[Paper Review] Convergence of Payoff-Based Higher-Order Replicator Dynamics in Contractive Games

Hassan Abdelraouf, Vijay Gupta+1 more|arXiv (Cornell University)|Mar 18, 2026

Game Theory and Applications0 citations

TL;DR

The paper analyzes local and global convergence of payoff-based higher-order replicator dynamics in contractive games using passivity and incremental stability, showing local asymptotic convergence and global/incremental results in symmetric contractive settings.

ABSTRACT

We study the convergence properties of a payoff-based higher-order version of replicator dynamics, a widely studied model in evolutionary dynamics and game-theoretic learning, in contractive games. Recent work has introduced a control-theoretic perspective for analyzing the convergence of learning dynamics through passivity theory, leading to a classification of learning dynamics based on the passivity notion they satisfy, such as extdelta-passivity, equilibrium-independent passivity, and incremental passivity. We leverage this framework for the study of higher-order replicator dynamics for contractive games, which form the complement of passive learning dynamics. Standard replicator dynamics can be represented as a cascade interconnection between an integrator and the softmax mapping. Payoff-based higher-order replicator dynamics include a linear time-invariant (LTI) system in parallel with the existing integrator. First, we show that if this added system is strictly passive and asymptotically stable, then the resulting learning dynamics converge locally to the Nash equilibrium in contractive games. Second, we establish global convergence properties using incremental stability analysis for the special case of symmetric matrix contractive games.

Motivation & Objective

• Motivate the study of learning dynamics in population games through a control-theoretic lens.
• Extend replicator dynamics to higher-order, payoff-based settings and analyze convergence.
• Characterize local and global convergence using passivity (including strict passivity) and incremental stability.
• Establish conditions under which Nash equilibria are reached in contractive games, including symmetric matrix cases.

Proposed method

• Model higher-order replicator dynamics as a cascade interconnection between an LTI system (representing h(s)) and the softmax mapping.
• Show local convergence by proving Nash stationarity and using a Lyapunov/Lasalle argument with strict passivity.
• Use incremental stability analysis to derive global convergence results for symmetric matrix contractive games, linking passivity of G(s) to asymptotic or exponential convergence.
• Apply the KYP lemma and common quadratic Lyapunov functions to establish global incremental stability for the nonlinear closed-loop system.
• Provide examples (e.g., Rock–Paper–Scissors and a congestion game) illustrating local and global convergence properties.

Figure 1 : Block diagram of the linearized local dynamics.

Experimental results

Research questions

• RQ1 Under what conditions do payoff-based higher-order replicator dynamics converge locally to Nash equilibria in contractive games?
• RQ2 What global convergence guarantees (incremental asymptotic/exponential stability) can be established for symmetric matrix contractive games when the learning dynamics are driven by passive or strictly passive transfer functions?
• RQ3 How does the passivity property of the added LTI system influence convergence in the higher-order replicator framework?
• RQ4 Does Nash stationarity hold for these dynamics in the interior of the simplex, and how does it affect convergence results?
• RQ5 What do concrete game examples demonstrate about the practical convergence behavior of these dynamics?

Key findings

• If the added LTI system h(s) is strictly passive, the mixed Nash equilibrium in contractive games is locally asymptotically stable.
• In symmetric matrix contractive games, global incremental stability holds when the learning dynamics are modeled as G(s)I_n cascaded with the softmax mapping, with asymptotic or exponential convergence depending on passivity level (passive vs strictly passive).
• A common quadratic Lyapunov function can certify global incremental stability for all trajectory linearizations, enabling global convergence results under Nash stationarity.
• The KYP lemma and contraction analysis are used to show stability of the nonlinear closed-loop system when G(s) is passive or strictly passive.
• Concrete examples (Rock–Paper–Scissors and a congestion game) illustrate local convergence for strictly passive h(s) and global/incremental stability for passive/strictly passive G(s).

Figure 2 : Local convergence of payoff-based higher-order replicator dynamics for $h(s)=\tfrac{2s+3}{s^{2}+3s+2}$ in the RPS game.