# Match Statistics

**Home * Engine Testing * Match Statistics**

**Match Statistics**,

the statistics of chess tournaments and matches, that is a collection of chess games and the presentation, analysis, and interpretation of game related data, most common game results to determine the relative playing strength of chess playing entities, here with focus on chess engines. To apply match statistics, beside considering statistical population, it is conventional to hypothesize a statistical model describing a set of probability distributions.

## Contents

# Ratios / Operating Figures

Common tools, ratios and figures to illustrate a tournament outcome and provide a base for its interpretation.

## Number of games

The total number of games played by an engine in a tournament.

```
N = wins + draws + losses
```

## Score

The score is a representation of the tournament-outcome from the viewpoint of a certain engine.

```
score_difference = wins - losses
```

```
score = wins + draws/2
```

## Win & Draw Ratio

```
win_ratio = score/N
```

```
draw_ratio = draws/N
```

These two ratios depend on the strength difference between the competitors, the average strength level, the color and the drawishness of the opening book-line. Due to the second reason given, these ratios are very much influenced by the timecontrol, what is also confirmed by the published statistics of the testing orgnisations CCRL and CEGT, showing an increase of the draw rate at longer time controls. This correlation was also shown by Kirill Kryukov, who was analyzing statistics of his test-games ^{[2]} . The program playing white seems to be more supported by the additional level of strength. So, although one would expect with increasing draw rates the win ratio to approach 50%, in fact it is remaining about equal.

Timecontrol | Draw Ratio | Win Ratio (white) | Source |
---|---|---|---|

40/4 | 30.9% | 55.0% | CEGT |

40/20 | 35.6% | 54.6% | CEGT |

40/120 | 41.3% | 55.4% | CEGT |

40/120 (4cpu) | 45.2% | 55.9% | CEGT |

Timecontrol | Draw Ratio | Win Ratio (white) | Source |
---|---|---|---|

40/4 | 31.0% | 54.1% | CCRL |

40/40 | 37.2% | 54.6% | CCRL |

**Doubling Time Control**
As posted in October 2016 ^{[3]} , Andreas Strangmüller conducted an experiment with Komodo 9.3, time control doubling matches under Cutechess-cli, playing 3000 games with 1500 opening positions each, without pondering, learning, and tablebases, Intel i5-750 @ 3.5 GHz, 1 Core, 128 MB Hash ^{[4]} , see also Kai Laskos' 2013 results with Houdini 3 ^{[5]} and Diminishing Returns:

Time Control | 2 vs 1 |
20+0.2 10+0.1 |
40+0.4 20+0.2 |
80+0.8 40+0.4 |
160+1.6 80+0.8 |
320+3.2 160+1.6 |
640+6.4 320+3.2 |
1280+12.8 640+6.4 |
2560+25.6 1280+12.8 |
---|---|---|---|---|---|---|---|---|---|

Elo | 144 | 133 | 112 | 101 | 93 | 73 | 59 | 51 | |

Win | 44.97% | 41.27% | 36.67% | 32.67% | 30.47% | 25.17% | 21.77% | 18.97% | |

Draw | 49.20% | 54.00% | 57.93% | 63.03% | 65.33% | 70.47% | 73.17% | 76.63% | |

Loss | 5.83% | 4.73% | 5.40% | 4.30% | 4.20% | 4.37% | 5.07% | 4.40% |

## Elo-Rating & Win-Probability

*see Pawn Advantage, Win Percentage, and Elo*

```
Expected win_ratio, win_probability (E)
```

```
Elo Rating Difference (Δ) = Elo_Player1 - Elo_Player2
```

```
E = 1 / ( 1 + 10-Δ/400)
```

```
Δ = 400 log10(E / (1 - E))
```

Generalization of the Elo-Formula:
*win_probability of player i in a tournament with n players*

```
Ei = 10Eloi / (10Elo1 + 10Elo2 + ... + 10Elon-1 + 10Elon)
```

## Likelihood of Superiority

*See LOS Table*

The likelihood of superiority (LOS) denotes how likely it would be for two players of the same strength to reach a certain result - in other fields called a p-value, a measure of statistical significance of a departure from the null hypothesis ^{[6]}. Doing this analysis after the tournament one has to differentiate between the case where one knows that a certain engine is either stronger or equally strong (directional or one-tailed test) or the case where one has no information of whether the other engine is stronger or weaker (non-directional or two-tailed test). The latter due to the reduced information results in larger confidence intervals.

**Two-tailed Test**

Null- and alternative hypothesis:

H0 : Elo_Player1 = Elo_Player2 H1 : Elo_Player1 ≠ Elo_Player2

```
LOS = P(Score > score of 2 programs with equal strength)
```

The probability of the null hypothesis being true can be calculated given the tournament outcome. In other words, how likely would it be for two players of the same strength to reach a certain result. The LOS would then be the inverse, 1 - the resulting probability.

For this type of analysis the trinomial distribution, a generalization of the binomial distribution, is needed. Whilest the binomial distribution can only calculate the probability to reach a certain outcome with two possible events, the trinominal distribution can account for all three possible events (win, draw, loss).

The following functions gives the probability of a certain game outcome assuming both players were of equal strength:

```
win_probability = (1 - draw_ratio) / 2
```

```
P(wins,draws,losses) = N!/(wins! draws! losses!) win_probabilitywins draw_ratiodrwas win_probabilitylosses
```

This calculation becomes very inefficient for larger number of games. In this case the standard normal distribution can give a good approximation:

**N**(N/2, N(1-draw_ratio))

where N(1 - draw_ratio) is the sum of wins and losses:

**N**(N/2, wins + losses)

To calculate the LOS one needs the cumulative distribution function of the given normal distribution. However, as pointed out by Rémi Coulom, calculation can be done cleverly, and the normal approximation is not really required ^{[7]} . As further emphasized by Kai Laskos ^{[8]} and Rémi Coulom ^{[9]} ^{[10]} , draws do not count in LOS calculation and don't make a difference whether the game results were obtained when playing Black or White. It is a good approximation when the two players played the same number of games with each color:

