The paper shows that complex, chaotic behavior in a simple spatial game can be traced back to the stability of a few small local patterns, or motifs. The authors build a map of dynamical behavior across the two payoff parameters that define canonical 2×2 games. They find that whether the whole system becomes ordered or chaotic is set by which local motifs become unstable at their interfaces. That link lets them predict where chaos appears from simple payoff-balance rules at those small interfaces.

The model is a deterministic cellular automaton on a square grid. Each site is either a cooperator (C) or defector (D) and plays with its four nearest neighbors. Payoffs depend on two parameters u and v (differences of the usual payoff matrix entries) restricted to −1≤u,v≤1. All sites update synchronously by an “imitate-the-best” rule: each site adopts the strategy in its neighborhood that earned the highest payoff. To deal with the fact that this rule is not smooth, the authors use a Boolean linearization (the Boolean derivative) in the limit where initial perturbations are rare. That gives a linear operator (the Boolean Jacobian) that governs first-order perturbation growth and lets them derive instability thresholds for motifs such as single invaders, cooperative pairs, stripe interfaces, and cooperative cores.

To measure chaos they use two complementary diagnostics. The normalized Hamming distance measures the fraction of sites that differ between two nearly identical runs. The Derrida slope measures the average one-step amplification of a very small perturbation. By combining these measures they divide the payoff plane into four regions: ordered (no spread of damage), transient-chaotic (temporary complex behavior that dies out), sustained-chaotic (small perturbations grow and persist), and subcritical-chaotic. The subcritical-chaotic region is notable: there the Derrida slope is less than one (infinitesimal perturbations shrink) but the long-term Hamming distance stays positive, meaning finite-size perturbations can still sustain chaos.