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championship runs, chess, chess analytics, chess history, chess metrics, engine analysis, Fischer, Karpov, Stockfish, world championship

1. CHESS ANALYTICS 01 part 1/3: the 11th World-Champion, Robert Fischer

I treated Robert James Fischer as “main player” throughout, and used the CSVs as the source tables, with the markdown report as the readable cross-check. The report confirms the run package combines match-level Stockfish 18 WDL reports and defines the core WDL accuracy as 100 × (1 − WDL expected-score loss).

1.1. Overall verdict

Across the played games in the 1971–1972 run, Fischer’s measured edge is real but not mainly a huge raw-accuracy gap. The headline WDL Accuracy edge is only:

Metric	Fischer	Opponents	Edge
WDL Accuracy	97.847	97.058	+0.788
Performance Quality / PQ	95.313	94.534	+0.779
Mean ES Loss	0.0215	0.0294	Fischer 26.8% lower
RMS ES Loss	0.0636	0.0844	Fischer 24.6% lower
WDL Volatility	0.0225	0.0307	Fischer 26.6% lower
Dominance	+0.803	−0.803	+1.606 difference
Conversion	+5.194	−5.194	+10.388 difference
Expected Score	25.806	15.194	+10.612
Actual Score	31.0	10.0	+21.0

So the run is best explained as:

small but persistent move-quality edge → lower losses and lower volatility → repeated positional/technical dominance → very large conversion surplus.

The most score-explanatory families are therefore:

Expected-score edge / dominance family: Fischer’s engine-WDL expected score was already +10.612 across the run.
Conversion family: he scored +5.194 above expectation, while the opponents scored −5.194. This doubled the practical score gap from a large expected advantage into a crushing actual score margin.
Loss/volatility family: Fischer’s mean ES loss, RMS ES loss, and volatility were consistently lower. This is the “why” behind the expected-score and conversion edges.
Raw WDL Accuracy / PQ family: useful, but the numerical percentage gaps are small because both sides were very strong. These metrics are less dramatic than the loss and conversion metrics.

For signed metrics like Dominance and Conversion, the ratios are mathematically unstable or semantically misleading, so the differences are the correct reading.

1.2. Match-by-match headline

Match	Score	WDL Acc. edge	PQ edge	Dominance diff.	Mean ES Loss ratio	RMS ES Loss ratio	Volatility ratio	Conversion diff.
Fischer–Taimanov	6–0	+0.972	+1.035	+2.124	0.642	0.640	0.638	+3.256
Fischer–Larsen	6–0	+1.222	+1.206	+2.483	0.630	0.699	0.639	+2.569
Fischer–Petrosian	6.5–2.5	+0.624	+0.502	+1.050	0.820	0.864	0.822	+2.324
Fischer–Spassky	12.5–7.5	+0.334	+0.373	+0.768	0.853	0.847	0.843	+2.239

Interpretation: the four matches form a clean gradient. The two 6–0 matches show large relative move-quality/loss advantages. Petrosian and Spassky narrow the raw-quality gap, but Fischer still keeps a measurable edge and continues to convert above expectation.

1.3. Fischer–Taimanov, 6–0

This is the clearest “complete control” match after Larsen, but with slightly less dominance than Larsen by some measures.

Metric	Fischer	Taimanov	Reading
Expected Score	4.372	1.628	Fischer was engine-favored even before conversion.
Actual Score	6.0	0.0	Result far exceeded expectation.
Conversion	+1.628	−1.628	Massive practical overperformance.
WDL Accuracy	98.257 ± 6.412	97.285 ± 8.733	Fischer +0.972 and more stable.
PQ	95.867 ± 1.959	94.832 ± 2.128	Fischer +1.035.
Dominance	+1.062	−1.062	Strong positive engine-WDL pressure.
Mean ES Loss	0.0174	0.0271	Fischer’s average loss was only 64.2% of Taimanov’s.
RMS ES Loss	0.0593	0.0927	Fischer’s larger-error profile was much better.
Volatility	0.0179	0.0281	Fischer was much less swingy.
Error Concentration	3.345	3.453	Roughly similar, slight Fischer edge.
HardRAP	575.2	0.0	Sweep gives opponent zero.
SoftRAP	575.2	284.5	Fischer about 2.02×.

The important SD pattern: Fischer’s WDL Accuracy SD was 6.412 vs 8.733, SD ratio 0.734, so he was not only better but also more stable. His volatility SD ratio was also about 0.735. The only caveat is RMS ES Loss SD ratio 1.482, suggesting Fischer’s rare larger errors varied more in size, even though his average RMS loss was much lower.

Main explanation of the 6–0: Taimanov did not collapse by raw accuracy alone; rather, Fischer’s lower mean loss, lower volatility, and positive conversion turned a 4.37–1.63 expected-score match into a perfect 6–0.

1.4. Fischer–Larsen, 6–0

By the metric package, this is probably the most forceful pure match-level superiority.

Metric	Fischer	Larsen	Reading
Expected Score	4.715	1.285	Strongest expected-score margin of the two 6–0s.
Actual Score	6.0	0.0	Again converted to perfection.
Conversion	+1.285	−1.285	Still huge.
WDL Accuracy	97.922 ± 6.786	96.700 ± 8.907	Fischer +1.222.
PQ	95.180 ± 1.856	93.973 ± 1.923	Fischer +1.206.
Dominance	+1.241	−1.241	Largest dominance among the four matches.
Mean ES Loss	0.0208	0.0330	Fischer only 63.0% of Larsen’s mean loss.
RMS ES Loss	0.0649	0.0929	Fischer only 69.9% of Larsen’s RMS loss.
Volatility	0.0215	0.0336	Fischer only 63.9% of Larsen’s volatility.
Error Concentration	3.085	2.870	Larsen slightly better by this one metric.
HardRAP	571.1	0.0	Sweep makes HardRAP one-sided.
SoftRAP	571.1	281.9	Fischer about 2.03×.

This match has the largest WDL Accuracy edge (+1.222), largest PQ edge (+1.206), and largest dominance difference (+2.483). Fischer was also more stable in WDL Accuracy: SD ratio 0.762.