LOS = ϕ((wins - losses)/√(wins + losses)) LOS = ½[1 + erf((wins - losses)/√(2wins + 2losses))]

^{[11]} ^{[12]} ^{[13]}

**One-tailed Test**

Null- and alternative hypothesis:

H0 : Elo_Player1 ≤ Elo_Player2 H1 : Elo_Player1 > Elo_Player2

**Sample Program**

A tiny C++11 program to compute Elo difference and LOS from W/L/D counts was given by Álvaro Begué ^{[14]} :

#include <cstdio> #include <cstdlib> #include <cmath> int main(int argc, char **argv) { if (argc != 4) { std::printf("Wrong number of arguments.\n\nUsage:%s <wins> <losses> <draws>\n", argv[0]); return 1; } int wins = std::atoi(argv[1]); int losses = std::atoi(argv[2]); int draws = std::atoi(argv[3]); double games = wins + losses + draws; std::printf("Number of games: %g\n", games); double winning_fraction = (wins + 0.5*draws) / games; std::printf("Winning fraction: %g\n", winning_fraction); double elo_difference = -std::log(1.0/winning_fraction-1.0)*400.0/std::log(10.0); std::printf("Elo difference: %+g\n", elo_difference); double los = .5 + .5 * std::erf((wins-losses)/std::sqrt(2.0*(wins+losses))); std::printf("LOS: %g\n", los); }

## Statistical Analysis

**The trinomial versus the 5-nomial model**

As indicated above a match between two engines is usually modeled as a sequence of independent trials taken from a trinomial distribution with probabilities (win_ratio,draw_ratio,loss_ratio). This model is appropriate for a match with randomly selected opening positions and randomly assigned colors (to maintain fairness). However one may show that under reasonable elo models the trinomial model is not correct in case games are played in pairs with reversed colors (as is commonly the case) and unbalanced opening positions are used.

This was also empirically observed by Kai Laskos ^{[15]} . He noted that the statistical predictions of the trinomial model do not match reality very well in the case of paired games. In particular he observed that for some data sets the variance of the match score as predicted by the trinomial model greatly exceeds the variance as calculated by the jackknife estimator. The jackknife estimator is a non-parametric estimator, so it does not depend on any particular statistical model. It appears the mismatch may even occur for balanced opening positions, an effect which can only be explained by the existence of correlations between paired games - something not considered by any elo model.

Over estimating the variance of the match score implies that derived quantities such as the number of games required to establish the superiority of one engine over another with a given level of significance are also over estimated. To obtain agreement between statistical predictions and actual measurements one may adopt the more general 5-nomial model. In the 5-nomial model the outcome of paired games is assumed to follow a 5-nomial distribution with probabilities

```
(p0, p1/2, p1, p3/2, p2)
```

These unknown probabilities may be estimated from the outcome frequencies of the paired games and then subsequently be used to compute an estimate for the variance of the match score. Summarizing: in the case of paired games the 5-nomial model handles the following effects correctly which the trinomial model does not:

- Unbalanced openings
- Correlations between paired games

For further discussion on the potential use of unbalanced opening positions in engine testing see the posting by Kai Laskos ^{[16]} .

## SPRT

The sequential probability ratio test (SPRT) is a specific sequential hypothesis test - a statistical analysis where the sample size is not fixed in advance - developed by Abraham Wald ^{[17]} . While originally developed for use in quality control studies in the realm of manufacturing, SPRT has been formulated for use in the computerized testing of human examinees as a termination criterion ^{[18]}. As mentioned by Arthur Guez in this 2015 Ph.D. thesis *Sample-based Search Methods for Bayes-Adaptive Planning* ^{[19]}, Alan Turing assisted by Jack Good used a similar sequential testing technique to help decipher enigma codes at Bletchley Park ^{[20]}. SPRT is applied in Stockfish testing to terminate self-testing series early if the result is likely outside a given elo-window ^{[21]} . In August 2016, Michel Van den Bergh posted following Python code in CCC to implement the SPRT a la Cutechess-cli or Fishtest: ^{[22]} ^{[23]}

from __future__ import division import math def LL(x): return 1/(1+10**(-x/400)) def LLR(W,D,L,elo0,elo1): """ This function computes the log likelihood ratio of H0:elo_diff=elo0 versus H1:elo_diff=elo1 under the logistic elo model expected_score=1/(1+10**(-elo_diff/400)). W/D/L are respectively the Win/Draw/Loss count. It is assumed that the outcomes of the games follow a trinomial distribution with probabilities (w,d,l). Technically this is not quite an SPRT but a so-called GSPRT as the full set of parameters (w,d,l) cannot be derived from elo_diff, only w+(1/2)d. For a description and properties of the GSPRT (which are very similar to those of the SPRT) see http://stat.columbia.edu/~jcliu/paper/GSPRT_SQA3.pdf This function uses the convenient approximation for log likelihood ratios derived here: http://hardy.uhasselt.be/Toga/GSPRT_approximation.pdf The previous link also discusses how to adapt the code to the 5-nomial model discussed above. """ # avoid division by zero if W==0 or D==0 or L==0: return 0.0 N=W+D+L w,d,l=W/N,D/N,L/N s=w+d/2 m2=w+d/4 var=m2-s**2 var_s=var/N s0=LL(elo0) s1=LL(elo1) return (s1-s0)*(2*s-s0-s1)/var_s/2.0 def SPRT(W,D,L,elo0,elo1,alpha,beta): """ This function sequentially tests the hypothesis H0:elo_diff=elo0 versus the hypothesis H1:elo_diff=elo1 for elo0<elo1. It should be called after each game until it returns either 'H0' or 'H1' in which case the test stops and the returned hypothesis is accepted. alpha is the probability that H1 is accepted while H0 is true (a false positive) and beta is the probability that H0 is accepted while H1 is true (a false negative). W/D/L are the current win/draw/loss counts, as before. """ LLR_=LLR(W,D,L,elo0,elo1) LA=math.log(beta/(1-alpha)) LB=math.log((1-beta)/alpha) if LLR_>LB: return 'H1' elif LLR_<LA: return 'H0' else: return ''

Beside the above SPRT implementation using pentanomial frequencies and a simulation tool in Python ^{[24]} ^{[25]}, Michel Van den Bergh wrote a much faster multi-threaded C version ^{[26]} ^{[27]}.

