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Elo Ratings in the Presence of Intransitivity

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that when a game's win probabilities are intransitive, the final Elo ratings depend on which players are scheduled to play whom, not just on their skill.

desk verdict The fixed-point theorem is real and clean, but the paper oversells it as a statement about observed Elo ratings when the stochastic convergence bridge is not proved. read the letter →

arxiv 2412.14427 v1 pith:Q2MZ2LST submitted 2024-12-19 math.PR

Adam H. Hamilton , Matthew Roughan , Anna Kalenkova This is my paper · ORCID
classification math.PR MSC 60G9900X0091A99
keywords EloratingsystemIntransitivitySelectionmatrixStabilityequationHodgedecompositionBrouwerfixedpointBradley-TerrymodelRock-Paper-Scissors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks what happens to the Elo rating system when the game being rated is intransitive, meaning that player A may usually beat B, B usually beats C, yet C usually beats A. It proves that in that case the final Elo ratings are not a property of the players alone: they depend on the selection matrix, the probabilities with which each pair of players is scheduled to play. At the same time, it shows that for any fixed schedule with a connected comparison graph there is exactly one rating vector satisfying the stability equation, so Elo still has a well-defined fixed point. The paper also defines a scalar measure of intransitivity based on the Hodge decomposition of the win-probability matrix, and validates it on simulated Rock-Paper-Scissors data.

What carries the argument

The load-bearing object is the stability equation $\operatorname{div}(Q \odot P) = \operatorname{div}(Q \odot \sigma(\operatorname{grad}(r)))$, which states that at a final Elo score the conditionally expected one-step change in ratings is zero, where $Q$ is the selection matrix, $P$ the win-probability matrix, $\sigma$ the logistic function, and $\operatorname{grad}$ and $\operatorname{div}$ the combinatorial gradient and divergence from Hodge theory. Existence of a solution follows from applying Brouwer's fixed-point theorem to the expected-update map $f(r) = r + \eta \operatorname{div}(Q \odot (P - \sigma(\operatorname{grad}(r))))$ on a large enough hypersphere. Uniqueness is proved by showing the map $g(r) = \operatorname{div}(Q \odot \sigma(\operatorname{grad}(r)))$ is injective when $Q$'s graph is strongly connected, using the orthogonality of the kernel of $\operatorname{div}$ to the image of $\operatorname{grad}$. The schedule-dependence result uses spanning trees: on a tree, ratings are forced by summing $\sigma^{-1}(P)$ along the unique paths, and path-dependence of these sums is exactly intransitivity.

What would settle it

Run a long simulation of an intransitive game with a fixed connected selection matrix Q, record the time-averaged Elo ratings, and compare them to the unique solution of the stability equation; if the averages do not track the fixed point and instead cycle or converge elsewhere, the paper's notion of long-term behaviour is not the behaviour the system exhibits. A second check: compute the final ratings for two different spanning-tree selection matrices and verify empirically that the long-run averages land at the two different fixed points, as Theorem 4.1 predicts.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.1: if the advantage matrix $M = \sigma^{-1}(P)$ is not a strongly transitive additive comparison matrix, meaning the win probabilities cannot be written as $\sigma(r_i - r_j)$ for any skill vector $r$, then there exist two selection matrices $Q_1$ and $Q_2$ whose associated final Elo scores are different. The proof constructs two spanning interaction trees that contain different paths between the same pair of players; because the sums of $\sigma^{-1}(P)$ along the paths differ, the ratings forced by the stability equation differ. Sections 5 and 6 show the complementary result: for any fixed selection matrix whose graph has one strongly connected component, the stability equation has a unique solution, found by Brouwer's fixed-point theorem and an injectivity argument. Thus an intransitive game has a whole set of possible final ratings, called the Elotope, and Elo selects one point of it depending on the fixture.

Load-bearing premise

The paper defines the final Elo score as a fixed point of the expected one-step update and assumes this deterministic object describes where the stochastic Elo Markov chain really goes in the long run, even though it does not prove the chain has a limiting distribution or that it converges to this fixed point.

