Replicator-Mutator Dynamics with Three Strategies

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This Demonstration shows the phase portrait in an equilateral triangle of the replicator-mutator dynamics with three strategies. The parameters denote the payoff that an -strategist obtains in an interaction with a -strategist. The parameter denotes the total fraction of mutants or entrants in the population. The parameters denote the weights given by mutants to strategy , so is the fraction of mutants that adopt strategy . This Demonstration also calculates the critical points of the system and their corresponding eigenvalues, which are helpful in assessing the dynamic stability of the critical point.

Contributed by: Luis R. Izquierdo and Segismundo S. Izquierdo (July 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Snapshot 1: system analyzed in [1]

Snapshot 2: if the payoff matrix is symmetric (which corresponds to the standard population-genetic model of natural selection on a large diploid population), a global Lyapunov function can be found that excludes cyclic behavior and guarantees that all orbits converge to the set of fixed points; see [2]

Snapshot 3: rock-scissors-paper game

Snapshot 4: strictly dominated strategies in the replicator-mutator dynamics can have an influence on the location of the limit rest points for small mutation; see [3]

References

[1] L. A. Imhof, D. Fudenberg, and M. A. Nowak, "Evolutionary Cycles of Cooperation and Deflection," in Proceedings of the National Academy of Sciences, 102(31), 2005 pp. 10797–10800. http://www.pnas.org/content/102/31/10797.

[2] J. Hofbauer, "The Selection Mutation Equation," Journal of Mathematical Biology, 23, 1985 pp. 41–53, http://dx.doi.org/10.1007/BF00276557.

[3] S. S. Izquierdo and L. R. Izquierdo, "Strictly Dominated Strategies in the Replicator-Mutator Dynamics," Games 2(3), 2011 pp. 355–364, http://dx.doi.org/10.3390/g2030355.



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