# American Institute of Mathematical Sciences

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

## Hamiltonian evolutionary games

 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.
Citation: Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics and 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. [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. [3] J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, 242, Birkhäuser Verlag, Basel, 2005. [4] I. Eshel and E. Akin, Coevolutionary instability and mixed Nash solutions, J. Math. Biol., 18 (1983), 123-133. doi: 10.1007/BF00280661. [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. [6] J. Hofbauer, Evolutionary dynamics for bimatrix games: A Hamiltonian system?, J. Math. Biol., 34 (1996), 675-688. doi: 10.1007/BF02409754. [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. [8] J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179. [9] J. T. Howson, Jr., Equilibria of polymatrix games, Management Sci., 18 (1971/72), 312-318. [10] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982. [11] J. Nash, Non-cooperative games, Ann. of Math. (2), 54 (1951), 286-295. doi: 10.2307/1969529. [12] G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games, J. Math. Biol., 19 (1984), 329-334. doi: 10.1007/BF00277103. [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. [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. [15] M. Plank, Hamiltonian structures for the $n$-dimensional Lotka-Volterra equations, J. Math. Phys., 36 (1995), 3520-3534. doi: 10.1063/1.530978. [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. [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. [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. [19] K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games, Econometrica, 63 (1995), 1371-1399. doi: 10.2307/2171774. [20] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010. [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. [22] V. Volterra, Leçons Sur La Théorie Mathématique de la Lutte Pour La Vie, Éditions Jacques Gabay, Sceaux, 1990.

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##### References:
 [1] E. Akin and V. Losert, Evolutionary dynamics of zero-sum games, J. Math. Biol., 20 (1984), 231-258. doi: 10.1007/BF00275987. [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. [3] J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, 242, Birkhäuser Verlag, Basel, 2005. [4] I. Eshel and E. Akin, Coevolutionary instability and mixed Nash solutions, J. Math. Biol., 18 (1983), 123-133. doi: 10.1007/BF00280661. [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. [6] J. Hofbauer, Evolutionary dynamics for bimatrix games: A Hamiltonian system?, J. Math. Biol., 34 (1996), 675-688. doi: 10.1007/BF02409754. [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. [8] J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179. [9] J. T. Howson, Jr., Equilibria of polymatrix games, Management Sci., 18 (1971/72), 312-318. [10] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, 1982. [11] J. Nash, Non-cooperative games, Ann. of Math. (2), 54 (1951), 286-295. doi: 10.2307/1969529. [12] G. Palm, Evolutionary stable strategies and game dynamics for $n$-person games, J. Math. Biol., 19 (1984), 329-334. doi: 10.1007/BF00277103. [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. [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. [15] M. Plank, Hamiltonian structures for the $n$-dimensional Lotka-Volterra equations, J. Math. Phys., 36 (1995), 3520-3534. doi: 10.1063/1.530978. [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. [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. [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. [19] K. Ritzberger and J. Weibull, Evolutionary selection in normal-form games, Econometrica, 63 (1995), 1371-1399. doi: 10.2307/2171774. [20] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, Cambridge, MA, 2010. [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. [22] V. Volterra, Leçons Sur La Théorie Mathématique de la Lutte Pour La Vie, Éditions Jacques Gabay, Sceaux, 1990.
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