What the paper does
Setting. Two kinds of evaluation data. Agent-vs-agent (AvA): a matrix of win probabilities p_ij, turned into the antisymmetric log-odds matrix A_ij = log(p_ij / p_ji). Agent-vs-task (AvT): a score matrix S, agents by tasks, embedded in an antisymmetric (m + n)-square matrix. On the complete graph, the paper takes the Hodge decomposition from Jiang et al. (the record’s LIT-1439): every antisymmetric matrix is grad(div A) plus a rotational part, orthogonally under the Frobenius inner product (p. 4). div A is the row mean.
Elo, read algebraically.
- Proposition 1. Batch Elo is at a fixed point exactly when the predicted and observed win probabilities have the same row sums, so many win matrices share one rating, as many joints share marginals (p. 3).
- Proposition 3. If p_ij = σ(r_i − r_j), then div(logit P) = r. An Elo rating generating P exists if and only if curl(logit P) = 0, that is, log-odds around every triangle sum to zero. “Elo is, essentially, a uniform average in logit space” (p. 4). In rock-paper-scissors every rating is equal and Elo has no predictive power (p. 3).
- Copies. “an agent’s Elo rating can be inflated by instantiating many copies of an agent it beats (or conversely)” (p. 2). Appendix A’s example is rock-paper-scissors at 0.9/0.1 with C copied, where the paper reports Elo moving from (0, 0, 0) to (−63, 63, 0, 0). The reading recomputed the batch fixed point of Proposition 1 at the stated α and found ±71.9. The sign and the point stand; the magnitude printed is not the fixed point.
Multidimensional Elo. The rotational part is approximated by a low-rank antisymmetric C Ω Cᵀ from the Schur decomposition, giving p̂_ij = σ(r_i − r_j + c_iᵀ Ω c_j). On eight Go programs from AlphaGo’s table, mElo₂ cuts the Frobenius error from 0.85 to 0.35 and gets every likely winner right among αv, αp and Zen, where Elo gets two wrong (p. 5). For AvT, Proposition 2 makes the SVD of the score residual and the Schur decomposition the same thing: latent abilities of agents and latent problems posed by tasks (p. 5).
Nash averaging. Desired properties (p. 6): P1, adding redundant copies makes no difference; P2, robustness to small changes; P3, intuitive results in basic cases. The meta-game: two players each choose a distribution over agents, with payoff pᵀAq. Proposition 4: the game has a unique symmetric Nash equilibrium of maximum entropy. The Nash average is n = A p*, each agent’s expected log-odds against the maximum-entropy Nash team. Theorem 1:
- P1: invariant to redundancies;
- P2: a Nash for  is an ε-Nash for A, ε = ‖A − Â‖_max (proved, p. 15);
- P3: p* is uniform exactly when div A = 0, and uniform on the top-rated agents when A = grad(r) (proved, p. 15). Appendix D does the same for AvT, with one meta-player choosing agents and the other environments.
Atari (pp. 7–8). 20 agents (including human, random and no-op) on 54 environments, scores rescaled to [0, 1] per environment. The support of the equilibrium is five agents (distribDQN, rainbow, human, prior, popart) and five environments. All supported agents tie, by the indifference principle, “suggesting better-than-human performance has not yet been achieved” (p. 2). But “We find a Nash equilibrium using an LP-solver; it should be possible to find the maxent Nash using the algorithm in [71, 72]” (p. 7). The equilibrium shown is not the one Theorem 1 is about.
What the authors concede (p. 8). “Firstly, it is only as good as the data: garbage in, garbage out.” “Secondly, for good or ill, Nash averaging removes control from the user.” “Finally, Nash is a harsh master”: it is adversarial. Whether other invariant evaluations exist is left open.
What it establishes and what it does not
Proved, and re-derived here. Propositions 1–4, Theorem 1’s P2 and P3, and Lemma 1, the key step for P1: since the game’s value is zero, p is a Nash equilibrium exactly when every agent scores at most zero against it, so splitting a copied agent’s mass in any proportion preserves the condition, and merging copies reverses it.
