The Higher Replicator

Standard replicator dynamics model how a population of strategies evolves: successful strategies grow, unsuccessful ones shrink. Mathematically, the dynamics decompose into a cascade — an integrator (which accumulates payoffs) feeding through a softmax mapping (which converts accumulated payoffs to strategy frequencies). The system has no memory beyond the accumulated payoff totals.

Higher-order replicator dynamics add memory. A linear time-invariant system runs in parallel with the integrator, processing the same payoff signals through a filter before combining them with the raw accumulated payoffs. The filter can weight recent payoffs differently from old ones, implement momentum, or smooth out noise. The strategy frequencies still come from softmax, but they respond to filtered payoffs, not just raw accumulations.

The question: does this richer system still converge to Nash equilibrium?

For contractive games — games where the payoff map is a contraction, pulling trajectories together — the answer is yes, under two conditions on the added filter: it must be strictly passive (a stability condition from control theory) and asymptotically stable. Convergence is local: starting near enough to the Nash equilibrium, the dynamics reach it.

For the special case of symmetric matrix contractive games, convergence is global — the dynamics converge from any starting point. The proof uses incremental stability analysis, which bounds the distance between two trajectories rather than the distance from a trajectory to the equilibrium.

The passivity framework provides the structural insight. Standard replicator dynamics are themselves passive in a specific sense. The higher-order extension preserves passivity because a passive system in cascade with a strictly passive filter remains passive. Convergence follows from the interaction between the passive dynamics and the contractive game — the game pulls the dynamics toward equilibrium, and passivity ensures they don’t oscillate away.