American Institute of Mathematical Sciences

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January  2015, 2(1): 33-49. doi: 10.3934/jdg.2015.2.33

Hamiltonian evolutionary games

Hassan Najafi Alishah 1, and  Pedro Duarte 2,

1. 

Departamento de Matemática and CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2. 

Departamento de Matemática and CMAF, Faculdade de Ciências,Universidade de Lisboa, Campo Grande, Edi ficio C6, Piso 2, 1749-016 Lisboa, Portugal

Received  May 2014 Revised  February 2015 Published  June 2015

We introduce a class of o.d.e.'s that generalizes to polymatrix games the replicator equations on symmetric and asymmetric games. We also introduce a new class of Poisson structures on the phase space of these systems, and characterize the corresponding subclass of Hamiltonian polymatrix replicator systems. This extends known results for symmetric and asymmetric replicator systems.
Keywords: Hamiltonian polymatrix replicator equation, Evolutionary games, poisson Hamiltonian systems..
Mathematics Subject Classification: Primary: 91A22, 37C10; Secondary: 34G20, 53D1.
Citation: Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33
References:
[1]

E. Akin and V. Losert, Evolutionary dynamics of zero-sum games, J. Math. Biol., 20 (1984), 231-258. doi: 10.1007/BF00275987.  Google Scholar

[2]

P. Duarte, R. L. Fernandes and W. M. Oliva, Dynamics of the attractor in the Lotka-Volterra equations, J. Differential Equations, 149 (1998), 143-189. doi: 10.1006/jdeq.1998.3443.  Google Scholar

[3]

J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, 242, Birkhäuser Verlag, Basel, 2005.  Google Scholar

[4]

I. Eshel and E. Akin, Coevolutionary instability and mixed Nash solutions, J. Math. Biol., 18 (1983), 123-133. doi: 10.1007/BF00280661.  Google Scholar

[5]

R. L. Fernandes, J.-P. Oretga and T. S. Ratiu, The momentum map in Poisson geometry, Amer. J. Math., 131 (2009), 1261-1310. doi: 10.1353/ajm.0.0068.  Google Scholar

[6]

J. Hofbauer, Evolutionary dynamics for bimatrix games: A Hamiltonian system?, J. Math. Biol., 34 (1996), 675-688. doi: 10.1007/BF02409754.  Google Scholar

[7]

J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal., 5 (1981), 1003-1007. doi: 10.1016/0362-546X(81)90059-6.  Google Scholar

[8]

J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.  Google Scholar

[9]

J. T. Howson, Jr., Equilibria of polymatrix games,, Management Sci., 18 (): 312.   Google Scholar

[10]

J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982. Google Scholar

[11]

J. Nash, Non-cooperative games, Ann. of Math. (2), 54 (1951), 286-295. doi: 10.2307/1969529.  Google Scholar

[12]

G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games, J. Math. Biol., 19 (1984), 329-334. doi: 10.1007/BF00277103.  Google Scholar

[13]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[14]

M. Plank, Bi-Hamiltonian systems and Lotka-Volterra equations: A three-dimensional classification, Nonlinearity, 9 (1996), 887-896. doi: 10.1088/0951-7715/9/4/004.  Google Scholar

[15]

M. Plank, Hamiltonian structures for the $n$-dimensional Lotka-Volterra equations, J. Math. Phys., 36 (1995), 3520-3534. doi: 10.1063/1.530978.  Google Scholar

[16]

M. Plank, Some qualitative differences between the replicator dynamics of two player and $n$ player games, Nonlinear Anal., 30 (1997), 1411-1417. doi: 10.1016/S0362-546X(97)00202-2.  Google Scholar

[17]

R. Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable, J. Differential Equations, 82 (1989), 251-268. doi: 10.1016/0022-0396(89)90133-2.  Google Scholar

[18]

R. Redheffer and W. Walter, Solution of the stability problem for a class of generalized Volterra prey-predator systems, J. Differential Equations, 52 (1984), 245-263. doi: 10.1016/0022-0396(84)90179-7.  Google Scholar

[19]

K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games, Econometrica, 63 (1995), 1371-1399. doi: 10.2307/2171774.  Google Scholar

[20]

W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010.  Google Scholar

[21]

