Standard replicator dynamics — the evolutionary update rule where successful strategies reproduce proportionally — can be decomposed as an integrator feeding through a softmax mapping. The integrator accumulates payoff information; the softmax converts it into a strategy distribution. This is the simplest learning dynamic in games, and it works well in passive games where the payoff structure cooperates with gradient-like learning.
Contractive games are the complement — games where the payoff structure actively fights gradient-based learning. The standard replicator fails here because pure integration doesn't provide enough structure to overcome the game's contractiveness.
The fix is to add a linear time-invariant system in parallel with the integrator. This creates a higher-order learner — one that doesn't just accumulate payoff but processes it through a dynamical filter before updating. The paper shows that if this added system is strictly passive and asymptotically stable, the combined dynamics converge locally to Nash equilibrium in contractive games. For the special case of symmetric matrix contractive games, convergence is global.
The structural point is about what makes learning work against adversarial structure. Simple memory (integration) is insufficient. You need dynamical filtering — a mechanism that transforms the payoff signal before it drives the update. The passivity requirement on the filter means it must not inject energy into the learning dynamics; it must only reshape the signal. The game is contractive; the filter must be dissipative. The two negatives produce convergence.
(arXiv:2603.18219)