# Tournaments

# See also

- Automated Tuning
- Bishop versus Knight - Winning Percentages
- Depth | Diminishing Returns
- Draw
- Engine Similarity
- LOS Table
- Pawn Advantage, Win Percentage, and Elo
- Playing Strength
- Search Statistics
- The Technology Curve
- Time Controls
- Who is the Master?

# Publications

## 1920 ...

- L. L. Thurstone (
**1927**).*A law of comparative judgement*. Psychological Review, Vol. 34, No. 4^{[28]} - Ernst Zermelo (
**1929**).*Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung*. pdf (German)

## 1930 ...

- Alexander Aitken (
**1935**).*On Least Squares and Linear Combinations of Observations*. Proceedings of the Royal Society of Edinburgh

## 1940 ...

- Abraham Wald (
**1945**).*Sequential Tests of Statistical Hypotheses*. Annals of Mathematical Statistics, Vol. 16, No. 2, doi: 10.1214/aoms/1177731118 - Abraham Wald (
**1947**).*Sequential Analysis*. John Wiley and Sons, AbeBooks

## 1950 ...

- Frederick Mosteller (
**1951**).*Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations*. Psychometrika, Vol. 16, No. 1 - Ralph A. Bradley, Milton E. Terry (
**1952**).*Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons*. Biometrika, Vol. 39, Nos. 3/4

## 1960 ...

- William A. Glenn, Herbert A. David (
**1960**).*Ties in Paired-Comparison Experiments Using a Modified Thurstone-Mosteller Model*. Biometrics, Vol. 16, No. 1 - Florence Nightingale David (
**1962**).*Games, Gods & Gambling: A History of Probability and Statistical Ideas*. Dover Publications - P. V. Rao, L. L. Kupper (
**1967**).*Ties in Paired-Comparison Experiments: A Generalization of the Bradley-Terry Model*. Journal of the American Statistical Association, Vol. 62, No. 317

## 1970 ...

- Roger R. Davidson (
**1970**).*On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments*. Journal of the American Statistical Association, Vol. 64, No. 329 - F. Donald Bloss (
**1973**).*Rate your own Chess*. Van Nostrand Reinhold Inc. - Tony Marsland, Paul Rushton (
**1973**).*Mechanisms for Comparing Chess Programs.*ACM Annual Conference, pdf - James Gillogly (
**1978**).*Performance Analysis of the Technology Chess Program*. Ph.D. Thesis. Tech. Report CMU-CS-78-189, Carnegie Mellon University, CMU-CS-77 pdf » Tech - Arpad Elo (
**1978**).*The Rating of Chessplayers, Past and Present*. Arco Publications^{[29]} - David Cahlander (
**1979**).*Strength of a Chess Playing Computer*. ICCA Newsletter, Vol. 2, No. 1 - Jack Good (
**1979**).*On the Grading of Chess Players*. Personal Computing, Vol. 3, No. 3, pp. 47 - Gary L. Ratliff (
**1979**).*Practical Rating Program*. Personal Computing, Vol. 3, No. 9, pp. 62 » Bloss - Frieder Schwenkel (
**1979**).*Berating the ratings system*. Personal Computing, Vol. 3, No. 11, pp. 77 » Ratliff, Bloss - John Shaposka (
**1979**).*"JS" Takes the Bloss Test*. Personal Computing, Vol. 3, No. 12, pp. 75 » Ratliff, Bloss

## 1980 ...

- Floyd R. Kirk (
**1980**).*Bloss Flunks Test*. Personal Computing, Vol. 4, No. 8, pp. 72 » Ratliff, Bloss - John F. White (
**1981**).*Survey-Chess Games*. Your Computer, August/September 1981^{[30]} - Ken Thompson (
**1982**).*Computer Chess Strength*. Advances in Computer Chess 3 - David Siegmund (
**1985**).*Sequential Analysis. Tests and confidence intervals*. Springer - Hans Berliner, Gordon Goetsch, Murray Campbell, Carl Ebeling (
**1989**).*Measuring the Performance Potential of Chess Programs*, Advances in Computer Chess 5 - Eric Hallsworth (
**1989**).*Playing Levels*. Computer Chess News Sheet 23, pp 2, pdf hosted by Mike Watters

## 1990 ...

- Hans Berliner, Gordon Goetsch, Murray Campbell, Carl Ebeling (
**1990**).*Measuring the Performance Potential of Chess Programs.*Artificial Intelligence, Vol. 43, No. 1 - Hal Stern (
**1990**).*Are all Linear Paired Comparison Models Equivalent*. pdf - Eric Hallsworth (
**1990**).*Speed, Processors and Ratings*. Computer Chess News Sheet 25, pp 6, pdf hosted by Mike Watters - Hans Berliner, Danny Kopec, Ed Northam (
**1991**).*A taxonomy of concepts for evaluating chess strength: examples from two difficult categories*. Advances in Computer Chess 6, pdf - Steve Maughan (
**1992**).*Are You Sure It's Better?*Selective Search 40, pp. 21, pdf hosted by Mike Watters - Warren D. Smith (
**1993**).*Rating Systems for Gameplayers, and Learning*. ps - Mark E. Glickman (
**1993**).*Paired Comparison Models with Time Varying Parameters*. Ph.D. thesis, Harvard University, advisor Hal Stern, pdf - Mark E. Glickman (
**1995**).*A Comprehensive Guide To Chess Ratings*. pdf - Mark E. Glickman, Christopher Chabris (
**1996**).*Using Chess Ratings as Data in Psychological Research*. pdf - Robert Hyatt, Monroe Newborn (
**1997**).*CRAFTY Goes Deep*. ICCA Journal, Vol. 20, No. 2 » Crafty - Mark E. Glickman, Albyn C. Jones (
**1999**).*Rating the Chess Rating System*. pdf

