Oscillatory dynamics in evolutionary games are suppressed by heterogeneous adaptation rates of players

Game dynamics in which three or more strategies are cyclically competitive, as represented by the rock-scissors-paper game, have attracted practical and theoretical interests. In evolutionary dynamics, cyclic competition results in oscillatory dynamics of densities of individual strategists. In finite-size populations, it is known that oscillations blow up until all but one strategies are eradicated if without mutation. In the present paper, we formalize replicator dynamics with players that have different adaptation rates. We show analytically and numerically that the heterogeneous adaptation rate suppresses the oscillation amplitude. In social dilemma games with cyclically competing strategies and homogeneous adaptation rates, altruistic strategies are often relatively weak and cannot survive in finite-size populations. In such situations, heterogeneous adaptation rates save coexistence of different strategies and hence promote altruism. When one strategy dominates the others without cyclic competition, fast adaptors earn more than slow adaptors. When not, mixture of fast and slow adaptors stabilizes population dynamics, and slow adaptation does not imply inefficiency for a player.

💡 Research Summary

The paper addresses a fundamental problem in evolutionary game theory: cyclic competition among three or more strategies (the classic rock‑scissors‑paper, RSP, scenario) typically generates oscillations in strategy frequencies. In finite populations without mutation, these oscillations tend to amplify until only a single strategy survives, leading to loss of biodiversity or altruistic behavior. Prior work has shown that spatial structure (e.g., lattices, small‑world networks) or heterogeneous contact numbers (hubs versus peripheral nodes) can dampen such oscillations, but these mechanisms rely on specific network topologies that may not be present in many real‑world systems.

The authors propose a different, more general stabilizing factor: heterogeneity in players’ adaptation rates. Some individuals are fast learners, quickly copying successful strategies, while others adapt more slowly. To formalize this, they extend the standard replicator dynamics, which is usually written as

 dx_i/dt = x_i (f_i – \bar f),

where x_i is the proportion of strategy i, f_i its fitness, and \bar f the population‑averaged fitness. They split the population into two sub‑populations characterized by distinct adaptation rates β_x and β_y. The proportions of strategy i in the fast and slow groups are denoted x_i and y_i, respectively, with ∑_i x_i = ∑_i y_i = 1. The transition (imitation) rate from strategy j to i is a smooth, monotone function G(f_i – f_j) that satisfies G′(0)=½, G(x)–G(–x)=x, and G(x)→0 as x→–∞. This choice allows for occasional “mistakes” (imitation of a less fit strategy) and ensures that even small fitness differences can generate non‑zero flow.

The resulting extended replicator equations are

 (1/β_x) dx_i/dt = x_i( f_i – ∑_j f_j x_j ) + y_i ∑_j G(f_i – f_j) x_j – x_i ∑_j G(f_j – f_i) y_j,

 (1/β_y) dy_i/dt = y_i( f_i – ∑_j f_j y_j ) + x_i ∑_j G(f_i – f_j) y_j – y_i ∑_j G(f_j – f_i) x_j.

These equations conserve the total mass in each sub‑population and reduce to the classic replicator dynamics when β_x = β_y (the ratio can be absorbed into a time rescaling).

Theoretical analysis focuses on a symmetric RSP game with payoff matrix

 A =