Synchronization and extinction in cyclic games with mixed strategies

TL;DR

This paper studies cyclic games with evolving strategies where agents adapt based on losses, revealing how stability and synchronization depend on the number of strategies, system size, and probability discretization.

Contribution

It introduces a model of strategy adaptation using urns and analyzes stability and synchronization in cyclic games with multiple strategies through equations and simulations.

Findings

Transition from neutral to stable dynamics in three-strategy models with discretization change.

Large urns approximate continuous distributions, leading to synchronized oscillations.

Four-strategy systems remain neutrally stable with regime variations based on size and discretization.

Abstract

We consider cyclic Lotka-Volterra models with three and four strategies where at every interaction agents play a strategy using a time-dependent probability distribution. Agents learn from a loss by reducing the probability to play a losing strategy at the next interaction. For that, an agent is described as an urn containing $\beta$ balls of three respectively four types where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy. Using both mean-field rate equations and numerical simulations, we investigate a range of quantities that allow us to characterize the properties of these cyclic models with time-dependent probability distributions. For the three-strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution. For large values of $\beta$, yielding a good approximation to a continuous distribution, spatially synchronized temporal oscillations dominate the system. For the four-strategy game the system is always neutrally stable, but different regimes emerge, depending on the size of the system and the level of discretization.