## 2000 ...

- Ernst A. Heinz (
**2000**).*New Self-Play Results in Computer Chess*. CG 2000 - Ernst A. Heinz (
**2001**).*Self-play Experiments in Computer Chess Revisited.*Advances in Computer Games 9 - Ernst A. Heinz (
**2001**).*Modeling the “Go Deep” Behaviour of CRAFTY and DARK THOUGHT.*Advances in Computer Games 9 » Crafty, Dark Thought - Ernst A. Heinz (
**2001**).*Self-Play, Deep Search and Diminishing Returns.*ICGA Journal, Vol. 24, No. 2 - Guy Haworth (
**2002**).*Self-play: Statistical Significance*. 7th Computer Olympiad Workshop - Guy Haworth (
**2003**).*Self-Play: Statistical Significance*. ICGA Journal, Vol. 26, No. 2 - Ernst A. Heinz (
**2003**).*Follow-Up on Self-Play, Deep Search, and Diminishing Returns.*ICGA Journal, Vol. 26, No. 2 - David R. Hunter (
**2004**).*MM Algorithms for Generalized Bradley-Terry Models*. The Annals of Statistics, Vol. 32, No. 1, 384–406, pdf^{[31]}^{[32]}^{[33]}^{[34]}

## 2005 ...

- Jan Renze Steenhuisen (
**2005**).*New Results in Deep-Search Behaviour*. ICGA Journal, Vol. 28, No. 4, pdf - Mark Levene, Judit Bar-Ilan (
**2005**).*Comparing Move Choices of Chess Search Engines*. ICGA Journal, Vol. 28, No. 2, pdf » Fritz, Junior - Mark Levene, Judit Bar-Ilan (
**2006**).*Comparing Typical Opening Move Choices Made by Humans and Chess Engines*. arXiv:cs/0610060 - Mark Levene, Judit Bar-Ilan (
**2007**).*Comparing Typical Opening Move Choices Made by Humans and Chess Engines*. The Computer Journal, Vol. 50, No. 5 - Matej Guid, Ivan Bratko (
**2007**).*Factors affecting diminishing returns for searching deeper*. CGW 2007 » Crafty, Rybka, Shredder, Diminishing Returns - Jeff Rollason (
**2007**).*Statistical Minefields with Version Testing*. AI Factory, Winter 2007 » Engine Testing - Shogo Takeuchi, Tomoyuki Kaneko, Kazunori Yamaguchi, Satoru Kawai (
**2007**).*Visualization and Adjustment of Evaluation Functions Based on Evaluation Values and Win Probability*. AAAI 2007, pdf - Rémi Coulom (
**2008**).*Whole-History Rating: A Bayesian Rating System for Players of Time-Varying Strength*. CG 2008, draft as pdf - Giuseppe Di Fatta, Guy McCrossan Haworth, Kenneth W. Regan (
**2009**).*Skill Rating by Bayesian Inference*. CIDM 2009, pdf^{[35]} - Guy McCrossan Haworth, Kenneth W. Regan, Giuseppe Di Fatta (
**2009**).*Performance and Prediction: Bayesian Modelling of Fallible Choice in Chess*. Advances in Computer Games 12, pdf

## 2010 ...

- Diogo R. Ferreira (
**2010**).*Predicting the Outcome of Chess Games based on Historical Data*. IST - Technical University of Lisbon^{[36]} - Kenneth W. Regan, Guy Haworth (
**2011**).*Intrinsic Chess Ratings*. AAAI 2011, pdf, slides as pdf^{[37]} - Kenneth W. Regan, Bartłomiej Macieja, Guy McCrossan Haworth (
**2011**).*Understanding Distributions of Chess Performances*. Advances in Computer Games 13, pdf - Trevor Fenner, Mark Levene, George Loizou (
**2011**).*A Discrete Evolutionary Model for Chess Players' Ratings*. arXiv:1103.1530 - Diogo R. Ferreira (
**2012**).*Determining the Strength of Chess Players based on actual Play*. ICGA Journal, Vol. 35, No. 1 - Daniel Shawul, Rémi Coulom (
**2013**).*Paired Comparisons with Ties: Modeling Game Outcomes in Chess*.^{[38]}^{[39]} - Diogo R. Ferreira (
**2013**).*The Impact of the Search Depth on Chess Playing Strength*. ICGA Journal, Vol. 36, No. 2 » Depth, Diminishing Returns, Playing Strength, Houdini^{[40]} - Miguel A. Ballicora (
**2014**).*ORDO v0.9.6 Ratings for chess and other games*. September 2014, pdf » Ordo^{[41]} - Don Dailey, Adam Hair, Mark Watkins (
**2014**).*Move Similarity Analysis in Chess Programs*. Entertainment Computing, Vol. 5, No. 3, preprint as pdf^{[42]} - Kenneth W. Regan, Tamal T. Biswas, Jason Zhou (
**2014**).*Human and Computer Preferences at Chess*. pdf - Erik Varend (
**2014**).*Quality of play in chess and methods for measuring*. pdf^{[43]}^{[44]} - Xiaoou Li, Jingchen Liu, Zhiliang Ying (
**2014**).*Generalized Sequential Probability Ratio Test for Separate Families of Hypotheses*. Sequential Analysis, Vol. 33, No. 4, pdf

## 2015 ...

- Tamal T. Biswas, Kenneth W. Regan (
**2015**).*Quantifying Depth and Complexity of Thinking and Knowledge*. ICAART 2015, pdf - Tamal T. Biswas, Kenneth W. Regan (
**2015**).*Measuring Level-K Reasoning, Satisficing, and Human Error in Game-Play Data*. IEEE ICMLA 2015, pdf preprint - Guy Haworth, Tamal T. Biswas, Kenneth W. Regan (
**2015**).*A Comparative Review of Skill Assessment: Performance, Prediction and Profiling*. Advances in Computer Games 14 - Shogo Takeuchi, Tomoyuki Kaneko (
**2015**).*Estimating Ratings of Computer Players by the Evaluation Scores and Principal Variations in Shogi*. ACIT-CSI - Jean-Marc Alliot (
**2017**).*Who is the Master*? ICGA Journal, Vol. 39, No. 1, draft as pdf » Stockfish, Who is the Master? - Ilan Adler, Yang Cao, Richard Karp, Erol A. Peköz, Sheldon M. Ross (
**2017**).*Random Knockout Tournaments*. Operations Research, Vol. 65, No. 6, arXiv:1612.04448 - Michel Van den Bergh (
**2017**).*A Practical Introduction to the GSPRT*. pdf - Leszek Szczecinski, Aymen Djebbi (
**2019**).*Understanding and Pushing the Limits of the Elo Rating Algorithm*. arXiv:1910.06081^{[45]} - Michel Van den Bergh (
**2019**).*The Generalized Maximum Likelihood Ratio for the Expectation Value of a Distribution*. pdf