The one exception is Error Concentration, where Fischer’s value is higher, 3.085 vs 2.870. Since lower is better, Larsen is technically better there. But this did not matter much because Fischer’s total losses, volatility, RMS loss, dominance, and conversion were all decisively superior.

Main explanation of the 6–0: this is the match where the quality edge itself was most decisive. Conversion mattered, but Fischer’s expected-score margin was already enormous: +3.431.

1.5. Fischer–Petrosian, 6.5–2.5

This is where the run becomes more competitive. Petrosian narrowed the raw quality gap, but Fischer still won the key loss and dominance families.

Metric	Fischer	Petrosian	Reading
Expected Score	5.338	3.662	Fischer expected edge +1.676.
Actual Score	6.5	2.5	Actual edge +4.0.
Conversion	+1.162	−1.162	Large overperformance.
WDL Accuracy	97.147 ± 7.426	96.522 ± 8.815	Fischer +0.624.
PQ	94.133 ± 2.730	93.631 ± 3.127	Fischer +0.502.
Dominance	+0.525	−0.525	Clear but much smaller than vs Taimanov/Larsen.
Mean ES Loss	0.0285	0.0348	Fischer 82.0% of Petrosian’s mean loss.
RMS ES Loss	0.0740	0.0857	Fischer 86.4% of Petrosian’s RMS loss.
Volatility	0.0301	0.0366	Fischer 82.2% of Petrosian’s volatility.
Error Concentration	2.651	2.573	Petrosian slightly better.
HardRAP	611.3	237.8	Fischer 2.57×.
SoftRAP	729.2	540.2	Fischer 1.35×.

Petrosian’s resistance appears in three places:

First, Fischer’s WDL Accuracy and PQ edges shrink to about half a point. Second, the SD ratios are closer to parity: WDL Accuracy SD ratio 0.842, PQ SD ratio 0.873, RMS SD ratio 0.870. Third, Error Concentration again slightly favors the opponent.

But Fischer’s practical edge remains strong: actual margin +4.0 versus expected margin +1.676. So the match was not just “Fischer was slightly more accurate”; it was Fischer converted the small-to-medium engine edge into a large match victory.

Main explanation of the 6.5–2.5: Petrosian reduced the quality gap, but Fischer still had lower losses and lower volatility, and conversion expanded the final result.

1.6. Fischer–Spassky, 12.5–7.5 in played games

This is the most balanced match by raw engine metrics, but Fischer still owns the cumulative edges.

Metric	Fischer	Spassky	Reading
Expected Score	11.381	8.619	Fischer expected edge +2.762.
Actual Score	12.5	7.5	Actual edge +5.0.
Conversion	+1.119	−1.119	Conversion again adds heavily.
WDL Accuracy	98.060 ± 6.832	97.726 ± 7.596	Fischer +0.334.
PQ	96.074 ± 2.981	95.701 ± 3.151	Fischer +0.373.
Dominance	+0.384	−0.384	Smallest dominance gap, but still positive.
Mean ES Loss	0.0194	0.0227	Fischer 85.3% of Spassky’s mean loss.
RMS ES Loss	0.0560	0.0661	Fischer 84.7% of Spassky’s RMS loss.
Volatility	0.0207	0.0245	Fischer 84.3% of Spassky’s volatility.
Error Concentration	3.171	3.134	essentially equal, slight Spassky edge.
HardRAP	1198.2	725.2	Fischer 1.65×.
SoftRAP	1559.9	1319.6	Fischer 1.18×.

This is a very important interpretive case. Spassky was close in WDL Accuracy and PQ: Fischer’s accuracy ratio is only 1.0034, and PQ ratio only 1.0039. But the loss metrics remain directionally consistent: Fischer loses less, has lower RMS loss, and lower volatility.

The SD ratios also show this was not a blowout of consistency:

SD ratio	Value
WDL Accuracy SD ratio	0.900
PQ SD ratio	0.946
Mean ES Loss SD ratio	0.900
RMS ES Loss SD ratio	0.961
Volatility SD ratio	0.894

So Fischer was still steadier, but Spassky was the closest opponent. The result was produced by small persistent advantages plus conversion, not by an overwhelming average accuracy gap.

Main explanation of the 12.5–7.5: Spassky nearly matched Fischer’s raw accuracy, but Fischer’s losses were smaller, his volatility lower, and he converted the expected edge better.

1.7. Game-level findings

Across the 41 played games, Fischer had the better side of the direct per-game comparison in:

Metric family	Fischer better in games
Accuracy	29 / 41
PQ	29 / 41
Mean ES Loss	29 / 41
RMS ES Loss	30 / 41
Volatility	29 / 41
Dominance	29 / 41

By match:

Match	Accuracy/PQ/Dominance better games	RMS better games
Taimanov	6/6	6/6
Larsen	6/6	6/6
Petrosian	6/9	6/9
Spassky	11/20	12/20

That is a useful compressed summary of the run: Fischer did not merely win many games; he usually won the metric comparison inside the games too. Against Spassky, however, the game-by-game edge was only slight, which fits the much narrower match-level accuracy gap.

The strongest game-level correlations with Fischer’s score were:

Game-level predictor	Correlation with Fischer game score
PQ difference	~0.897
Accuracy / dominance difference	~0.895
Mean ES Loss advantage	~0.895
Conversion	~0.827
RMS ES Loss advantage	~0.816
Volatility advantage	~0.814

The interesting negative finding: game accuracy and mutual accuracy by themselves do not explain Fischer’s score well. High mutual accuracy often occurred in draws or in games where both sides played cleanly. What mattered more was relative advantage, not absolute game cleanliness.

1.8. What most strongly explains the scores?

1.8.A. Expected-score advantage explains the base result

Fischer’s total WDL expected score was:

25.806–15.194, a margin of +10.612.

That already predicts a dominant run. In other words, the engine-WDL data says Fischer was not merely “lucky” or only converting practical chances. He was objectively creating better expected-score positions/move sequences across the run.

1.8.B. Conversion explains why the result became historically crushing

Actual played-game score was:

31–10, a margin of +21.0.

Since:

Actual score = Expected score + Conversion

Fischer’s +5.194 conversion and the opponents’ −5.194 conversion produce a conversion swing of:

+10.388

So the final score margin is roughly:

+10.612 expected-score margin + +10.388 conversion swing = +21.000 actual score margin

That is probably the single cleanest mathematical explanation of the run.

