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REVIEW 4 major objections 4 minor 51 references

Inferring Piece Value in Chess and Chess Variants

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Piece values can be inferred from millions of chess games, and the data put a knight at 2.9 pawns, a bishop at 3.2, a rook at 4.6, and a queen at 9.6 pawns.

desk verdict Useful and honest empirical study of chess-variant piece values, but the headline numbers sit on an outcome-dependent filter that shifts them 25–30% — treat them as provisional. read the letter →

arxiv 2509.04691 v3 pith:ZQS3DSFN submitted 2025-09-04 stat.AP

Steven E. Pav This is my paper · ORCID
classification stat.AP
keywords chesspiecevalueslogisticregressionLichessGlicko-2SIMEXmeasurementerrorvariantsvalue
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

This paper tries to show that the relative value of chess pieces can be estimated empirically from real games, not just assumed from tradition. Using millions of Lichess games and controlling for player skill, the author finds that classic 3-3-5-9 valuations are roughly right, but with a twist: bishops are worth slightly more than knights. The same regression method applied to chess variants shows that piece values shrink in Atomic and Antichess, where position matters more than material, and even turn negative in Antichess, where losing pieces is the goal. The analysis also yields concrete numbers for how many rating points a pawn, knight, bishop, rook, or queen is worth when giving odds. If the method holds up, it gives a data-driven way to calibrate material evaluation in any chess-like game.

What carries the argument

The engine is ordinary logistic regression of game outcome on material differences plus a Glicko-2 rating-term, where the coefficients are converted to Elo-like units. Because published ratings are noisy proxies for skill, the paper applies SIMEX (simulation-extrapolation): it adds controlled Gaussian noise to rating differences, re-fits the regression, and extrapolates back to zero noise. A no-capture filter on snapshots serves as a crude quiescence filter, and the final piece values are reported as ratios of the fitted piece coefficient to the fitted pawn coefficient.

What would settle it

Re-run the full regression pipeline on a fresh Lichess sample filtered by an explicit quiescence check, such as requiring that the position evaluation is stable across a one-ply engine search. If the bishop-over-knight gap disappears, reverses sign, or moves by more than a few percent, the reported piece values are artifacts of the crude quiescence filter rather than true material worth.

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Extended reading notes

Core claim

The central claim is that a simple logistic regression on material imbalance, with Glicko-2 rating differences as a covariate and SIMEX-based correction for rating noise, recovers meaningful piece values. For standard chess, after filtering out positions with an immediate capture, the estimated values are knight 2.9 pawns, bishop 3.2, rook 4.6, and queen 9.6. These are close to classical systems, except that bishops edge out knights, and the values lie on the low end of historical estimates, implying a relatively high pawn value. In Chess 960 the same values come out slightly depressed (2.7, 3.0, 4.2, 8.7), in Atomic chess they are much smaller in absolute terms but similar relative to a paw

Load-bearing premise

The whole analysis hinges on the assumption that a single noise level for Glicko-2 ratings, chosen by trial and error, plus a filter for positions with no immediate capture, removes the confounding effects of player skill and non-quiescent positions.

Editorial extensions

If this is right

  • Standard chess material evaluation can be checked and updated from observed play, giving baseline values of roughly 3, 3, 5, 10 pawns for knight, bishop, rook, queen, with bishop slightly above knight.
  • In Chess 960, piece values track standard chess but run slightly lower for each piece, suggesting the same hand-evaluation system transfers to random openings.
  • In Atomic and Horde chess, major pieces are worth fewer pawns, meaning positional factors dominate material the way players often assume.
  • In Antichess, most pieces carry negative value, and a king is worth nearly zero on average, so equalizing odds by removing pieces from the stronger player is a quantitative, not merely intuitive, exercise.
  • The reported equalizer values convert material directly into rating points, giving a concrete way to set up fair games between mismatched opponents.