# Forum & Blog Postings

## 1996 ...

- Theoretical chess rating question... by Cyber Linguist, rgcc, April 17, 1996
- Statistical validity of medium-length match results by Bruce Moreland, rgcc, February 15, 1997
- Small number statistics and small differences by Daniel Homan, CCC, August 14, 1998
- Re: Waltzing Matilda (was: statistics, 10 events tell us what ? by Daniel Homan, CCC, August 17, 1998
- ELO performance? by Stefan Meyer-Kahlen, CCC, May 22, 1999 » Pawn Advantage, Win Percentage, and Elo, Playing Strength

## 2000 ...

- Some thougths about statistics by Martin Schubert, CCC, February 24, 2000
- Who is better? Some statistics... by Gian-Carlo Pascutto, CCC, June 11, 2001
- Simulating the result of a single game by random numbers by Christoph Fieberg, CCC, July 03, 2001
- Simulating the result of a single game by random numbers - Update! by Christoph Fieberg, CCC, July 10, 2001
- Simulating the result of a single game by random numbers - Update! by Christoph Fieberg, CCC, August 02, 2001
- ELO & statistics question by Gian-Carlo Pascutto, CCC, April 26, 2002
- Statistical significance of a match result by Rémi Coulom, CCC, November 23, 2002
- Value of playing different versions of a program against each other by Tom King, CCC, January 06, 2003
- A question about statistics... by Roger Brown, CCC, January 04, 2004
- New tool to estimate the statistical significance of match results by Rémi Coulom, CCC, July 17, 2004
- ELOStat algorithm ? by Rémi Coulom, Winboard Forum, December 10, 2004 » EloStat

## 2005 ...

- bayeselo: new Elo-rating tool, applied to CCT7 by Rémi Coulom, CCC, February 13, 2005 » CCT7
- table for detecting significant difference between two engines by Joseph Ciarrochi, CCC, February 03, 2006
^{[46]} - Observator bias or... by Alessandro Scotti, CCC, May 29, 2007
- a beat b,b beat c,c beat a question by Uri Blass, CCC, May 16, 2007
- how to do a proper statistical test by Rein Halbersma, CCC, September 19, 2007
- Comparing two version of the same engine by Fermin Serrano, CCC, October 26, 2008
- Debate: testing at fast time controls by Fermin Serrano, CCC, December 15, 2008
- Elo Calcuation by Edmund Moshammer, CCC, April 19, 2009
- Likelihood of superiority by Marco Costalba, CCC, November 15, 2009

## 2010 ...

- Engine Testing - Statistics by Edmund Moshammer, CCC, January 14, 2010

- Re: Engine Testing - Statistics by John Major, CCC, January 14, 2010

- Chess Statistics by Edmund Moshammer, CCC, June 17, 2010
- Do You really need 1000s of games for testing? by Jouni Uski, CCC, November 04, 2010
- GUI idea: Testing until certainty by Albert Silver, CCC, December 07, 2010
- SPRT and Engine testing by Adam Hair, CCC, December 13, 2010 » SPRT

**2011**

- Testing very small changes ( <= 5 ELO points of gain) by Fermin Serrano, CCC, April 08, 2011
- Pairwise Analysis of Chess Engine Move Selections by Adam Hair, CCC, April 17, 2011
- Ply vs ELO by Andriy Dzyben, CCC, June 28, 2011
- One billion random games by Steven Edwards, CCC, August 27, 2011
- Increase in Elo ..Question For The Experts by Steve B, CCC, December 05, 2011

**2012**

- Are all test the same? by Fermin Serrano, CCC, January 17, 2012
- Advantage for White; Bayeselo (to Rémi Coulom) by Edmund Moshammer, CCC, March 03, 2012
- Pairwise Analysis of Chess Engine Move Selections Revisited by Adam Hair, CCC, March 04, 2012
- Human Elo ratings: averages and standard deviations by Jesús Muñoz, CCC, March 18, 2012
^{[47]} - Elo uncertainties calculator by Jesús Muñoz, CCC, March 24, 2012
- Elo versus speed by Peter Österlund, CCC, April 02, 2012
- Pawn Advantage, Win Percentage, and Elo by Adam Hair, CCC, April 15, 2012 » Pawn Advantage, Win Percentage, and Elo
- Elo Increase per Doubling by Adam Hair, CCC, May 07, 2012
- Rybka odds matches and the strength of engines by Kai Laskos, CCC, June 09, 2012 » Rybka
- A new way to compare chess programs by Larry Kaufman, CCC, June 21, 2012 » Komodo
- EloStat, Bayeselo and Ordo by Kai Laskos, CCC, June 24, 2012 » EloStat, Bayeselo, Ordo
- about error margins? by Fermin Serrano, CCC, August 01, 2012
- normal vs logistic curve for elo model by Daniel Shawul, CCC, August 02, 2012
- Derivation of bayeselo formula by Rémi Coulom, CCC, August 07, 2012
^{[48]} - Yet Another Testing Question by Brian Richardson, CCC, September 15, 2012
- margin of error by Larry Kaufman, CCC, September 16, 2012
- Average number of plies in {1-0, ½-½, 0-1} by Jesús Muñoz, CCC, September 21, 2012
- Another testing question by Larry Kaufman, CCC, September 23, 2012
- LOS calculation: Does the same result is always the same? by Marco Costalba, CCC, October 01, 2012
- LOS (again) by Ed Schröder, CCC, October 30, 2012
- Elo points gain from doubling time by Kai Laskos, CCC, December 10, 2012
- A word for casual testers by Don Dailey, CCC, December 25, 2012

