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Persistence probability of a random polynomial arising from evolutionary game theory

Part of: Geometric function theory Game theory Polynomials, rational functions

Published online by Cambridge University Press:  01 October 2019

Van Hao Can ,
Manh Hong Duong  and
Viet Viet Hung Pham
Van Hao Can*
Affiliation:
Vietnam Academy of Science and Technology, and Kyoto University
Manh Hong Duong*
Affiliation:
University of Birmingham
Viet Viet Hung Pham*
Affiliation:
Vietnam Academy of Science and Technology
*
*Postal address: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Street, 10307 Hanoi, Vietnam.
**Postal address: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK.
*Postal address: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Street, 10307 Hanoi, Vietnam.
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Abstract

We obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.

Keywords

Random polynomialevolutionary game theorypersistence probabilityequilibrium pointGaussian process

MSC classification

Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Secondary: 26C10: Polynomials 91A22: Evolutionary games
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 870 - 890
Copyright
© Applied Probability Trust 2019 

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References

Bender, C. M. and Orszag, S. A. (1999). Advanced Mathematical Methods for Scientists And Engineers. I: Asymptotic Methods and Perturbation Theory. Springer, New York.CrossRefGoogle Scholar
Bharucha-Reid, A. T. and Sambandham, M. (1986). Random Polynomials. Probability and Mathematical Statistics. Academic Press, Orlando, FL.Google Scholar
Bloch, A. and Pólya, G. (1932). On the roots of certain algebraic equations. Proc. London Math. Soc. 33, 102–114.CrossRefGoogle Scholar
Broom, M., Cannings, C. and Vickers, G.T. (1997). Multi-player matrix games. Bull. Math. Biol. 59, 931–952.CrossRefGoogle ScholarPubMed
Butez, R. and Zeitouni, O. (2017). Universal large deviations for Kac polynomials. Electron. Commun. Probab. 22, 1–10.CrossRefGoogle Scholar
Can, V. H. and Pham, V. H. (2017). Persistence probability of random Weyl polynomial. Preprint, arXiv:1710.01090.Google Scholar
Dembo, A. and Mukherjee, S. (2015). No zero-crossings for random polynomials and the heat equation. Ann. Probab. 43, 85–118.CrossRefGoogle Scholar
Dembo, A. and Mukherjee, S. (2017). Persistence of Gaussian processes: non-summable correlations. Prob. Theory Relat. Fields 169, 1007–1039.CrossRefGoogle Scholar
Dembo, A., Poonen, B., Shao, Q.-M. and Zeitouni, O. (2002). Random polynomials having few or no real zeros. J. Amer. Math. Soc. 15, 857–892.CrossRefGoogle Scholar
Do, Y. and Vu, V. (2017) Central limit theorems for the real zeros of Weyl polynomials. Preprint, arXiv:1707.09276.Google Scholar
Duong, M. H. and Han, T. A. (2016). On the expected number of equilibria in a multi-player multi-strategy evolutionary game. Dyn. Games Appl. 6, 324–346.CrossRefGoogle Scholar
Duong, M. H. and Han, T. A. (2016). Analysis of the expected density of internal equilibria in random evolutionary multi-player multi-strategy games. J. Math, Biol , 73, 1727–1760.CrossRefGoogle ScholarPubMed
Duong, M. H., Tran, H. M. and Han, T. A. (2017). On the distribution of the number of internal equilibria in random evolutionary games. Preprint, arXiv:1711.03848.Google Scholar
Duong, M. H., Tran, H. M. and Han, T. A. (2017). On the expected number of internal equilibria in random evolutionary games with correlated payoff matrix. Preprint, arXiv:1708.01672.Google Scholar
Edelman, A. and Kostlan, E. (1995). How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. (N.S.) 32, 1–37.CrossRefGoogle Scholar
Farahmand, K. (1998). Topics in Random Polynomials. Chapman & Hall/CRC Research Notes in Mathematics Series. Taylor & Francis, London.Google Scholar
Fudenberg, D. and Harris, C. (1992). Evolutionary dynamics with aggregate shocks. J. Econom. Theory 57, 420–441.CrossRefGoogle Scholar
Fyodorov, Y. V. (2004). Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices. Phys. Rev. Lett. 92, 240601.CrossRefGoogle ScholarPubMed
Fyodorov, Y. V. and Khoruzhenko, B. A. (2016). Nonlinear analogue of the May–Wigner instability transition. Proc. Nat. Acad. Sci. 113 6827–6832.CrossRefGoogle ScholarPubMed
Fyodorov, Y. V. and Nadal, C. (2012). Critical behavior of the number of minima of a random landscape at the glass transition point and the Tracy–Widom distribution. Phys. Rev. Lett. 109, 167203.CrossRefGoogle ScholarPubMed
Gokhale, C. S. and Traulsen, A. (2010). Evolutionary games in the multiverse. Proc. Natl. Acad. Sci. U.S.A. 107, 5500–5504.CrossRefGoogle ScholarPubMed
Gokhale, C. S. and Traulsen, A. (2014). Evolutionary multiplayer games. Dyn. Games Appl. 4, 468–488.CrossRefGoogle Scholar
Gross, T, Rudolf, L., Levin, S. A. and Dieckmann, U. (2009). Generalized models reveal stabilizing factors in food webs. Science 325, 747–750.CrossRefGoogle ScholarPubMed
Han, T. A., Traulsen, A. and Gokhale, C. S. (2012). On equilibrium properties of evolutionary multi-player games with random payoff matrices. Theoret. Pop. Biol. 81, 264–272.CrossRefGoogle ScholarPubMed
Hofbauer, J. and Sigmund, K. (1998). Evolutionary Games and Population Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Li, W. V. and Shao, Q.-M. (2005). Recent developments on lower tail probabilities for Gaussian processes. Cosmos 1, 95–106.CrossRefGoogle Scholar
Littlewood, J. E. and Offord, A. C. (1939). On the number of real roots of a random algebraic equation. II. Math. Proc. Cambridge Philosoph. Soc. 35, 133–148.CrossRefGoogle Scholar
Littlewood, J. E. and Offord, A. C. (1948). On the distribution of zeros and a-values of a random integral function (II). Ann. Math. 49, 885–952.CrossRefGoogle Scholar
Lubinsky, D. S., Pritsker, I. E. and Xie, X. (2018). Expected number of real zeros for random orthogonal polynomials. Math. Proc. Cambridge Philosoph. Soc. 164, 47–66.CrossRefGoogle Scholar
May, R. M. (2001). Stability and Complexity in Model Ecosystems, Vol. 6. Princeton University Press.CrossRefGoogle Scholar
Maynard Smith, J. and Price, G. R. (1973). The logic of animal conflict. Nature 246, 15–18.CrossRefGoogle Scholar
Nguyen, H., Nguyen, O. and Vu, V. (2016). On the number of real roots of random polynomials. Commun. Contemp. Math. 18, 1550052.CrossRefGoogle Scholar
Schehr, G. and Majumdar, S. (2008). Real roots of random polynomials and zero crossing properties of diffusion equation. J. Statist. Phys. 132, 235–273.CrossRefGoogle Scholar
Sigmund, K. (2010). The Calculus of Selfishness. Princeton University Press.CrossRefGoogle Scholar
Tao, T. and Vu, V. (2015). Local universality of zeroes of random polynomials. Internat. Math. Res. Not. 2015, 5053–5139.CrossRefGoogle Scholar
Wang, X.-S. and Wong, R. (2012). Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. 10, 215–235.CrossRefGoogle Scholar