1.8.C. Loss and volatility metrics explain the engine edge

Fischer’s loss-side metrics are consistently better:

Metric	Fischer / Opponent ratio	Interpretation
Mean ES Loss	0.732	Fischer lost about 26.8% less expected score per move/unit.
RMS ES Loss	0.754	Fischer’s larger mistakes were much smaller overall.
WDL Volatility	0.734	Fischer’s games/move sequences were less destabilizing for himself.
Total WDL Volatility	0.771	Same story cumulatively.

These are more explanatory than raw accuracy percentages because WDL Accuracy compresses strong play into a narrow 96–98% band. The loss metrics expose the practical difference more clearly.

1.8.D. Error Concentration is not a primary explanation

Overall Error Concentration is:

Player	Error Concentration
Fischer	3.063
Opponents	3.008

Because lower is better, this slightly favors the opponents. Match-by-match, Fischer only clearly wins Error Concentration vs Taimanov. Larsen, Petrosian, and Spassky are roughly equal or slightly better.

So the run was not mainly explained by Fischer having less concentrated errors. Rather, his errors were generally smaller and less costly, while the opponents’ losses and volatility were higher.

1.8.E. RAP metrics mostly encode score + quality dominance

HardRAP and SoftRAP heavily favor Fischer:

Metric	Fischer	Opponents	Ratio
HardRAP	2955.8	963.0	3.07×
SoftRAP	3435.4	2426.3	1.42×

HardRAP becomes extreme in sweep matches because the opponent’s score/result component collapses. SoftRAP is more informative across competitive matches: it still gives Fischer a large edge, but less explosively.

1.9. Overall chess interpretation

Fischer’s run looks like a combination of four qualities:

1. Technical accuracy under pressure.
The raw accuracy edge is modest, but it appears in every match.

2. Lower average damage.
Mean ES Loss is consistently lower. Fischer’s ordinary inaccuracies were less harmful.

3. Fewer severe practical collapses.
RMS ES Loss is consistently lower. This suggests the opponents’ worst mistakes were more costly.

4. Superior conversion of favorable WDL states.
This is the decisive match-score amplifier. Fischer turned expected advantages into full points at a rate that greatly exceeded the opponents.

So, in compressed form:

Fischer’s superiority was not “he played 2–3% more accurately.”
It was: he lost less expected score, allowed less volatility, built small-to-large WDL edges more often, and converted those edges with exceptional efficiency.

That is why the match results are much more dramatic than the raw WDL Accuracy gaps alone would suggest.

2. CHESS ANALYTICS 01 part 2/3: the 12th World-Champion, Anatoly Karpov

I treated Anatoly Karpov as the “main player” throughout. The uploaded report identifies this as the Karpov 1974 World-Championship Run, combining the Polugaevsky, Spassky, and Korchnoi candidate matches under Stockfish 18 WDL expected-score analysis.

2.1. Overall verdict

Karpov’s 1974 run is not a “huge raw accuracy gap” story. It is a small but very stable superiority story:

Overall metric	Karpov	Opponents	Edge / ratio
Score	25.0	18.0	+7.0
Expected Score	23.383	19.617	+3.765
Conversion	+1.617	−1.617	+3.235 difference
WDL Accuracy	98.604	98.301	+0.303
PQ	97.143	96.835	+0.308
Dominance	+0.314	−0.314	+0.629 difference
Mean ES Loss	0.0140	0.0170	Karpov 17.9% lower
RMS ES Loss	0.0402	0.0524	Karpov 23.4% lower
Error Concentration	2.750	3.024	Karpov 9.1% lower
WDL Volatility	0.0149	0.0179	Karpov 16.9% lower
Total WDL Volatility	30.061	34.562	Karpov 13.0% lower
HardRAP	2428.3	1744.7	Karpov 1.392×
SoftRAP	3299.2	2952.0	Karpov 1.118×

The key formulaic explanation is:

Actual score margin = Expected-score margin + Conversion swing

So:

+7.000 actual score margin = +3.765 expected-score margin + +3.235 conversion swing

That is the cleanest mathematical summary of the run. Karpov’s engine-WDL advantage already explains more than half the match-score edge, and his conversion explains the rest.

2.2. Overall stability and SD reading

The SDs show that Karpov was generally more stable than the opponent pool, especially in raw WDL accuracy and loss/volatility metrics.

Metric	Karpov mean of match SDs	Opponent mean of match SDs	SD ratio
WDL Accuracy	5.156	6.305	0.818
Mean ES Loss	0.0516	0.0630	0.818
RMS ES Loss	0.0331	0.0354	0.936
WDL Volatility	0.0517	0.0632	0.817
PQ	2.207	2.294	0.962
Error Concentration	0.746	1.102	0.677

So Karpov’s edge is twofold:

He was slightly better on average.
He was usually less unstable.

This is very “Karpovian”: the superiority appears less as tactical fireworks and more as reduced leakage, reduced volatility, and steady pressure.

2.3. Match-by-match summary

Match	Score	Exp. Score	Conversion	WDL Acc. edge	PQ edge	Dominance diff.	Mean Loss ratio	RMS Loss ratio	Volatility ratio
Karpov–Polugaevsky	5.5–2.5	4.545–3.455	+0.955	+0.490	+0.436	+0.881	0.704	0.516	0.724
Karpov–Spassky	7–4	6.556–4.444	+0.444	+0.300	+0.346	+0.703	0.826	0.814	0.847
Karpov–Korchnoi	12.5–11.5	12.281–11.719	+0.219	+0.120	+0.142	+0.301	0.930	0.971	0.916

This table shows a clear narrowing:

Polugaevsky: large technical superiority
→ Spassky: moderate but clear superiority
→ Korchnoi: very small, almost level superiority

The Korchnoi match was extremely close by engine-WDL terms, while the Polugaevsky match was Karpov’s clearest performance.

2.4. Karpov–Polugaevsky, 5.5–2.5

This was Karpov’s cleanest match in the run.