**2013**

- A poor man's testing environment by Ed Schröder, CCC, January 04, 2013 » Engine Testing
- Noise in ELO estimators: a quantitative approach by Marco Costalba, CCC, January 06, 2013
- Updated Dendrogram by Kai Laskos, CCC, February 02, 2013
- Experiment: influence of colours at fixed depth by Jesús Muñoz, CCC, March 10, 2013
- LOS by BB+, OpenChess Forum, March 31, 2013
- Fishtest Distributed Testing Framework by Marco Costalba, CCC, May 01, 2013
- The influence of the length of openings by Kai Laskos, CCC, July 14, 2013
- Scaling at 2x nodes (or doubling time control) by Kai Laskos, CCC, July 23, 2013 » Doubling TC, Diminishing Returns, Playing Strength, Houdini
- Type I error in LOS based early stopping rule by Kai Laskos, CCC, August 06, 2013
^{[49]} - How much elo is pondering worth by Michel Van den Bergh, CCC, August 07, 2013 » Pondering
- Contempt and the ELO model by Michel Van den Bergh, CCC, September 05, 2013 » Contempt Factor
- 1 draw=1 win + 1 loss (always!) by Michel Van den Bergh, CCC, September 19, 2013
- SPRT and narrowing of (elo1 - elo0) difference by Jesús Muñoz, CCC, October 05, 2013 » SPRT
- sprt and margin of error by Larry Kaufman, CCC, October 15, 2013 » SPRT
- How (not) to use SPRT ? by BB+, OpenChess Forum, October 19, 2013
- Houdini, much weaker engines, and Arpad Elo by Kai Laskos, CCC, November 29, 2013 » Houdini, Pawn Advantage, Win Percentage, and Elo
^{[50]} - Testing on time control versus nodes | ply by Ed Schröder, CCC, December 04, 2013

**2014**

- Calculating the LOS (likelihood of superiority) from results by Robert Tournevisse, CCC, January 22, 2014
- LOS --> Draws are irrelevant by User923005, OpenChess Forum, January 24, 2014
- Empirically 1 win + 1 loss ~ 2 draws by Kai Laskos, CCC, June 24, 2014
- Ordo 0.9.6 by Miguel A. Ballicora, CCC, September 10, 2014 » Ordo
- Using the Stockfish position evaluation score to predict victory probability by unavoidablegrain, Tumblr, September 28, 2014 » Pawn Advantage, Win Percentage, and Elo, Stockfish
- Elo estimation using quasi-Monte Carlo integration by Branko Radovanovic, CCC, September 30, 2014
- SPRT question by Robert Hyatt, CCC, November 13, 2014 » SPRT
- Usage sprt / cutechess-cli by Michael Hoffmann, CCC, November 16, 2014 » Cutechess-cli, SPRT

## 2015 ...

- 2-SPRT by Michel Van den Bergh, CCC, January 28, 2015 » SPRT
- Script for computing SPRT probabilities by Michel Van den Bergh, CCC, April 05, 2015
- Maximum ELO gain per test game played? by Forrest Hoch, CCC, April 20, 2015
- Getting SPRT right by Alexandru Mosoi, CCC, April 22, 2015 » SPRT
- SPRT questions by Uri Blass, CCC, May 15, 2015 » SPRT
- Adam Hair's article on Pairwise comparison of engines by Charles Roberson, CCC, May 19, 2015
- computing elo of multiple chess engines by Alexandru Mosoi, CCC, August 09, 2015
- Some musings about search by Ed Schröder, CCC, August 14, 2015 » Automated Tuning, Search
- Bullet vs regular time control, say 40/4m CCRL/CEGT by Ed Schröder, CCC, August 29, 2015
- The SPRT without draw model, elo model or whatever... by Michel Van den Bergh, CCC, September 01, 2015 » SPRT

- Re: The SPRT without draw model, elo model or whatever.. by Michel Van den Bergh, CCC, August 18, 2016

- Name for elo without draws? by Marcel van Kervinck, CCC, September 02, 2015
- The future of chess and elo ratings by Larry Kaufman, CCC, September 20, 2015 » Opening Book
- Depth of Satisficing by Ken Regan, Gödel's Lost Letter and P=NP, October 06, 2015 » Depth, Match Statistics, Pawn Advantage, Win Percentage, and Elo, Stockfish, Komodo
^{[51]} - ELO error margin by Fabio Gobbato, CCC, October 17, 2015
- testing multiple versions & elo calculation by Folkert van Heusden, CCC, October 27, 2015
- A simple expression by Kai Laskos, CCC, December 09, 2015
- Counting 1 win + 1 loss as 2 draws by Kai Laskos, CCC, December 15, 2015

**2016**

- A Chess Firewall at Zero? by Ken Regan, Gödel's Lost Letter and P=NP, January 21, 2016
- Ordo 1.0.9 (new features for testers) by Miguel A. Ballicora, CCC, January 25, 2016
- Why the errorbar is wrong ... simple example! by Frank Quisinsky, CCC, February 23, 2016
- a direct comparison of FIDE and CCRL rating systems by Erik Varend, CCC, February 22, 2016 » FIDE, CCRL
- Some properties of the Type I error in p-value stopping rule by Kai Laskos, CCC, March 01, 2016
- A Visual Look at 2 Million Chess Games - Thinking Through the Party by Buğra Fırat, March 02, 2016
- Type I error for p-value stopping: Balanced and Unbalanced by Kai Laskos, CCC, June 16, 2016
- Empirically Logistic ELO model better suited than Gaussian by Kai Laskos, CCC, July 12, 2016
- Testing resolution and combining results by Daniel José Queraltó, CCC, July 28, 2016
- Error margins via resampling (jackknifing) by Kai Laskos, CCC, August 12, 2016
^{[52]}^{[53]} - Properties of unbalanced openings using Bayeselo model by Kai Laskos, CCC, August 27, 2016 » Opening Book
- ELO inflation ha ha ha by Henk van den Belt, CCC, September 16, 2016 » Delphil, Stockfish, Playing Strength, TCEC Season 9
^{[54]}