P. Schuster , K. Sigmund, J. Hofbauer, R. Gottlieb and P. Merz, Self-regulation of behaviour in animal societies. III. Games between two populations with self-interaction, Biol. Cybernet., 40 (1981), 17-25. doi: 10.1007/BF00326677.  Google Scholar

[22]

V. Volterra, Leçons Sur La Théorie Mathématique de la Lutte Pour La Vie, Éditions Jacques Gabay, Sceaux, 1990.  Google Scholar

show all references

References:
[1]

E. Akin and V. Losert, Evolutionary dynamics of zero-sum games, J. Math. Biol., 20 (1984), 231-258. doi: 10.1007/BF00275987.  Google Scholar

[2]

P. Duarte, R. L. Fernandes and W. M. Oliva, Dynamics of the attractor in the Lotka-Volterra equations, J. Differential Equations, 149 (1998), 143-189. doi: 10.1006/jdeq.1998.3443.  Google Scholar

[3]

J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, 242, Birkhäuser Verlag, Basel, 2005.  Google Scholar

[4]

I. Eshel and E. Akin, Coevolutionary instability and mixed Nash solutions, J. Math. Biol., 18 (1983), 123-133. doi: 10.1007/BF00280661.  Google Scholar

[5]

R. L. Fernandes, J.-P. Oretga and T. S. Ratiu, The momentum map in Poisson geometry, Amer. J. Math., 131 (2009), 1261-1310. doi: 10.1353/ajm.0.0068.  Google Scholar

[6]

J. Hofbauer, Evolutionary dynamics for bimatrix games: A Hamiltonian system?, J. Math. Biol., 34 (1996), 675-688. doi: 10.1007/BF02409754.  Google Scholar

[7]

J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal., 5 (1981), 1003-1007. doi: 10.1016/0362-546X(81)90059-6.  Google Scholar

[8]

J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.  Google Scholar

[9]

J. T. Howson, Jr., Equilibria of polymatrix games,, Management Sci., 18 (): 312.   Google Scholar

[10]

J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982. Google Scholar

[11]

J. Nash, Non-cooperative games, Ann. of Math. (2), 54 (1951), 286-295. doi: 10.2307/1969529.  Google Scholar

[12]

G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games, J. Math. Biol., 19 (1984), 329-334. doi: 10.1007/BF00277103.  Google Scholar

[13]

A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-9257-5.  Google Scholar

[14]

M. Plank, Bi-Hamiltonian systems and Lotka-Volterra equations: A three-dimensional classification, Nonlinearity, 9 (1996), 887-896. doi: 10.1088/0951-7715/9/4/004.  Google Scholar

[15]

M. Plank, Hamiltonian structures for the $n$-dimensional Lotka-Volterra equations, J. Math. Phys., 36 (1995), 3520-3534. doi: 10.1063/1.530978.  Google Scholar

[16]

M. Plank, Some qualitative differences between the replicator dynamics of two player and $n$ player games, Nonlinear Anal., 30 (1997), 1411-1417. doi: 10.1016/S0362-546X(97)00202-2.  Google Scholar

[17]

R. Redheffer, A new class of Volterra differential equations for which the solutions are globally asymptotically stable, J. Differential Equations, 82 (1989), 251-268. doi: 10.1016/0022-0396(89)90133-2.  Google Scholar

[18]

R. Redheffer and W. Walter, Solution of the stability problem for a class of generalized Volterra prey-predator systems, J. Differential Equations, 52 (1984), 245-263. doi: 10.1016/0022-0396(84)90179-7.  Google Scholar

[19]

K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games, Econometrica, 63 (1995), 1371-1399. doi: 10.2307/2171774.  Google Scholar

[20]

W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010.  Google Scholar

[21]

P. Schuster , K. Sigmund, J. Hofbauer, R. Gottlieb and P. Merz, Self-regulation of behaviour in animal societies. III. Games between two populations with self-interaction, Biol. Cybernet., 40 (1981), 17-25. doi: 10.1007/BF00326677.  Google Scholar

[22]

V. Volterra, Leçons Sur La Théorie Mathématique de la Lutte Pour La Vie, Éditions Jacques Gabay, Sceaux, 1990.  Google Scholar

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