Editorial extensions

If this is right

  • In any game whose win probabilities are intransitive, Elo ratings are schedule-dependent: the same set of players can receive different final scores under different match fixtures.
  • Despite that, Elo still has a unique final rating for each fixed, connected schedule, so the system does not lose its fixed-point property when transitivity fails.
  • The set of all possible final ratings over all schedules, the Elotope, grows with the cyclic component of the advantage matrix, so the spread of possible ratings is a measure of how much intransitivity distorts the ratings.
  • The schedule-dependence explains why adaptive matchmaking, where opponents are chosen from current ratings, can prevent convergence: changing $Q$ moves the target fixed point while the process is chasing it.
  • The proposed intransitivity measure $I(A)$ stays near or below 1 for predominantly transitive games and rises above 1 when the cyclic component dominates, matching what one would expect on Rock-Paper-Scissors examples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The schedule-dependence mechanism is not specific to Elo: any pairwise scalar-rating update whose fixed point is a weighted sum of edge contributions will inherit the dependence on the edge weights of the schedule.
  • A more direct index of rating instability than $I(A)$ would be the diameter of the Elotope, the maximum difference between final scores achievable by varying $Q$; the paper describes the Elotope but does not formalize this diameter.
  • Because the paper proves only that the mean-field fixed point exists and is unique, not that the stochastic Elo chain converges to it, the practical relevance depends on whether time averages of real Elo updates track the fixed point; this could be tested by simulation across different learning rates $\eta$ and selection matrices.
  • The intransitivity measure requires full pairwise win probabilities, which are rarely available in real sports data; a useful extension would be to estimate $I(A)$ from the sparse match data that Elo itself uses, avoiding the circularity the paper notes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies the Elo rating system when pairwise win probabilities are not consistent with a transitive, additive skill model. It defines the "final Elo score" as a fixed point of the conditional expected one-step update (the stability equation), proves that this fixed point depends on the schedule of matches when the advantage matrix is not a strongly transitive additive comparison matrix (Theorem 4.1), claims existence and uniqueness of the fixed point for connected selection matrices (Theorems 5.1 and 6.1), and introduces an intransitivity measure I(A) validated on simulated Rock-Paper-Scissors data. The spanning-tree construction behind Theorem 4.1 is elegant, and the proposed measure is clearly described and behaves sensibly in the simulations. The main gaps are the missing connection between the mean-field fixed point and the actual stochastic Elo chain, and several incomplete proofs in the existence and uniqueness arguments.

Significance. If the results were fully established, the paper would make an important point for practitioners: scalar Elo ratings in intransitive games are not determined by the players' abilities alone, and the match schedule can change the final ratings. The uniqueness claim would also provide a useful stability property. The paper correctly identifies the gap between the fixed point of the expected update and the long-run behavior of the stochastic chain, but it does not bridge that gap. The intransitivity measure and its simulations are a useful contribution, though they validate only the metric, not the convergence claims. The use of combinatorial Hodge theory is appropriate, and the authors are honest about several limitations, but the advertised practical conclusion currently exceeds what the proofs establish.

major comments (4)
  1. [Section 3.2.1 and Theorem 4.1] The central practical claim that "estimated ratings are dependent on who plays who" is not established for the stochastic Elo process. The paper defines the final Elo score as a fixed point of the conditional expected one-step update and explicitly states that no proof of stationarity, recurrence, or convergence to this fixed point is given. With a constant gain eta, the Markov chain cannot converge to a point; at best one would need a stationary-distribution concentration result as eta -> 0, and no such result or bound on the spread is provided. Theorem 4.1 therefore describes the deterministic mean-field equation, not necessarily "the ratings users observe." The simulations in Section 7 validate the intransitivity measure, not this bridge. The authors should either prove a concentration statement, state clearly that all results concern the mean-field fixed point, or both.
  2. [Section 5.3, Lemma 5.4 and Step 1.3] The existence proof via Brouwer's fixed-point theorem is incomplete. Lemma 5.4 is justified by "can be verified visually" for m = 3, with no rigorous argument for general m. Step 1.3 reasons about the limit as the radius of K approaches infinity and then asserts that the inward components "will stay bounded from below" without proof; the passage from asymptotic behavior to a fixed finite radius is not justified. Section 5.4 then extends the radius by sqrt(2 eta) and asserts that the inward-direction property persists on the inner shell, which does not follow from the boundary argument alone. As written, Theorem 5.1's existence claim lacks a complete proof.
  3. [Section 6, Theorem 6.1] The uniqueness proof is not valid as written. The argument that tangent vectors to sigma(im(grad)) and vectors in ker(div) lie in different orthants is unsupported: Euclidean orthogonality of the two subspaces does not imply sign-opposition of the entries of their vectors. The stated sufficient condition for global injectivity of a projection is only a local condition; without a properness or covering argument it does not imply global injectivity. The "if and only if" direction for disconnected Q is asserted rather than proved. The theorem may be true and can be proved by a monotonicity argument: for connected Q, <g(r)-g(s), r-s> > 0 for r != s because (x-y)(sigma(x)-sigma(y)) > 0 for the logistic function. I recommend replacing the current proof with such an argument.
  4. [Abstract and Introduction] The claim that the fixed point "represents the maximum likelihood estimator of the Elo ratings, given the observed data" is not proven anywhere in the manuscript. The stability equation is an expected score equation, and connecting it to the MLE for realized data requires a likelihood calculation or an explicit stochastic-approximation theorem. As written, this advertised contribution is unsupported and should be either proved or removed.
minor comments (6)
  1. [Title] The title contains a typo: "Presense" should be "Presence."
  2. [Section 3.2, Eq. (4)] With div(A) defined as the row-mean in Section 2, Eq. (3) equals m * eta * div(Q circ (P - sigma(grad(r)))), not eta * div(...). The factor m cancels in the stability equation, but the displayed equality is false as written.
  3. [Section 5.1] The sum index in "the subspace sum_{i=0}^m r_i = 0" should be i=1, not i=0.
  4. [Section 6] The text uses "simply connected interaction network" to mean a connected graph and refers to "strongly connected component" for an undirected graph; the standard term is "connected component."
  5. [Sections 2 and 7] The symbol A denotes both the cyclic component in the Hodge decomposition and the advantage matrix in the intransitivity measure (Eq. (8)); this overloaded notation is confusing and should be changed.
  6. [Section 7.1.2, Figure 9] The "vertical asymptote at t = 1" is a limiting statement because logit(0) and logit(1) are infinite; the text should describe it as a limit of the ground-truth calculation rather than a property that the finite simulations display.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are proven from explicit stability-equation assumptions, with no fitted input renamed as prediction.