- About expected scores and draw ratios by Jesús Muñoz, CCC, September 17, 2016

- The scaling with time of opening books by Kai Laskos, CCC, September 23, 2016 » Opening Book
- Perfect play by Patrik Karlsson, CCC, September 28, 2016
- Stockfish underpromotes much more often than Komodo by Kai Laskos, CCC, October 05, 2016 » Komodo, Promotions, Stockfish
- Differences between top engines related to "style" by Kai Laskos, October 07, 2016
- SPRT when not used for self testing by Andrew Grant, CCC, October 21, 2016
- Doubling of time control by Andreas Strangmüller, CCC, October 21, 2016 » Doubling TC, Diminishing Returns, Playing Strength, Komodo
- Stockfish 8 - Double time control vs. 2 threads by Andreas Strangmüller, CCC, November 15, 2016 » Doubling TC, Diminishing Returns, Playing Strength, Stockfish
- When Data Serves Turkey by Ken Regan, Gödel's Lost Letter and P=NP, November 30, 2016
- Magnus and the Turkey Grinder by Ken Regan, Gödel's Lost Letter and P=NP, December 08, 2016 » Pawn Advantage, Win Percentage, and Elo
^{[55]} - Regan's conundrum by Carl Lumma, CCC, December 09, 2016
- Statistical Interpretation by Dennis Sceviour, CCC, December 10, 2016
- Absolute ELO scale by Nicu Ionita, CCC, December 17, 2016
- A question about SPRT by Andrew Grant, CCC, December 25, 2016 » SPRT
- Diminishing returns and hyperthreading by Kai Laskos, CCC, December 27, 2016 » Diminishing Returns, Playing Strength, Thread

**2017**

- Progress in 30 years by four intervals of 7-8 years by Kai Laskos, CCC, January 19, 2017 » Playing Strength
- sprt tourney manager by Richard Delorme, CCC, January 24, 2017 » Amoeba Tournament Manager, SPRT
- Binomial distribution for chess statistics by Lyudmil Antonov, CCC, March 03, 2017
- Higher than expected by me efficiency of Ponder ON by Kai Laskos, CCC, March 06, 2017 » Pondering
- What can be said about 1 - 0 score? by Kai Laskos, CCC, March 28, 2017
- 6-men Syzygy from HDD and USB 3.0 by Kai Laskos, CCC, April 04, 2017 » Komodo, Playing Strength, Syzygy Bases, USB 3.0
- Scaling of engines from FGRL rating list by Kai Laskos, CCC, April 07, 2017 » FGRL
- Low impact of opening phase in engine play? by Kai Laskos, CCC, April 18, 2017 » Opening
- How to simulate a game outcome given Elo difference? by Nicu Ionita, CCC, April 25, 2017
- Wilo rating properties from FGRL rating lists by Kai Laskos, CCC, May 01, 2017 » FGRL
- MATCH sanity by Ed Schroder, CCC, May 03, 2017 » Portable Game Notation

- Re: MATCH sanity by Salvatore Giannotti, CCC, May 03, 2017

- Symmetric multiprocessing (SMP) scaling - SF8 and K10.4 by Andreas Strangmüller, CCC, May 05, 2017 » Lazy SMP, Komodo, Stockfish
- Symmetric multiprocessing (SMP) scaling - K10.4 Contempt=0 by Andreas Strangmüller, CCC, May 11, 2017 » SMP, Komodo, Contempt Factor
- Symmetric multiprocessing (SMP) scaling - SF8 Contempt=10 by Andreas Strangmüller, CCC, May 13, 2017 » SMP, Stockfish, Contempt Factor
- Likelihood Of Success (LOS) in the real world by Kai Laskos, CCC, May 26, 2017
- Opening testing suites efficiency by Kai Laskos, CCC, June 21, 2017 » Engine Testing, Opening
- Testing A against B by playing a pool of others by Andrew Grant, CCC, June 24, 2017
- Engine testing & error margin ? by Mahmoud Uthman, CCC, July 05, 2017
- Invariance with time control of rating schemes by Kai Laskos, CCC, July 22, 2017
^{[56]} - Ways to avoid "Draw Death" in Computer Chess by Kai Laskos, CCC, July 25, 2017
- SMP NPS measurements by Peter Österlund, CCC, August 06, 2017 » Lazy SMP, Parallel Search, Nodes per Second

- ELO measurements by Peter Österlund, CCC, August 06, 2017 » Playing Strength

- What is a Match? by Henk van den Belt, CCC, September 01, 2017
- Scaling from FGRL results with top 3 engines by Kai Laskos, CCC, September 26, 2017 » FGRL, Houdini, Komodo, Stockfish
- Statistical interpretation of search and eval scores by J. Wesley Cleveland, CCC, November 18, 2017 » Pawn Advantage, Win Percentage, and Elo, Score
- "Intrinsic Chess Ratings" by Regan, Haworth -- seq by Kai Middleton, CCC, November 19, 2017

- Re: "Intrinsic Chess Ratings" by Regan, Haworth -- by Kenneth Regan, CCC, November 20, 2017 » Who is the Master?

