Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics

Abstract

This note shows, by means of two simple, three-strategy games, the existence of stable periodic orbits and of multiple, interior steady states in a smooth version of the Best Response Dynamics, the Logit Dynamics. The main …nding is that, unlike Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles, occur under the Logit Dynamics, even for three strategy games. We also show that the Logit Dynamics displays another bifurcation which cannot occur under the Replicator Dynamics: the fold bifurcation, with non-monotonic creation and disappearance of steady states. JEL classi…cation: C72, C73Keywords: Evolutionary games, Logit dynamics, Hopf bifurcation, fold bifurcation

Social cycling and conditional responses in the Rock-Paper-Scissors game

“…This model is characterized by a set Γ ≡ { W − , W + ; T − , T + ; L − , L + } of six CR parameters. Notice this CR model differs qualitatively from the discrete-time logit dynamics model2425 used in Ref. 28, which assumes each player has global information about the population's social state.…”

“…On the other hand, evolutionary game theory drops the infinite rationality assumption and looks at the RPS game from the angle of evolution and adaption131415161718. Evolutionary models based on various microscopic learning rules (such as the replicator dynamics12192021, the best response dynamics2223 and the logit dynamics2425) generally predict cyclic evolution patterns for the action marginal distribution (mixed strategy) of each player, especially in finite populations.…”

Cycles and Instability in a Rock-Paper-Scissors Population Game: A Continuous Time Experiment

“…We find the critical value of 0.18 by numerical analysis. The existence of a supercritical Hopf bifurcation for RPS games has been established by Hommes and Ochea (2012). This result is less complete than Proposition 1, in that it does not give a complete picture of PBR cycles away from equilibrium.…”

Transition matrix model for evolutionary game dynamics

“…Thus the replicator dynamic does not admit spontaneous generation of species, which is not surprising as it was constructed as a reproductive model; as Taylor and Jonker state, "the simplest hypothesis is that of exponential growth or decay" [1]. Several authors have distinguished between two different classes of such equations: imitative dynamics vs pairwise comparison dynamics [6,13,22]. To clarify our classification of the various strategy evolution equations, we distinguish between two types, along similar lines to Sandholm's classification of revision protocols [6]:…”