Source
Sato and Crutchfield, LIT-1520, pp. 2–3 and Figures 1–3 of arXiv v2, read in NOTE-1144.
What was actually shown
- No decay, zero-sum (ε_X = −ε_Y). Quasiperiodic tori only at ε = 0; for ε_X = −ε_Y > 0, Hamiltonian chaos, with a periodic orbit arbitrarily close to any chaotic one and the mixed Nash equilibrium unstable (Fig. 1).
- No decay, not zero-sum. Transients to heteroclinic cycles: time spent near the vertices grows when ε_X + ε_Y < 0, and chaotic transients persist when ε_X + ε_Y > 0.
- Decay (α_X = α_Y = 0.01, ε_X = 0.5). Limit cycles and chaotic attractors (Fig. 2). Across ε_Y ∈ [−1, 1] the largest Lyapunov exponent is “positive across a significant fraction of parameter space; indicating that chaos is common” (p. 3). Near zero-sum the agents can reach the stable Nash equilibrium.
What this does not say
- That chaos is common in multi-agent learning generally. That is the authors’ expectation for “high-dimensional and heterogeneous populations”, stated, not shown.
- How large “a significant fraction” is. The figure was not inspected and the text gives no number.
Connections
- CLAIM-451. Persistent cycling and chaos from fixed, self-interested learning rules, with no regime switching and no preference instability. It also warns that behaviour that looks like switching between cooperation and competition can be one deterministic chaotic regime.
- THEORY-894. The ε = 0 game is harmonic, and its orbits never converge. How the tie rewards divide between potential and harmonic parts is not computed here or in the paper.