Metric	Karpov	Polugaevsky	Reading
Score	5.5	2.5	+3.0 result margin
Expected Score	4.545	3.455	+1.090 engine-WDL margin
Conversion	+0.955	−0.955	Result exceeded expectation strongly
WDL Accuracy	98.836 ± 3.698	98.346 ± 6.189	Karpov +0.490; much stabler
PQ	97.584 ± 1.791	97.148 ± 1.694	Karpov +0.436
Dominance	+0.440	−0.440	+0.881 difference
Mean ES Loss	0.0116	0.0165	Karpov 29.6% lower
RMS ES Loss	0.0269	0.0522	Karpov 48.4% lower
Error Concentration	2.220	3.061	Karpov 27.5% lower
WDL Volatility	0.0124	0.0171	Karpov 27.6% lower
HardRAP	537.8	242.8	Karpov 2.215×
SoftRAP	659.2	510.0	Karpov 1.293×

The most striking feature is RMS ES Loss: Karpov’s value was only 51.6% of Polugaevsky’s. That means the larger-error profile was dramatically better for Karpov. Error Concentration also strongly favored Karpov, unlike in some Fischer-run matches where Error Concentration was less explanatory.

SD-wise, Karpov’s WDL Accuracy SD ratio was 0.597, Mean ES Loss SD ratio 0.597, and Volatility SD ratio 0.599. This means Polugaevsky was not only worse on average, but also much more variable.

Main explanation: Karpov’s result came from a broad technical edge: lower mean loss, much lower RMS loss, lower volatility, lower error concentration, and strong conversion.

2.5. Karpov–Spassky, 7–4

This was Karpov’s most objectively dominant match by expected-score margin, even though the raw accuracy edge was smaller than against Polugaevsky.

Metric	Karpov	Spassky	Reading
Score	7.0	4.0	+3.0 result margin
Expected Score	6.556	4.444	+2.112 engine-WDL margin
Conversion	+0.444	−0.444	Moderate practical overperformance
WDL Accuracy	98.575 ± 5.724	98.275 ± 6.327	Karpov +0.300
PQ	97.145 ± 1.715	96.799 ± 1.744	Karpov +0.346
Dominance	+0.352	−0.352	+0.703 difference
Mean ES Loss	0.0142	0.0172	Karpov 17.4% lower
RMS ES Loss	0.0444	0.0545	Karpov 18.6% lower
Error Concentration	2.962	3.112	Karpov 4.8% lower
WDL Volatility	0.0154	0.0182	Karpov 15.3% lower
HardRAP	681.9	387.6	Karpov 1.760×
SoftRAP	875.3	726.2	Karpov 1.205×

Against Spassky, the most explanatory metric family is probably Expected Score / Dominance, not Conversion. Karpov’s expected-score margin was +2.112, while conversion added only +0.888 to the score difference.

So unlike Polugaevsky, where conversion was very important, against Spassky the result looks more like sustained objective pressure.

Stability is also favorable but less extreme than vs Polugaevsky. Karpov’s WDL Accuracy SD ratio was 0.905, Mean ES Loss SD ratio 0.905, and Volatility SD ratio 0.907. This means Spassky was fairly stable too, but Karpov retained a clean edge.

Main explanation: Karpov beat Spassky mostly by objective WDL superiority: better expected score, lower loss, lower volatility, and a clear but not enormous conversion surplus.

2.6. Karpov–Korchnoi, 12.5–11.5

This was by far the closest match. The raw metric gaps are tiny.

Metric	Karpov	Korchnoi	Reading
Score	12.5	11.5	+1.0 result margin
Expected Score	12.281	11.719	+0.563 engine-WDL margin
Conversion	+0.219	−0.219	Small conversion surplus
WDL Accuracy	98.401 ± 6.046	98.281 ± 6.398	Karpov +0.120
PQ	96.700 ± 3.115	96.558 ± 3.445	Karpov +0.142
Dominance	+0.151	−0.151	+0.301 difference
Mean ES Loss	0.0160	0.0172	Karpov 7.0% lower
RMS ES Loss	0.0492	0.0507	Karpov only 2.9% lower
Error Concentration	3.067	2.898	Korchnoi better by 5.8%
WDL Volatility	0.0168	0.0183	Karpov 8.4% lower
HardRAP	1208.6	1114.4	Karpov 1.085×
SoftRAP	1764.7	1715.9	Karpov 1.028×

This is nearly even. Karpov’s edge exists, but it is small:

WDL Accuracy ratio: 1.0012
PQ ratio: 1.0015
Expected Score ratio: 1.048
SoftRAP ratio: 1.028
RMS ES Loss ratio: 0.971

Korchnoi actually wins Error Concentration, 2.898 vs Karpov’s 3.067. That suggests Karpov’s errors may have been somewhat more concentrated, even though his total loss/volatility profile was still slightly better.

The SD readings are close too:

SD ratio	Value
WDL Accuracy SD ratio	0.945
PQ SD ratio	0.904
Mean ES Loss SD ratio	0.945
RMS ES Loss SD ratio	0.916
Volatility SD ratio	0.938
Error Concentration SD ratio	0.923

So Karpov was slightly more stable, but not overwhelmingly. This match was essentially decided by small cumulative advantages plus a small conversion edge.

Main explanation: Karpov–Korchnoi was a razor-thin technical match. Karpov’s lower volatility and slightly lower ES loss explain the small expected-score margin; conversion explains why that became a one-point match win.

2.7. Game Accuracy and Mutual Accuracy

The match-level overall table has blank player-split values for Game Accuracy and Mutual Accuracy, but the game summary provides game-level values.

Match	Game Accuracy avg.	Mutual Accuracy avg.	Interpretation
Karpov–Polugaevsky	98.670	97.365	Cleanest/highest-quality match
Karpov–Spassky	98.471	96.971	Very high quality, slightly more unstable
Karpov–Korchnoi	98.285	96.627	Still extremely high, but most difficult/volatile

These values show that the whole 1974 run was played at a very high average technical level. However, Game Accuracy and Mutual Accuracy alone do not explain Karpov’s score well.

