Scale-free memory model for multiagent reinforcement learning. Mean field approximation and rock-paper-scissors dynamics

A continuous time model for multiagent systems governed by reinforcement learning with scale-free memory is developed. The agents are assumed to act independently of one another in optimizing their choice of possible actions via trial-and-error search. To gain awareness about the action value the agents accumulate in their memory the rewards obtained from taking a specific action at each moment of time. The contribution of the rewards in the past to the agent current perception of action value is described by an integral operator with a power-law kernel. Finally a fractional differential equation governing the system dynamics is obtained. The agents are considered to interact with one another implicitly via the reward of one agent depending on the choice of the other agents. The pairwise interaction model is adopted to describe this effect. As a specific example of systems with non-transitive interactions, a two agent and three agent systems of the rock-paper-scissors type are analyzed in detail, including the stability analysis and numerical simulation. Scale-free memory is demonstrated to cause complex dynamics of the systems at hand. In particular, it is shown that there can be simultaneously two modes of the system instability undergoing subcritical and supercritical bifurcation, with the latter one exhibiting anomalous oscillations with the amplitude and period growing with time. Besides, the instability onset via this supercritical mode may be regarded as “altruism self-organization”. For the three agent system the instability dynamics is found to be rather irregular and can be composed of alternate fragments of oscillations different in their properties.

💡 Research Summary

The paper introduces a continuous‑time reinforcement‑learning framework in which agents possess a scale‑free (power‑law) memory of past rewards. Instead of the usual exponential decay, the contribution of a reward received at time τ to the perceived value of an action at the current time t is weighted by a kernel K(t‑τ) ∝ (t‑τ)^{‑γ} with 0 < γ < 1. This kernel leads, in the limit of continuous time, to a fractional (Caputo) derivative D_t^{γ} in the governing equations, thereby endowing the system with long‑range temporal correlations and non‑Markovian dynamics.

Agents act independently, but the reward of each agent depends on the actions chosen by the others. The authors model this interdependence through a pairwise interaction term, which for the concrete case of a rock‑paper‑scissors (RPS) game yields a cyclic, non‑transitive payoff matrix. By applying a mean‑field approximation, the high‑dimensional stochastic process collapses to a set of coupled fractional differential equations for the action probabilities p_i(t). The generic form is

  D_t^{γ} p_i(t) = p_i(t)