Dimensional Thresholds and Exact Obstructions for Evolutionary Stability under Replicator Equivalences

Abstract

Stefano Allesina asked whether every globally stable interior equilibrium of a replicator equation becomes an evolutionarily stable strategy (ESS) after a common-payoff shift and positive projective rescaling. This preprint gives an exact obstruction theory and determines the dimensional threshold. For a specified full-support equilibrium, an ESS-equivalent representative exists exactly when there is no nonzero positive-semidefinite matrix (Y) with (Y\mathbf 1=0) and(\operatorname{diag}(YA)\ge 0). The result is a simplex-tangent specialization of the classical Barker–Berman–Plemmons/Gordan–Stiemke alternative and yields normalized primal and dual semidefinite margins. Every globally asymptotically stable full-support equilibrium of a two-strategy game is already an ESS. A three-strategy integer example has a globally asymptotically stable uniform equilibrium but cannot be made an ESS under any allowed equivalence. This example is degenerate, has exactly two Nash equilibria, and is zero-margin, so it is consistent with Allesina's generic three-strategy theorem. Explicit positive-parameter counterexample families are given for every number of strategies (n\ge3). Theunconditional threshold is therefore three strategies, while the first ambient-open failures occur at four. A strict four-strategy competitive Lotka–Volterra construction is certified using Jansen permanence and the Baigent–Hou split-Lyapunov condition. Exact Schur-complement and eigenpair-preserving perturbations lift the strict certificate to ambient-open counterexample sets in every (n\ge4). These sets contain models with entrywise-positive competitive interaction matrices and infinitely many pairwise nonproportional rational and integer payoff matrices. The deposit includes the article, an exact-arithmetic SymPy verifier, and its machine-readable JSON output. No empirical data are used.