In the game-level correlations I computed:

Predictor	Correlation with Karpov game score
Karpov Expected Score	~0.880
PQ difference	~0.862
Accuracy difference	~0.855
Mean ES Loss advantage	~0.855
Dominance difference	~0.855
Volatility advantage	~0.833
RMS ES Loss advantage	~0.802
Karpov Conversion	~0.728
Game Accuracy	~0.005
Mutual Accuracy	~0.004
Game Volatility	~−0.003

This is important: absolute game quality does not determine who scores. Relative advantage determines who scores. A very accurate draw can have high Game Accuracy and high Mutual Accuracy, but still not produce a Karpov point. The decisive variables are the differences: Karpov accuracy minus opponent accuracy, Karpov PQ minus opponent PQ, and opponent ES loss minus Karpov ES loss.

2.8. Game-by-game relative edge

Across all 43 games, Karpov was better than his opponent in the direct game comparison as follows:

Metric family	Karpov better in games
Accuracy	25 / 43
PQ	26 / 43
Dominance	26 / 43
Mean ES Loss	25 / 43
RMS ES Loss	25 / 43
Volatility	28 / 43

By match:

Match	Accuracy better	PQ better	Dominance better	Mean Loss better	RMS better	Volatility better
Polugaevsky, 8 games	7	7	7	7	7	7
Spassky, 11 games	6	6	6	6	7	6
Korchnoi, 24 games	12	13	13	12	11	15

This is another way to see the progression:

Polugaevsky: Karpov clearly outplayed him in almost every game-level metric.
Spassky: Karpov had a moderate game-by-game majority.
Korchnoi: the match was almost exactly balanced, with Karpov’s best game-level signal being lower volatility: 15/24 games.

2.9. Which metric families best explain the run?

2.9.1. Expected Score and Dominance

Karpov’s expected score was 23.383–19.617, a margin of +3.765. This is the objective engine-WDL foundation of the match results.

Dominance was +0.314 vs −0.314, a difference of +0.629. Since Dominance is signed, the difference is much more meaningful than the ratio.

This family explains the base result: Karpov was creating slightly better WDL states across the run.

2.9.2. Conversion

Karpov’s conversion was +1.617, while the opponents’ was −1.617. The difference is +3.235.

This is nearly as important as the expected-score edge. Without conversion, the run would look like a strong but narrower Karpov success. With conversion, it becomes a comfortable 25–18 total result.

Conversion was strongest against Polugaevsky, moderate against Spassky, and small against Korchnoi.

2.9.3. Mean ES Loss, RMS ES Loss, and Volatility

These are probably the most “style-revealing” metrics.

Metric	Karpov	Opponents	Interpretation
Mean ES Loss	0.0140	0.0170	Karpov leaked less expected score.
RMS ES Loss	0.0402	0.0524	Karpov’s larger errors were smaller.
WDL Volatility	0.0149	0.0179	Karpov’s play was less swingy.

This is very consistent with a technical, positional profile: Karpov’s edge was not necessarily spectacular, but he made fewer destabilizing concessions.

2.9.4. Error Concentration

Unlike Fischer’s previous run analysis, Karpov’s Error Concentration is quite meaningful here:

Match	Karpov	Opponent	Better
Polugaevsky	2.220	3.061	Karpov
Spassky	2.962	3.112	Karpov
Korchnoi	3.067	2.898	Korchnoi
Overall	2.750	3.024	Karpov

Overall, Karpov’s errors were less concentrated. Against Korchnoi, however, this reverses, which supports the idea that the final match was much more uncomfortable and closely contested.

2.9.5. RAP metrics

HardRAP and SoftRAP both favor Karpov:

Match	HardRAP ratio	SoftRAP ratio
Polugaevsky	2.215×	1.293×
Spassky	1.760×	1.205×
Korchnoi	1.085×	1.028×
Overall	1.392×	1.118×

HardRAP magnifies the match-score advantage more heavily. SoftRAP is more restrained and probably better for judging close matches like Karpov–Korchnoi.

2.10. Chess interpretation

Karpov’s 1974 run looks like a textbook case of technical accumulation.

He did not crush the field through gigantic raw WDL Accuracy margins. His overall WDL Accuracy edge was only +0.303, and his PQ edge only +0.308. But he combined that small superiority with:

Lower mean expected-score loss
Lower RMS expected-score loss
Lower volatility
Lower overall error concentration
Positive conversion
Slight but persistent dominance

In chess terms, this suggests:

Karpov’s 1974 strength was not mainly “brilliancy frequency.”
It was position-by-position risk control, small expected-score accumulation, and unusually efficient conversion of small advantages.

The run also has a clear internal arc:

Polugaevsky match: Karpov’s most complete technical superiority.
Spassky match: strongest expected-score/dominance win.
Korchnoi match: almost level, decided by small margins and slightly better stability.

So the best compressed verdict is:

Karpov 1974 was a high-accuracy, low-volatility, high-conversion run.
His edge was small per move, but persistent enough to become decisive over 43 games.

3. CHESS ANALYTICS 01 part 3/3: 11th vs. 12th World-Championship run

Fischer 1971–72 compared with Karpov 1974

The two uploaded run reports define the comparison set as Fischer’s 1971–1972 World-Championship road and Karpov’s 1974 World-Championship road, both measured with Stockfish 18 WDL expected score, where ES = W + 0.5D and WDL move accuracy is based on expected-score loss.

3.1. The shortest verdict

Fischer 1971–72 was the more dominant run.
Karpov 1974 was the cleaner, lower-volatility run.

Fischer produced the larger score margin, larger expected-score margin, larger dominance, larger conversion surplus, and larger RAP superiority. Karpov produced higher raw WDL Accuracy, higher PQ, lower mean ES loss, lower RMS ES loss, lower volatility, and better error concentration.

So the rough distinction is:

Player-run	Statistical personality
Fischer 1971–72	More forceful, more dominant, more result-explosive, higher conversion, larger separation from the field.
Karpov 1974	Cleaner, quieter, lower-loss, lower-volatility, more technically compressed, smaller but very stable advantages.

In chess-style language:

Fischer’s run looks like maximum competitive domination.
Karpov’s run looks like maximum technical control.

