nucleation

THEORY-1233: For the replicator flow of a finite game, every asymptotically stable sink chain component contains at least one sink strongly connected component of the weakly-better-response graph, a Markov-Conley chain

1 min read

Properties4
StatusProposed
Tagsgame-theory, complex-systems
SourceLIT-1516 — α-Rank: Multi-Agent Evaluation by Evolution
ExtendsTHEORY-942 — The α-Rank chain over pure strategy profiles, built from Fermi-imitation fixation probabilities in finite populations with rare mutation, has a unique stationary distribution at every finite ranking intensity; as the intensity grows without bound its transition matrix converges to that of a chain that moves only along weakly better unilateral deviations, whose closed classes are the sink strongly connected components of the game's weakly-better-response graph, on each of which it is a Markov-Conley chain, and each of which lies inside a single chain component of the replicator flow

Graph View

Source

Omidshafiei et al., LIT-1516, read in NOTE-1156 from arXiv v4: Theorem 2.4.24 (p. 14, proof p. 44).

What is shown and what is not

The paper argues this. The interior replicator flow preserves volume after a change of coordinates, so an asymptotically stable set must meet the boundary. Inductively it contains a vertex, and then the set of profiles reachable from that vertex by weakly better replies, which is an MCC. The change of coordinates is easy. In log-ratio coordinates the interior field has divergence 0, since no population’s payoff differences depend on its own mix, and a script at the status review of 2026-10-30 confirmed it. What is not carried out is the induction: that the trace of an attractor on a face is again an attractor there, and that the vertex’s weakly-better closure is pulled in. THEORY-942 holds the other half, that each MCC lies inside one chain component, re-derived. The converse of this entry, that every MCC lies in an asymptotically stable component, is false: in matching pennies the whole square is one component (NOTE-1156).

Backlinks