Reading between the lines

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

  • The method could be extended to non-material positional features, like knight outposts, king safety, or castling status, by adding them as regression terms, potentially yielding a fully data-driven static evaluation function.
  • The discrepancy between human-play estimates and Stockfish self-play values suggests piece values may depend on the skill of the players; a natural test is to run the same regression on games filtered by rating band to see if the piece coefficients shift with playing strength.
  • The sigma_0 noise estimate for Glicko-2 ratings, which varies across variants, could itself be estimated jointly with the piece values using a measurement-error model, rather than fixed by trial and error.
  • A cleaner falsifiable prediction is that the ratio of bishop-to-knight value should remain above 1 across time controls; running the regression on bullet or blitz games would test whether the effect is an artifact of longer time controls.
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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 / 4 minor

Summary. The paper estimates chess piece values by fitting logistic regressions to large samples of Lichess games, using Glicko-2 rating differences as a skill control and a SIMEX/regression-calibration procedure to correct attenuation from rating noise. Standard chess, Chess 960, Atomic, Antichess, and Horde are analyzed. The headline standard-chess results, based on the 'no capture next 1' sample, are relative values of approximately knight 2.9, bishop 3.2, rook 4.6, and queen 9.6 pawns, with bishops rated slightly above knights. The paper also reports Stockfish self-play ablations that yield more compressed values and explicitly notes this discrepancy. The paper is candid about limitations: the no-capture filter is called a crude quiescence proxy, sigma_0 is fitted by trial and error, and confidence intervals are described as misleading because filter changes move estimates more than the intervals.

Significance. If the results hold, the paper would provide large-sample, publicly grounded evidence on classical piece-value questions and on how piece values change across chess variants. Strengths include the use of millions of games, a transparent measurement-error-correction exercise with Monte Carlo validation, and honest discussion of confounding and selection issues. The variant comparisons and the equalizer-value side results are potentially useful. However, the central quantitative claims are heavily dependent on a filter whose endogeneity the paper itself concedes, and the paper's own sensitivity analyses show that the estimates move by more than the reported bootstrap uncertainty. The significance is therefore conditional on resolving the selection-bias and residual-confounder concerns.

major comments (4)
  1. [§4.1, Table 3] The headline values are taken from the 'No Capture Next 1' column, but this filter conditions on the realized next move rather than on a static property of the position. A position with a winning capture is excluded if the player captures and included if the player overlooks the capture, so the sample is enriched for mistakes and for positions where material is not tactically contested. This is a collider-style selection rule, not a quiescence filter. The magnitude of the concern is visible in the table: moving from 'No Filtering' to 'No Capture Next 1' changes the material coefficients by 25–36% (e.g., knight 147 to 195, queen 479 to 650). The paper concedes the filter is crude and calls for a better one, but still publishes the filtered numbers as the central result. The identifying assumption is not merely 'positions are quiescent' but that the next-move capture choice is ignorable af
  2. [§3.1, Figure 2; §4.1] The measurement-error correction relies on sigma_0 = 58 for standard chess, selected 'via trial and error' to make Monte Carlo and SIMEX curves coincide. No uncertainty is attached to this tuning parameter, and the rating coefficient after correction remains 0.94 rather than the calibrated 1.0. The paper attributes the residual to lopsided positions, but that is the same unmodeled confounding that could bias material coefficients. The Monte Carlo validation in §3.2 covers only a single piece coefficient in a synthetic setting where the mechanism is known; it does not validate the full specification or the no-capture filter. The authors should provide a sensitivity analysis over sigma_0 and should address why the post-correction rating coefficient near 0.94 does not indicate remaining skill confounding.
  3. [§4.1.1] The paper reports bootstrap confidence intervals for the no-capture-filter model but then states that many alternative filtering choices fall far outside those intervals and concludes that 'confidence intervals are somewhat misleading in this study.' Since the no-filter versus one-filter change is 25–30% for several material coefficients, the central estimates have a large model-selection uncertainty that is never quantified. The claim that bishops are slightly more valuable than knights is supported by a 19-point difference in Table 4 with narrow marginal intervals, but the paper itself notes that non-overlap of marginal intervals is not a significance test and does not report the covariance. Without an assessment of the filter-selection uncertainty, the precision of the headline ratios is overstated.
  4. [§5, Tables 15–16] The Stockfish self-play experiments give piece values that are strongly compressed relative to the Lichess-based estimates: roughly knight 2.7–2.9, bishop 2.9–3.1, rook 3.5–3.6, and queen 4.9–5.6 pawns, versus 2.9, 3.2, 4.6, and 9.6 from the main analysis. The paper presents this as a 'contrasting view' and lists possible explanations, but the discrepancy is large enough to undermine the general framing that these are 'piece values' rather than estimates specific to a particular population and selection rule. The discussion should either substantially temper the causal interpretation or provide a test that distinguishes the competing explanations, for example by running the same Lichess filter on Stockfish self-play positions.
minor comments (4)
  1. [Appendix A] Typo: 'frist player' should be 'first player' in the sentence 'Both these differences are with respect to the frist player.'
  2. [§4.1 / Table 4] The table title says 'under filtering for immediate capture,' which is ambiguous: the filter actually removes positions where the next move is a capture. Rephrase to 'excluding positions with an immediate capture.'
  3. [§4.2] The estimated sigma_0 for Chess 960 (34) is nearly half the standard-chess value (58), and the paper says it is 'not clear why.' Since the SIMEX correction is central, this unexplained variation across datasets deserves more than a parenthetical remark; even a brief discussion of possible causes (rating volatility, time window, player pool) would help.
  4. [Tables 3–14] The tables report 'num observations' for what are often multiple snapshots per game, and the text sometimes says 'games' and sometimes 'positions.' Clarify whether the unit is games, positions, or snapshots, and whether the standard errors (when reported) account for repeated snapshots from the same game.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: piece values are estimated from observed game outcomes, and the acknowledged limitations are validity threats, not definitional loops.