3.2. Overall comparison table

Metric	Fischer 1971–72	Karpov 1974	Edge
Games	41	43	Similar sample size
Score	31–10	25–18	Fischer much larger
Score %	75.61%	58.14%	Fischer
Expected Score	25.806–15.194	23.383–19.617	Fischer larger margin
Expected Score %	62.94%	54.38%	Fischer
Conversion	+5.194	+1.617	Fischer much larger
WDL Accuracy	97.847	98.604	Karpov
Opponent WDL Accuracy	97.058	98.301	Karpov faced cleaner aggregate opposition / cleaner games
WDL Accuracy edge	+0.788	+0.303	Fischer larger relative edge
PQ	95.313	97.143	Karpov
PQ edge	+0.779	+0.308	Fischer larger relative edge
Dominance	+0.803	+0.314	Fischer
Mean ES Loss	0.0215	0.0140	Karpov cleaner
RMS ES Loss	0.0636	0.0402	Karpov cleaner
Error Concentration	3.063	2.750	Karpov
WDL Volatility	0.0225	0.0149	Karpov lower
Total WDL Volatility	40.603	30.061	Karpov lower
HardRAP	2955.8	2428.3	Fischer
SoftRAP	3435.4	3299.2	Fischer, but close

This table immediately shows the paradox:

Karpov has the better “clean chess” metrics. Fischer has the better “domination” metrics.

3.3. The core mathematical explanation

For both runs, the score margin can be decomposed cleanly:

Fischer

Actual score margin = +21.000

This comes from:

Expected-score margin + Conversion swing

= +10.612 + +10.388 = +21.000

Karpov

Actual score margin = +7.000

This comes from:

Expected-score margin + Conversion swing

= +3.765 + +3.235 = +7.000

So Fischer’s run was about three times larger in actual score margin, and also about 2.8 times larger in expected-score margin:

Margin type	Fischer	Karpov	Fischer / Karpov
Actual score margin	+21.000	+7.000	3.00×
Expected-score margin	+10.612	+3.765	2.82×
Conversion swing	+10.388	+3.235	3.21×
Dominance difference	+1.606	+0.629	2.56×

This is probably the most important statistical distinction.

Karpov’s run was cleaner; Fischer’s run separated from the opponents much more violently.

3.4. Accuracy: Karpov higher, Fischer more separated

At first glance, Karpov looks better by raw WDL Accuracy:

Metric	Fischer	Karpov
Main WDL Accuracy	97.847	98.604
Main PQ	95.313	97.143
Main Mean ES Loss	0.0215	0.0140
Main RMS ES Loss	0.0636	0.0402

Karpov’s games were, on average, cleaner. His move-level expected-score losses were lower.

But the relative edge over the opponent pool tells a different story:

Metric edge over opponents	Fischer	Karpov
WDL Accuracy difference	+0.788	+0.303
PQ difference	+0.779	+0.308
Mean ES Loss ratio	0.732	0.821
RMS ES Loss ratio	0.754	0.766
WDL Volatility ratio	0.734	0.831

Here Fischer is generally more dominant. His own absolute accuracy was lower, but the gap between him and his opponents was larger.

So the distinction is:

Karpov played cleaner chess. Fischer outdistanced his opposition more.

That may be the central sentence of the comparison.

3.5. Volatility and style

The volatility difference is one of the most style-revealing parts of the comparison.

Volatility metric	Fischer	Opponents vs Fischer	Karpov	Opponents vs Karpov
WDL Volatility	0.0225	0.0307	0.0149	0.0179
Total WDL Volatility	40.603	52.643	30.061	34.562

Karpov’s whole 1974 run was much less volatile than Fischer’s 1971–72 run. Karpov’s own WDL Volatility was about:

0.0149 / 0.0225 = 66.0% of Fischer’s

So Karpov’s run was roughly one-third less volatile.

This does not necessarily mean Karpov was “better.” It means the character of the games was different.

Fischer’s volatility profile

Fischer’s run contains more force, swing, and decisive pressure. His opponents had much higher volatility than he did, which implies that Fischer often pushed them into positions where their expected-score situation fluctuated more heavily.

Fischer’s style in this metric language:

Higher dominance
Higher conversion
Larger opponent destabilization
Larger score extraction
Less “clean” than Karpov in absolute terms, but more destructive in relative terms

Karpov’s volatility profile

Karpov’s run is lower-volatility on both sides. Even his opponents’ volatility against him was much lower than Fischer’s own volatility.

Karpov’s style in this metric language:

Lower leakage
Lower RMS loss
Lower error concentration
Lower volatility
Smaller but more compressed advantages
More technical containment than explosion

So the stylistic contrast could be phrased as:

Fischer created high-pressure separation.
Karpov created low-volatility suffocation.

3.6. Error concentration: a major difference

Error Concentration is one of the most discriminating differences between the two runs.

Metric	Fischer	Opponents	Karpov	Opponents
Error Concentration	3.063	3.008	2.750	3.024
Difference	+0.055		−0.274

Lower is better here.

For Fischer, Error Concentration does not really explain the run. His opponents were even slightly better overall by this one metric. Fischer’s domination came more from expected-score margin, dominance, RMS loss, volatility, and conversion.

For Karpov, Error Concentration does explain the run. Karpov’s value is clearly lower than his opponents’, especially against Polugaevsky and Spassky. That fits the intuitive Karpov image: his errors were less damagingly clustered; his play was more evenly controlled.

So:

Player	Error profile
Fischer	Not necessarily fewer concentrated errors, but more powerful superiority in position/result conversion.
Karpov	Fewer concentrated errors, cleaner technical stability, less self-damage.

3.7. Consistency and SDs

Both players were more stable than their opponents, but in different ways.

WDL Accuracy SD

Run	Main SD	Opponent SD	SD ratio
Fischer	6.864	8.513	0.806
Karpov	5.156	6.305	0.818

Very similar relative stability. Both players’ WDL accuracy varied about 18–19% less than their opponents’.

But Karpov’s absolute SD was lower. This means Karpov’s run was cleaner and less variable overall, while Fischer’s had more turbulence but still less than his opponents.

PQ SD

Run	Main SD	Opponent SD	SD ratio
Fischer	2.382	2.582	0.922
Karpov	2.207	2.294	0.962

Fischer’s PQ stability advantage over opponents is larger, but Karpov’s absolute PQ SD is slightly lower.