P1 as stated is not proved, and fails in general. Lemma 1 is about the Nash set. The maximum-entropy selection over the padded set maximizes H(p) + p_c·log 2, the copied agent’s mass picking up the entropy of its split. When the original equilibrium is unique, p_c is fixed, the even split is the maximum, and invariance holds. When the equilibria form a segment, the extra term moves the maximizer. The proof never deals with this. The appendix gestures at it: “If the copied row is not unique and receives positive mass under maxent Nash, then maxent Nash will already be spreading mass across the copies” (p. 14).
The reading’s counterexample, checked by grid search over the simplex and in closed form. Take rock, paper and scissors at ±1, a null agent D that ties all three, and an agent E that loses to all three and beats D. The Nash set is λ·(⅓, ⅓, ⅓) + (1 − λ)·D with λ ∈ [½, 1]. Maximum entropy gives λ = ¾, p* = (¼, ¼, ¼, ¼, 0), and E’s Nash average is −0.5. Copy D. The maximum-entropy equilibrium becomes (⅕, ⅕, ⅕, ⅕, 0, ⅕) instead of the “invariant” (¼, ¼, ¼, ⅛, 0, ⅛), and E’s Nash average rises to −0.2. D’s column differs from every other, as Definition 3 requires. So both the distribution and the Nash averages move under one copy. Equilibria are unique for generic matrices, so the failure needs ties, and an empirical matrix with draws or saturated win rates can have them.
What the invariance covers even where it holds. Exact copies: rows identical against everyone. Not near-copies, since P2 controls the payoff and not the distribution, which can jump (Example 2, p. 7). Not a new, non-redundant agent chosen to change the result: adding an agent that beats the current leader changes the equilibrium entirely, by design.
Cardinal. Everything is in log-odds. The transitive part is the row mean, a cardinal average, and the Hodge cyclic part can be nonzero on a record whose wins are acyclic. That is CLAIM-434’s point, and THEORY-894’s caveat applies here unchanged.
How it bears on the record
- CASE-124 and CASE-155: capture of the scoring layer. Uniform averaging over the current population, Elo included, is exactly a score that a party can move by changing who gets evaluated, without changing anything about itself. Nash averaging removes one such route: scoring against a maximum-entropy equilibrium team instead of the population present makes the score blind to duplicates, when the equilibrium is unique. That narrows CASE-124’s μ route, a regime that “changes the scoring itself”. The route through redundancy can be shut by construction. Other routes stay open, and Nash averaging does not address them: adding agents that are not copies; near-copies; the choice of tasks, since in AvT the scores depend on which tasks the data contain (“garbage in, garbage out”). And with ties, copying still moves the result. It is also distinct from CASE-155’s c. There a process’s score rises with its own share of authority. Here the score depends on who is in the evaluated set, not on how much authority each holds. A Nash-averaged score is share-independent only if the underlying head-to-head matrix is. Narrows.
- TERM-077 and THEORY-894. The paper’s “transitive” is the Hodge gradient, cardinal. Its condition for Elo, zero curl, is the cardinal counterpart of TERM-077’s condition for a potential, wins that never cycle. The two can disagree, and the record already holds the reason (CLAIM-434). The paper gives the record nothing ordinal. Irrelevant to the ordinal claim; consistent with THEORY-894.
- CLAIM-451. In a cyclic record, any scalar mean in the Elo family is uninformative: RPS gives every option the same rating. Nash averaging then reports that no option is better (P3(i)). In the claim’s vocabulary, both report that no ordering prevails, and neither distinguishes nonintegrability from neutrality. The claim’s feedback-arc measure is not among the paper’s tools. Complicates nothing; adds a negative instance.
- CLAIM-436 and CLAIM-440 (common currency). mElo shows how a cyclic record still has a low-dimensional representation, a scalar plus planar “phase” coordinates (appendix F). That is a representation of interactions, not a currency over options, so it does not bear on the currency claims beyond that.
- CLAIM-446 and CLAIM-445. Not addressed.