full rationale

The paper's central estimates are obtained by logistic regression of game outcomes on material imbalances, with Glicko-2 ratings used as a skill control. The piece-value coefficients are not assumed or fitted from prior piece-value tables; they are estimated directly from Lichess game data. The SIMEX/Monte Carlo procedure is a validation of the attenuation-correction method on synthetic data with known coefficients, and it does not set the final estimates. Moreover, the paper itself reports (Table 4) that the SIMEX calibration has little impact on the non-rating coefficients, so the headline piece values are essentially the raw logistic coefficients, not the output of a fitted parameter that encodes the answer. The author's self-citations (refs 30, 31) are background blog posts about atomic piece values; they are not invoked as authority for the method or as evidence for the results. The no-capture filter is a sample-selection rule, not a definitional equivalence: the paper explicitly calls it a 'crude way to approximately select only quiescent positions,' notes that the estimates change by 25-30% under it, and calls for a better filter (Section 4.1; Section 6 item 1). This is an endogeneity/validity concern, not a circular derivation. Similarly, the residual rating coefficient of 0.94 instead of 1.0 is acknowledged as possible confounding from lopsided positions, and the Stockfish self-play results are presented as a contrasting view with the paper itself noting that engines with internal piece values may bias such inferences (Section 1 item 4; Section 5). These are honest limitations that do not reduce the derivation to its inputs. No circular step can be exhibited with a specific equation or self-citation chain that forces the result.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The paper's estimates rely on a measurement-error model for Glicko-2 ratings whose noise level sigma_0 is fitted per variant by matching SIMEX curves; the main regression is a linear-logistic model of material imbalance. No new physical entities are introduced.

free parameters (6)
  • sigma_0 (standard chess) = 58
    Estimated by matching SIMEX curves to Monte Carlo, 'via trial and error' (Section 3.1).
  • sigma_0 (Chess 960) = 34
    Same procedure; Section 4.2.
  • sigma_0 (Atomic) = 33
    Same procedure; Section 4.3.
  • sigma_0 (Antichess) = 67
    Same procedure; Section 4.4.
  • sigma_0 (Horde) = 34
    Same procedure; Section 4.5.
  • Piece imbalance exclusion thresholds = |K|,|B|,|R| <= 3; |Q| <= 2
    Hand-chosen to remove 'unusual' imbalances (Section 4.1).
assumptions (7)
  • domain assumption Glicko-2 ratings are an unbiased, noisy proxy for true skill
    The causal graph in Section 3 assumes ratings are noisy but unbiased estimates of skill; the SIMEX correction relies on this.
  • standard math Elo logistic model (Eq 1) links rating difference to win probability
    Used to calibrate ratings and to define gamma; Section 3.
  • domain assumption Material imbalance has a linear effect on log-odds (Eq 3)
    The regression specification assumes piece-count differences enter linearly and additively.
  • ad hoc to paper Measurement error in rating differences is additive Gaussian with variance sigma_0^2
    The SIMEX calibration assumes this noise model; sigma_0 is fitted rather than independently measured.
  • ad hoc to paper The no-capture filter selects approximately quiescent positions
    Section 4.1 uses 'no capture in the next move' as a crude quiescence filter; the paper acknowledges it has 'obvious flaws'.
  • domain assumption Ratings below 1200 and short/unfinished games can be excluded without biasing piece values
    Filters on rating floor, game length, and ending condition are applied uniformly across variants (Section 4).
  • domain assumption Stockfish self-play from modified starting positions measures piece value independent of human heuristics
    Section 5 uses Stockfish to get a 'contrasting view'; assumes engine play is a valid benchmark.