Volatility SD

Run	Main SD	Opponent SD	SD ratio
Fischer	0.0685	0.0852	0.804
Karpov	0.0517	0.0632	0.817

Again, the pattern is almost identical relatively, but Karpov’s absolute values are lower.

Conclusion:
Both players were stabler than their opponents by similar proportions. But Fischer did it in a more volatile combat environment; Karpov did it in a cleaner, more controlled environment.

3.8. Match-run shape: Fischer expands, Karpov narrows

A very interesting structural difference:

Fischer’s route

Match	Score	WDL Acc. edge	Dominance	Conversion
Taimanov	6–0	+0.972	+1.062	+1.628
Larsen	6–0	+1.222	+1.241	+1.285
Petrosian	6.5–2.5	+0.624	+0.525	+1.162
Spassky	12.5–7.5	+0.334	+0.384	+1.119

Fischer’s route starts with two annihilations, then becomes more contested, but he never loses the metric edge.

Karpov’s route

Match	Score	WDL Acc. edge	Dominance	Conversion
Polugaevsky	5.5–2.5	+0.490	+0.440	+0.955
Spassky	7–4	+0.300	+0.352	+0.444
Korchnoi	12.5–11.5	+0.120	+0.151	+0.219

Karpov’s route progressively narrows until the Korchnoi match is nearly level.

So Fischer’s run has a more explosive historical profile: two perfect shutouts and then strong superiority over Petrosian and Spassky. Karpov’s run is more like a technical qualification path that becomes increasingly difficult, ending in a razor-thin win over Korchnoi.

3.9. Shared measuring-stick: Spassky 1972 vs Spassky 1974

Both players beat Spassky, which gives a useful, though imperfect, common reference.

Fischer–Spassky 1972

Metric	Fischer	Spassky
Score	12.5	7.5
Expected Score	11.381	8.619
Conversion	+1.119	−1.119
WDL Accuracy	98.060	97.726
PQ	96.074	95.701
Dominance	+0.384	−0.384
Mean ES Loss	0.0194	0.0227
RMS ES Loss	0.0560	0.0661
Error Concentration	3.171	3.134
WDL Volatility	0.0207	0.0245
Game Accuracy	97.910	—
Mutual Accuracy	95.885	—

Karpov–Spassky 1974

Metric	Karpov	Spassky
Score	7.0	4.0
Expected Score	6.556	4.444
Conversion	+0.444	−0.444
WDL Accuracy	98.575	98.275
PQ	97.145	96.799
Dominance	+0.352	−0.352
Mean ES Loss	0.0142	0.0172
RMS ES Loss	0.0444	0.0545
Error Concentration	2.962	3.112
WDL Volatility	0.0154	0.0182
Game Accuracy	98.471	—
Mutual Accuracy	96.971	—

What changed in Spassky?

Spassky’s 1974 numbers against Karpov are cleaner than his 1972 numbers against Fischer:

Spassky metric	vs Fischer 1972	vs Karpov 1974	Change
WDL Accuracy	97.726	98.275	+0.549
PQ	95.701	96.799	+1.099
Mean ES Loss	0.0227	0.0172	Better
RMS ES Loss	0.0661	0.0545	Better
WDL Volatility	0.0245	0.0182	Better
Error Concentration	3.134	3.112	Nearly same, tiny improvement
Mutual Accuracy of match	95.885	96.971	Higher-quality match

This suggests that Spassky’s play in the 1974 Candidates match was, by these metrics, cleaner and lower-volatility than in the 1972 title match.

But we should be cautious. There are at least three possible explanations:

Spassky may have played cleaner in 1974.
Karpov’s lower-volatility style may have produced cleaner games from both sides.
Fischer’s pressure may have created more destabilization, forcing Spassky into more volatile and costly decisions.

The third possibility is important. Spassky’s lower accuracy against Fischer does not automatically mean he was “worse” in 1972. It may mean Fischer created a more difficult, sharper, or more destabilizing problem-set.

Fischer vs Karpov through Spassky

Against Spassky:

Metric edge over Spassky	Fischer 1972	Karpov 1974
Score margin	+5.0 over 20 games	+3.0 over 11 games
Score %	62.5%	63.6%
Expected-score margin	+2.762	+2.112
Expected-score %	56.9%	59.6%
Conversion	+1.119	+0.444
WDL Accuracy edge	+0.334	+0.300
PQ edge	+0.373	+0.346
Dominance difference	+0.768	+0.703
Mean ES Loss ratio	0.853	0.826
RMS ES Loss ratio	0.847	0.814
Volatility ratio	0.843	0.847

This is strikingly close. Against the shared opponent, Fischer and Karpov had very similar relative edges.

Karpov’s Spassky match was cleaner in absolute terms, but Fischer’s result came in a longer world-title match and included a larger conversion surplus. Karpov’s expected-score percentage against Spassky was actually slightly better, but the sample was shorter.

A fair reading:

Against Spassky specifically, the metrics do not show a huge gap between Fischer and Karpov.
Fischer’s edge is more in conversion and historical match severity; Karpov’s edge is in cleaner loss metrics.

3.10. If Fischer and Karpov had played in 1975, who was statistically favored?

This must be cautious, because the two runs are not identical conditions:

Fischer’s data is from 1971–72.
Karpov’s data is from 1974.
Opponent pools differ.
Fischer’s later inactivity before 1975 is not captured directly by the run metrics.
Karpov was improving rapidly.
Style matchups matter.

But using only the uploaded WDL run statistics, I would estimate:

Metric-based favorite: Fischer, narrowly to moderately

Why?

Fischer’s advantages:

Category	Fischer edge
Score domination	Much larger
Expected-score margin	Much larger
Dominance	Much larger
Conversion	Much larger
Relative WDL Accuracy edge over opponents	Larger
Relative PQ edge over opponents	Larger
RAP separation	Larger
Game-by-game metric superiority	Stronger: 29/41 or 30/41 vs Karpov’s 25–28/43

Karpov’s advantages:

Category	Karpov edge
Absolute WDL Accuracy	Higher
Absolute PQ	Higher
Mean ES Loss	Lower
RMS ES Loss	Lower
Volatility	Lower
Error Concentration	Better
Technical cleanliness	Higher
Match stability	Very strong

So the question becomes:

Do we prefer Fischer’s larger relative domination, or Karpov’s cleaner absolute technical play?