full rationale

The paper's derivation chain is self-contained for what it proves. Theorem 4.1 takes the paper's own definition of final Elo score as a fixed point of the expected one-step update (Eq. 5) and shows by an explicit spanning-tree construction that a non-STACM advantage matrix yields two selection matrices with different fixed-point ratings; this is a mathematical property of Eq. 5, not an input to Eq. 5. Theorems 5.1 and 6.1 prove existence and uniqueness from Brouwer's fixed-point theorem and an injectivity argument, with no fitted constants. Section 7's intransitivity measure, Eq. (8), is a definition rather than a fitted prediction, and the simulations compare it to ground-truth values as a sanity check; the paper explicitly avoids estimating the advantage matrix through Elo because this would create a circular chain of reasoning (Section 7). The only self-citation, [19], supplies optional Elotope geometry and is not load-bearing. The gap the paper itself acknowledges—that it proves properties of the fixed point of the expected update, not stationarity or convergence of the stochastic Elo chain (Section 3.2.1)—is a scope limitation for the practical interpretation of the theorems, not circular reasoning.

Assumptions & free parameters 1 free parameters · 6 assumptions · 2 invented entities

The central theoretical results rely on the standard Bradley-Terry/Elo model with fixed selection matrix and on the mean-field fixed-point definition of long-term behavior. The only hand-chosen number is the additive constant in the proposed intransitivity measure. No fitted parameters enter the proofs of Theorems 4.1, 5.1, or 6.1.

free parameters (1)
  • Additive constant 1 in intransitivity measure I(A) = 1
    Introduced ad hoc in Eq. (8) to avoid division by zero when the transitive component norm is zero. The threshold between 'predominantly transitive' (I < 1) and 'effectively intransitive' (I > 1) depends on this arbitrary choice.
assumptions (6)
  • domain assumption The Elo model: P_ij = sigma(r_i - r_j) for a logistic sigmoid sigma, and ratings update by transferring a fraction of the difference between observed and expected outcome.
    Section 3.1. The entire analysis is conditional on this model of game outcomes.
  • ad hoc to paper Long-term behavior of the Elo Markov chain is characterized by the fixed point of the conditional expected one-step update (the stability equation).
    Section 3.2.1. The paper states the usual stationarity approach is unavailable and substitutes this weaker notion without proving that the stochastic process actually converges to or clusters around the fixed point.
  • domain assumption Selection matrix Q is constant and independent of game outcomes.
    Section 3.1. Adaptive selection is mentioned as a cause of non-convergence (citing Bolsinova et al. 2024) but not analyzed; all theorems assume fixed Q.
  • standard math Hodge decomposition of skew-symmetric matrices into gradient and cyclic components (Jiang et al. 2011).
    Section 2. Used for the Elotope analysis and the intransitivity measure in Eq. (8).
  • standard math Brouwer's fixed-point theorem.
    Theorem 5.2, used in the existence proof of a solution to the stability equation.
  • ad hoc to paper Global injectivity of the projection of sigma(grad(r)) onto the rating subspace follows from the tangent space avoiding the kernel of div.
    Section 6. The paper's orthant argument gives local injectivity; global injectivity needs an additional topological or monotonicity argument that is not supplied.
invented entities (2)
  • Elotope
    purpose: The set of all possible final Elo scores obtainable by varying the selection matrix Q for a fixed advantage matrix; used to define the degree of intransitivity via its size.
    A mathematical object defined in Section 4. It does not make falsifiable predictions outside the paper; it is a naming of a set rather than a new physical entity.
  • Intransitivity measure I(A)
    purpose: A statistic quantifying how much a game's advantage matrix deviates from transitivity; validated only on simulated data in Section 7.
    No independent empirical validation outside the paper's own simulations; the additive constants are chosen by hand.

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Pith. "Pith review of Elo Ratings in the Presence of Intransitivity." pith.science (2026). https://pith.science/paper/Q2MZ2LST

@misc{pith2026241214427,
  author       = {Pith},
  title        = {Pith review of: Elo Ratings in the Presence of Intransitivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2MZ2LST}},
  note         = {Machine review of arXiv:2412.14427}
}
read the original abstract

This paper studies how the Elo rating system behaves when the underlying modelling assumptions are not met.

Figures

Figures reproduced from arXiv: 2412.14427 by the authors.

Figure 1
Figure 1. A spannin playerwhose Elo rat . A spanning tree over [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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