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Cite this review

Pith. "Pith review of Inferring Piece Value in Chess and Chess Variants." pith.science (2026). https://pith.science/paper/ZQS3DSFN

@misc{pith2026250904691,
  author       = {Pith},
  title        = {Pith review of: Inferring Piece Value in Chess and Chess Variants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQS3DSFN}},
  note         = {Machine review of arXiv:2509.04691}
}
read the original abstract

We use logistic regression to estimate the value of the pieces in standard chess and several chess variants, namely Chess 960, Atomic chess, Antichess, and Horde chess. We perform our regressions on several years of data from Lichess, the free and open-source internet chess server. We use the published player ratings to control for the confounding effect of differential player skill. We adjust for the attenuation bias in regressions due to the noise in observed ratings. We find that major piece values, relative to the value of a pawn, are fairly consistent with historical valuation systems. However we find slightly higher value to bishops than knights. We find that piece values are smaller, in absolute value, in Atomic and Antichess than standard chess. We also present approximate values of the pieces to equalize odds when players of varying skill face off. We briefly consider self-play experiments using the Stockfish engine, which give a contrasting view of piece value.

Figures

Figures reproduced from arXiv: 2509.04691 by the authors.

Figure 1
Figure 1. The empirical probability of a win is plotted against the theoretical [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. The ratio of fit coefficient to γ is plotted for the Monte Carlo and SIMEX experiments. The SIMEX data are plotted using σ0 = 58. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The logistic regression fit values for the knight Monte Carlo experi [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The logistic regression fit values from Figure [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: The value of piece imbalance for standard chess is plotted versus mean [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: The calibrated estimated piece values, relative to a pawn, for standard [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: The value of piece imbalance for Chess 960 is plotted versus mean [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]
Figure 8
Figure 8. Figure 8: The values of piece imbalance for Chess 960 and standard chess are [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]
Figure 9
Figure 9. Figure 9: The value of piece imbalance for Atomic chess is plotted versus mean [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]
Figure 10
Figure 10. Figure 10: The values of piece imbalance for Atomic and standard chess are [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]
Figure 11
Figure 11. Figure 11: The value of piece imbalance for Antichess is plotted versus mean [PITH_FULL_IMAGE:figures/full_fig_p039_11.png]
Figure 12
Figure 12. Figure 12: The value of piece imbalance for Horde chess is plotted versus mean [PITH_FULL_IMAGE:figures/full_fig_p043_12.png]
Figure 13
Figure 13. Figure 13: The inferred value of each piece in the standard opening are shown. [PITH_FULL_IMAGE:figures/full_fig_p046_13.png]
Figure 14
Figure 14. Figure 14: The plot of Figure [PITH_FULL_IMAGE:figures/full_fig_p048_14.png]
Figure 15
Figure 15. Figure 15: The density of implied k values from the 4,106,242 standard chess games considered in Section 3.1 is plotted. A horizontal line is plotted at the median value, 22.83. To get some idea of the value of k, we consider the 4,106,242 standard chess games considered in Sect…
Figure 16
Figure 16. Figure 16: The standard error of ∆r implied by Equation 10 is plotted versus k for various values of p and ∆a. We plot a vertical line at the median value of k from the sample of standard games, 22.83 and a horizontal line at the implied standard error based on the SIMEX procedu…

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.