Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics

The paper proves (via Conley index and attractor–repeller duality) that the Kohlberg–Mertens 3×3 game admits no Type III dynamics (Theorem 5.1) and extends this to small-ε approximate equilibria and an open, positive-measure set of games (Theorem 6.2) . The candidate solution reaches the same conclusions but its core argument misapplies Lefschetz theory and attractor properties: (i) it incorrectly infers that CR(φ)=NE(g) implies NE(g) is a global attractor; the paper instead uses the attractor–repeller lattice and shows that a single chain-recurrent component must be the unique maximal attractor before deriving a contradiction via Conley index ; (ii) it asserts L(φ_T)=χ(NE(g)) for an attractor without establishing the required index localization and induced-map conditions; the paper avoids this by directly comparing Conley indices of the purported attractor and the whole space, invoking Corollary 4.6 to force a nonempty repeller . In Part B, the model further claims NE_ε(g) is an “annulus” (dimensionally incorrect in X=Δ^2×Δ^2) and relies on a non-justified Lipschitz neighborhood sandwich; the paper instead proves NE_ε(g) is homotopy equivalent to S^1 for sufficiently small ε by a careful polyhedral/BR-region analysis and then repeats the Conley-index obstruction, plus a clean positive-measure robustness argument .