From Griefing to Stability in Blockchain Mining Economies

The paper formalizes a Shmyrev-type convex program (SH-QCES), proves 1-Bregman convexity of its objective, derives Proportional Response (PR) as a mirror-descent step w.r.t. KL, and concludes convergence of PR to market equilibria for 0 < ρ_i ≤ 1. Key ingredients include the convex program F(b,w,p) with constraints ∑_i b_{ij} = p_j and ∑_j b_{ij} + w_i = K_i, the bound 0 ≤ dF(z',z) ≤ ∑_i (1/ρ_i) KL(b'_i || b_i), and a precise MD-to-PR derivation with two normalization cases, exactly yielding the PR-QCES update and Theorem 9 (convergence) . By contrast, the model’s (candidate solution’s) derivation contains a critical flaw: it claims the factors (p_k')^{-ρ_i} “cancel” in the blockwise KKT solution, leading directly to a PR formula that depends only on u_{ik} = (v_{ik} b_{ik})^{ρ_i}; in general they do not cancel. In the paper, the p_j^t terms are essential and only disappear after updating valuations using network aggregates (p_j ≡ X_j), which the candidate solution does not justify. The candidate also uses a block inequality with gradient evaluated at p' that depends on the new iterate, which is not a valid mirror-descent majorizer. Hence, the paper’s argument is correct and complete for 0 < ρ_i ≤ 1, while the model’s proof contains substantive errors in the MD step and normalization.