Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm
arXiv (Cornell University), 2017 · paper · AI and machine learning
Generalises the AlphaGo Zero approach into a single AlphaZero algorithm trained by self-play. Given only the game rules, it reached superhuman play in chess, shogi and Go within 24 hours and convincingly defeated a world-champion program in each.
Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
1 finding
Substance Main text p.4 says the exploration noise "is scaled in proportion to the typical number of legal moves for that game type", but Methods p.14 says "scaled in inverse proportion to the approximate number of legal moves in a typical position", and α = {0.3, 0.15, 0.03} for chess, shogi, Go falls as move count rises.
- p. 4 Main text, paragraph on hyper-parameters
this is scaled in proportion to the typical number of legal moves for that game type
- p. 14 Methods, Configuration
this was scaled in inverse proportion to the approximate number of legal moves
- p. 14 Methods, Configuration
this was scaled in inverse proportion to the approximate number of legal moves in a typical position
Judge's note: Checked both passages on the page images. The p.4 sentence says 'in proportion'. The p.14 Configuration paragraph says 'inverse proportion' and gives α values that fall from chess to shogi to Go (Go has the most legal moves). The main text is the reversed one. Fix: 'in inverse proportion'.
Second check: another model agreed
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9 minor (typos and the like; most checked by the judge only)
Reference or label P.15 says ratings were "computed by the BayesElo program (10)", but reference 10 is "R. Coulom. Whole-history rating: A Bayesian rating system for players of time-varying strength" (CG 2008).
- p. 15 Methods, Evaluation
computed by the BayesElo program (10)
- p. 8 Reference 10
R. Coulom. Whole-history rating: A Bayesian rating system for players of time-varying strength.
- p. 4 Main text, Figure 1 discussion
on an Elo scale (10)
- p. 8 Reference 10
Whole-history rating: A Bayesian rating system for players of time-varying strength
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: BayesElo is a separate rating program by Coulom, distinct from the Whole-History Rating method. Reference 10 on p.8 is the WHR paper as quoted, and the citation on p.15 names the BayesElo program. This is a loose citation that the authors should clarify. The p.4 citation 'on an Elo scale (10)' is less specific.
Second check: another model agreed
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Wording P.5 says "Each of these openings is independently discovered and played frequently by AlphaZero during self-play training", yet the Table 2 curves for E61 Kings Indian Defence and B30 Sicilian Defence stay close to 0% for the whole training run.
- p. 5 Main text, paragraph on Table 2
Each of these openings is independently discovered and played frequently by AlphaZero during self-play training.
- p. 6 Table 2, panels E61 Kings Indian Defence and B30 Sicilian Defence
E61: Kings Indian Defence
Judge's note: Viewed Table 2 on p.6. The E61 and B30 plots are flat near the 0% axis throughout, far below the 6% gridline, and B40 is also very low. 'Frequently' could be meant in absolute terms (44 million training games), but against these plotted proportions the sentence overstates things for several openings. This is a wording point.
Second check: another model agreed
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Wording P.13 prose lists "the repetition count for that position" among the "additional L constant-valued input planes", but Table S1 places Repetitions (2 planes for chess, 3 for shogi) in the first block, which the caption says is repeated for each of the T = 8 history steps.
- p. 13 Representation paragraph
There are an additional L constant-valued input planes denoting the player’s colour, the total move count, and the state of special rules
- p. 13 Table S1, first (history) block
Repetitions
Only the authors can say what is right here.
Judge's note: Checked Table S1 and the paragraph below it on p.13. The table totals (119 for chess, 362 for shogi) only work with Repetitions in the per-step block. With the prose placement, chess would total 105. The prose grouping therefore does not match the table. The data appear consistent with the table.
Second check: another model agreed
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Wording P.14 says "The number of games, positions, and thinking time varied per game ... and are shown in Table S3", but Table S3 lists only Mini-batches, Training Time, Training Games and Thinking Time, with no count of positions.
- p. 14 Configuration, first paragraph
The number of games, positions, and thinking time varied per game
- p. 15 Table S3
Selected statistics of AlphaZero training in Chess, Shogi and Go.
Judge's note: Viewed Table S3 on p.15. It has no row for positions. Either the text should drop 'positions' or the table needs a row for them.
Second check: another model agreed
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Wording P.15 gives p(a defeats b) = 1/(1+exp(c_elo(e(b)−e(a)))) "using the standard constant c_elo = 1/400". With a natural exponential, 1/400 does not give the conventional Elo scale.
- p. 15 Methods, Evaluation
using the standard constant celo = 1/400
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the conventional Elo logistic is 1/(1+10^((e(b)−e(a))/400)), which written with exp needs c_elo = ln(10)/400 ≈ 0.00576, not 0.0025. The printed formula and constant are as quoted on the p.15 image. The ratings were computed with BayesElo, so this may be a slip in how the formula is written rather than in the ratings. Only the authors can confirm which base was intended.
Second check: another model agreed
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Typo P.10: "In addition, a combination of domain-independent move ordering heuristics, such as killer heuristic, ... and also domain-dependent knowledge based on captures (SEE) and potential captures (MVV/LVA)." The sentence has no main verb.
- p. 10 Methods, Anatomy of a Computer Chess Program, move-ordering paragraph
In addition, a combination of domain-independent move ordering heuristics
Judge's note: The page image shows the sentence ending at '(MVV/LVA).' with no verb such as 'is used'. It is a fragment.
Second check: none (judge only)
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Typo P.11: "by learning to select expert human moves during alpha-beta serach". 'serach' should be 'search'.
- p. 11 Prior Work, Kaneko and Hoki paragraph
during alpha-beta serach
- p. 11
by learning to select expert human moves during alpha-beta serach
Judge's note: The misspelling is visible on the p.11 page image.
Second check: none (judge only)
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Typo P.11: "DeepChess (11) trained a neural network to performed pair-wise evaluations of positions". 'to performed' should be 'to perform'.
- p. 11 Prior Work, DeepChess paragraph
trained a neural network to performed pair-wise evaluations
- p. 11
trained a neural network to performed pair-wise evaluations of positions
Judge's note: The grammatical error is visible on the p.11 page image.
Second check: none (judge only)
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Typo P.14: "The first 64 planes encode 'queen moves' and the next 2 moves encode knight moves". 'moves' should be 'planes'.
- p. 14 Representation, shogi policy
and the next 2 moves encode knight moves
- p. 14 Representation, shogi policy paragraph
the next 2 moves encode knight moves
Judge's note: Checked on the p.14 image. The plane counts 64+2+64+2+7 = 139 match the 9×9×139 stack, and 81×139 = 11,259 matches the stated number of moves. So the '2' counts planes, not moves.
Second check: none (judge only)