Bottom Line: Our equation is defined on the positive orthant, instead of the simplex, but has the same equilibrium points as the replicator equation.Linear stability analysis produces the classical conditions for asymptotic stability of pure strategies, but the stability properties of internal equilibria can differ in the two frameworks.For example, in a two-strategy game with an internal equilibrium that is always stable under the replicator equation, the corresponding equilibrium can be unstable in the new framework resulting in a limit cycle.
Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria. firstname.lastname@example.orgShow MeSH
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Mentions: As an example, consider a game with uniform payoffs, . Then, every point on is an equilibrium, no matter how growth rates are chosen. For a=10, , and , the corresponding phase portrait is depicted in Fig. 3a. It shows that even with very disparate growth rates the stability properties of the pure equilibria cannot be changed in the degenerate case . Thus, the effect of a large discrepancy in growth rates is neutral with respect to equilibria, but leads to nearly horizontal trajectories in strategy density space, such that effectively only the fast-growing strategy changes its abundance when the dynamics converges to a continuum of equilibria. However, the slightest change in payoffs breaks the symmetry, such that the curve of equilibria collapses and the equilibrium with the higher payoff is approached, see Fig. 3b. Trajectories move towards a slow manifold in a short initial phase, during which strategy 2 hardly changes in abundance. When population size is saturated, the difference in growth rates becomes effective, such that strategy 1 is able to out-compete strategy 2 (compare the concepts of r- and K-selection, MacArthur and Wilson, 1967). Thus, even a highly increased growth rate cannot make up for a slightly worse payoff in the long run.
Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria. email@example.com