sandholm, Robust permanence and impermanence for the stochastic replicator dynamic, Journal of Biological Dynamics, 2 (2008), 180-195. 

[6]

J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996.

[7]

R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Springer-Science+Business Media, B. V., London, 1957. doi: 10.1007/978-94-011-2106-4.

[8]

S. Bruan and W. Flűckiger, Increased population of the aphid aphis pomi at a motorway. part 2-the effect of drought and deicing salt, Environmental Pollution Series A, Ecological and Biological, 36 (1984), 261-270.  doi: 10.1016/0143-1471(84)90007-2.

[9]

A. Cabrales, Stochastic replicator dynamics, International Economic Review, 41 (2000), 451-481.  doi: 10.1111/1468-2354.00071.

[10]

D. DownS. P. Meyn and R. L. Tweedie, Exponential and uniform ergodicity of markov processes, Annals of Applied Probability, 23 (1995), 1671-1691.  doi: 10.1214/aop/1176987798.

[11]

E. Dynkin, Markov Processes, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.

[12]

D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, Journal of Economic Theory, 57 (1992), 420-441.  doi: 10.1016/0022-0531(92)90044-I.

[13]

I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York, 1972.

[14]

F. Hanson and H. Tuckwell, Population growth with randomly distributed jumps, Journal of Mathematical Biology, 36 (1997), 169-187.  doi: 10.1007/s002850050096.

[15]

R. Z. Has'minski${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Rockville, Maryland, USA, 2004.

[16]

J. Hofbauer and L. Imhof, Time averages, recurrence and transience in the stochastic replicator dynamic, Annals of Applied Probability, 19 (2009), 1347-1368.  doi: 10.1214/08-AAP577.

[17]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[18]

L. Imhof, The long-run behavior of the stochastic replicator dynamics, Annals of Applied Probability, 15 (2005), 1019-1045.  doi: 10.1214/105051604000000837.

[19]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991.

[20]

R. Khasminskii and N. Potsepun, On the replicator dynamics behavior under Stratonovich type random perturbations, Stochastic and Dynamics, 6 (2006), 197-211.  doi: 10.1142/S0219493706001712.

[21]

R. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Netherlands, 1980.

[22]

H. Kushner, Stochastic Stability and Control, Academic Press Inc., New York, 1967.

[23]

H. Masuda, Ergodicity and exponential $β$-mixing bounds for multidimensional diffusions with jumps, Stochastic Processes and their Applications, 25 (2007), 35-56. 

[24]

M. Menotti-Raymond and S. O'Brien, Dating the genetic bottleneck of the african cheetah, Proc. Natl. Acad. Sci., 90 (1993), 3172-3176.  doi: 10.1073/pnas.90.8.3172.

[25]

P. Mertikopoulos and Y. Viossat, Imitation dynamics with payoff shocks, International Journal of Game Theory, 45 (2016), 291-320.  doi: 10.1007/s00182-015-0505-7.

[26]

S. P. Meyn and R. L. Tweedie, Stability of markovian processes Ⅲ: Foster-Lyapunov criteria for continuous-time processes, Annals of Applied Probability, 25 (1993), 518-548. 

[27]

S. P. Meyn and R. L. Tweedie, A survey of Foster-Lyapunov techniques for general state space Markov processes, Proc. Workshop Stochastic Stability and Stochastic Stabilization.

[28]

N. RabalaisR. Turner and W. Wiseman, Gulf of mexico hypoxia, a.k.a, "the dead zone", Ann Rev Ecol Sys, 33 (2002), 235-263.  doi: 10.1146/annurev.ecolsys.33.010802.150513.

[29]

W. Sandholm and M. Staudigl, Large deviations and stochastic stability in the small noise double limit, Theoretical Economics, 11 (2016), 279-355. 

[30]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.

[31]

A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society, Moscow, 1989.

[32]

K. Taira, Diffusion processes and partial differential equations, North-Holland Mathematics Studies, 98 (1984), 197-210.  doi: 10.1016/S0304-0208(08)71499-3.

[33]

K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer-Verlag, Berlin-Heidelberg, 2004.

[34]

H. C. Tuckwell, On the first-exit time problem for temporally homogeneous markov processes, Journal of Applied Probability, 13 (1976), 39-48. 

[35]

A. Vlasic, Stochastic replicator dynamics subject to Markovian switching, Journal of Mathematical Analysis and Applications, 427 (2015), 235-247.  doi: 10.1016/j.jmaa.2015.02.016.

[36]

P. Young and D. Foster, Stochastic evolutionary game dynamics, Theoretical Population Biololgy, 38 (1990), 219-232. 

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