For predicting a direct match, I would weigh relative domination, expected-score margin, dominance, conversion, and RAP more heavily than raw absolute accuracy, because raw accuracy is affected by game type and opponent-induced complexity.

Therefore, using these metrics alone:

Fischer should be considered the statistical favorite, but not by a landslide.

Rough match estimate

If mapped onto a long match, I would estimate something like:

Scenario	Estimated Fischer score
Fischer clearly retains 1971–72 form	55–60%
Karpov’s 1974 low-volatility style neutralizes Fischer	50–53% Fischer
Fischer’s inactivity/instability matters strongly	Karpov 52–56%

Purely from the run metrics, ignoring inactivity, I would place Fischer around:

Fischer 55% – Karpov 45%

In a 24-game match, that corresponds roughly to:

Fischer 13–11 Karpov, perhaps 12.5–11.5 if Karpov’s control style successfully dampened Fischer’s volatility.

But this is not a blowout forecast. The Karpov–Korchnoi match shows that Karpov’s edge could become extremely small against a maximally resistant opponent. The Fischer data suggests he was more dominant than Korchnoi-level opposition in 1971–72, but the time gap and stylistic clash make certainty impossible.

3.11. Style matchup: why Fischer–Karpov 1975 would have been fascinating

The metrics suggest a beautiful clash:

Fischer’s statistical weapons

Higher dominance
Huge conversion surplus
Greater opponent destabilization
Larger practical score extraction
Ability to turn small expected-score edges into full-point avalanches

Fischer’s danger to Karpov would be that he could create positions where Karpov’s low-volatility control was disrupted. If Fischer could raise the volatility of the match environment, his conversion and dominance metrics suggest he might begin to separate.

Karpov’s statistical weapons

Lower mean ES loss
Lower RMS ES loss
Lower volatility
Better error concentration
High technical accuracy
Ability to keep games clean and deny chaos

Karpov’s danger to Fischer would be that he could dampen Fischer’s forcing pressure. If Karpov kept the match in low-volatility technical channels, Fischer’s conversion edge might have fewer opportunities to operate.

So the match could be summarized as:

Fischer would try to increase separation.
Karpov would try to reduce fluctuation.

Or in your style-axis language:

Axis	Fischer	Karpov
Concrete vs abstract	More concrete-pressure dominant	More abstract-control dominant
Local vs global	Strong local forcing + global conversion	Global prophylaxis and long control
Reactive vs proactive	Highly proactive, forcing	Proactive but suppressive
Short-term vs long-term	Converts immediate pressure sharply	Accumulates long-term technical pressure
Tangibility vs vision	More tangible domination	More invisible positional restriction

3.12. Roughest differences

The most discriminative differences are:

3.12.1. Fischer’s result dominance is far larger

Fischer: 31–10
Karpov: 25–18

Fischer’s run is historically violent in score terms; Karpov’s is successful but much more contested.

3.12.2. Karpov’s technical cleanliness is higher

Karpov’s WDL Accuracy, PQ, Mean ES Loss, RMS ES Loss, and Volatility are all better in absolute terms.

3.12.3. Fischer’s relative separation is higher

Fischer’s opponent pool was farther behind him by accuracy, PQ, dominance, expected score, conversion, and RAP.

3.12.4. Karpov’s run is lower-volatility

Karpov’s games look more controlled, less destabilized, and less swingy.

3.12.5. Fischer’s conversion is historically extreme

Fischer’s +5.194 conversion is enormous. Karpov’s +1.617 is strong, but not comparable.

3.13. Closest similarities

The closest similarities are also important:

3.13.1. Both were more accurate than their opponents

Fischer +0.788 WDL Accuracy.
Karpov +0.303 WDL Accuracy.

3.13.2. Both had lower expected-score loss

Fischer’s Mean ES Loss ratio: 0.732.
Karpov’s Mean ES Loss ratio: 0.821.

3.13.3. Both had lower volatility than their opponents

Fischer Volatility ratio: 0.734.
Karpov Volatility ratio: 0.831.

3.13.4. Both scored above expectation

Fischer +5.194 conversion.
Karpov +1.617 conversion.

3.13.5. Both beat Spassky by similar relative WDL margins

Against Spassky, their WDL Accuracy and PQ edges are surprisingly close:

Edge over Spassky	Fischer	Karpov
WDL Accuracy edge	+0.334	+0.300
PQ edge	+0.373	+0.346
Dominance difference	+0.768	+0.703

This shared-opponent comparison is one of the strongest arguments that Karpov was already operating near the same elite zone as Fischer, even if Fischer’s full run was more dominant.

3.14. Final conclusion

The numbers suggest two different kinds of greatness.

Fischer 1971–72 was the greater domination run. His opponents were not merely beaten; they were separated from him by expected score, dominance, volatility, conversion, and final score. His run looks like an apex predator moving through the Candidates and then the World Championship: not always the cleanest in absolute engine-loss terms, but devastating in competitive effect.

Karpov 1974 was the cleaner technical run. His games were lower-loss, lower-volatility, and more controlled. He did not separate from the field as explosively as Fischer, but his statistical profile already shows the mature Karpov signature: small advantages, stability, reduced opponent counterplay, and excellent conversion of limited margins.

For the cancelled 1975 match, the fairest metric-based prediction is:

Fischer would be the narrow-to-moderate statistical favorite if he retained his 1971–72 form.
But Karpov’s lower-volatility, lower-loss profile makes him exactly the kind of opponent who could have reduced Fischer’s usual separation.

My best metric-based estimate:

Fischer 55% – Karpov 45%, with a plausible match score around 13–11 or 12.5–11.5 for Fischer in a 24-game format.

But if Fischer’s inactivity had reduced his sharpness, or if Karpov had successfully imposed his low-volatility positional regime, the balance could easily flip.

So the final article-style thesis could be:

Fischer’s road measured higher in domination; Karpov’s road measured higher in control.
Fischer looked more likely to break opponents. Karpov looked harder to break.
The cancelled 1975 match would likely have been decided by which force won: Fischer’s separation power, or Karpov’s volatility suppression.