- ELO progression measured by year by Ed Schroder, CCC, December 13, 2017

**2018**

- Wrong use of SPRT by Uri Blass, FishCooking, February 09, 2018 » Contempt Factor, SPRT
- Feed bayeselo with pure game results without PGN by Sergei S. Markoff, CCC, March 08, 2018
- Elo measurement of contempt in SF in self-play by Michel Van den Bergh, CCC, March 10, 2018 » Contempt, Playing Strength, Stockfish
- Time control envelope in top engines could be improved? by Kai Laskos, CCC, March 13, 2018 » Time Management
- LCZero: Progress and Scaling. Relation to CCRL Elo by Kai Laskos, CCC, March 28, 2018 » Leela Chess Zero
- Elostat Question by Michael Sherwin, CCC, March 30, 2018
- Issue with self play testing by Charles Roberson, CCC, May 18, 2018 » Engine Testing
- Why Lc0 eval (in cp) is asymmetric against AB engines? by Kai Laskos, CCC, July 25, 2018 » Asymmetric Evaluation, Leela Chess Zero, Pawn Advantage, Win Percentage, and Elo
- Are draws hard to predict? by Daniel Shawul, CCC, November 27, 2018 » Draw, Neural Networks
- testing consistency by Jon Dart, CCC, December 16, 2018
- Fixed nodes games and the pentanomial model by Michel Van den Bergh, CCC, December 29, 2018

**2019**

- Schizophrenic rating model for Leela by Kai Laskos, CCC, January 21, 2019 » Leela Chess Zero
- best way to determine elos of a group by Daniel Shawul, CCC, July 30, 2019

## 2020 ...

- Stockfish Reverts 5 Recent Patches by Deberger, CCC, February 01, 2020 » Stockfish

- Re: Stockfish Reverts 5 Recent Patches by Michel Van den Bergh, CCC, February 02, 2020

- repackaging of Fishtest's SPRT calculation code by Jon Dart, CCC, March 20, 2020 » SPRT
- Stockfish_dev is probably stronger than Sargon 1978 v1.00 by Kai Laskos, CCC, May 29, 2020 » Stockfish, Sargon
- Throwing out draws to calculate Elo by Dann Corbit, CCC, June 29, 2020
- Jackknife estimate of the variance of engine results for Arasan test suite by Kai Laskos, CCC, September 05, 2020 » Statistical Analysis

**2021**

- What K factor should be used if two players are in different K factor brackets? by Tamás Kuzmics, CCC, January 09, 2021
^{[57]}^{[58]} - Cutechess-cli and SPRT by Tom King, CCC, January 19, 2021 » Cutechess-cli, SPRT

# External Links

## Rating Systems

- Chessmetrics from Wikipedia
- Chess rating system from Wikipedia
- Elo rating system from Wikipedia
- Arpad Elo and the Elo Rating System by Dan Ross, ChessBase News, December 16, 2007
- Chess ratings - Elo versus the Rest of the World | Kaggle
- Elo Win Probability Calculator by François Labelle
- LOS Table by Joseph Ciarrochi from CEGT
^{[59]} - Kirr's Chess Engine Comparison KCEC - Draw rate » KCEC
- Chessanalysis homepage by Erik Varend
^{[60]} - Who is the Master? by Jean-Marc Alliot » Who is the Master?
- How Should Chess Players Be Rated? by Martin Koppe, CNRS News, April 25, 2017
- Ranking chess players according to the quality of their moves by Frederic Friedel, ChessBase News, April 27, 2017
- Rating List Stats by Ed Schroder » CCRL

## Tools

- BayesElo by Rémi Coulom builds tournament-statistics from a pgn-file
- Elostat by Frank Schubert builds tournament-statistics from a pgn-file
- Online Elo-Calculator by Pradu Kannan
- Ordo by Miguel A. Ballicora

## Statistics

- Statistics from Wikipedia
- Statistical assumption from Wikipedia
- Statistical inference from Wikipedia
- Probability theory from Wikipedia
- Probability from Wikipedia
- Probability density function from Wikipedia
- Likelihood function from Wikipedia
- p-value from Wikipedia
- Misunderstandings of p-values from Wikipedia
- Probability distribution from Wikipedia
- Binomial distribution from Wikipedia
- Cumulative distribution function from Wikipedia
- Multinomial distribution from Wikipedia
- Multivariate normal distribution from Wikipedia
- Normal distribution from Wikipedia
- Standard deviation from Wikipedia
- Standard error from Wikipedia
- Error bar from Wikipedia
- Margin of error from Wikipedia
- Confidence interval from Wikipedia
- Null hypothesis from Wikipedia
- Alternative hypothesis from Wikipedia
- Two-tailed test from Wikipedia
- Type I and type II errors from Wikipedia
- Type 1 Errors | Hypothesis testing with one sample | Khan Academy

## Hypothesis Testing

- Statistical hypothesis testing from Wikipedia
- Likelihood-ratio test from Wikipedia
- Score test from Wikipedia
- Wald test from Wikipedia

## Sequential Analysis

- Sequential analysis from Wikipedia
- Sequential probability ratio test from Wikipedia
- GitHub - vdbergh/pentanomial: SPRT for pentanomial frequencies and simulation tools by Michel Van den Bergh
- GitHub - vdbergh/simul: A multi-threaded pentanomial simulator by Michel Van den Bergh

## Data Visualization

- A Visual Look at 2 Million Chess Games - Thinking Through the Party by Buğra Fırat, March 02, 2016
^{[61]} - GitHub - ebemunk/chess-dataviz: chess visualization library written for d3.js by Buğra Fırat » JavaScript
- GitHub - ebemunk/pgnstats: parses PGN files and extracts statistics for chess games by Buğra Fırat » Go (Programming Language), Portable Game Notation

## Misc

- Björk - Possibly Maybe (1996), YouTube Video

# References

- ↑ Image based on Standard deviation diagram by Mwtoews, April 7, 2007 with R code given, CC BY 2.5, Wikimedia Commons, Normal distribution from Wikipedia
- ↑ Kirr's Chess Engine Comparison KCEC - Draw rate » KCEC
- ↑ Doubling of time control by Andreas Strangmüller, CCC, October 21, 2016
- ↑ K93-Doubling-TC.pdf
- ↑ Scaling at 2x nodes (or doubling time control) by Kai Laskos, CCC, July 23, 2013
- ↑ Re: Likelihood Of Success (LOS) in the real world by Álvaro Begué, CCC, May 26, 2017
- ↑ Re: Calculating the LOS (likelihood of superiority) from results by Rémi Coulom, CCC, January 23, 2014
- ↑ Re: Calculating the LOS (likelihood of superiority) from results by Kai Laskos, CCC, January 22, 2014
- ↑ Re: Likelihood of superiority by Rémi Coulom, CCC, November 15, 2009
- ↑ Re: Likelihood of superiority by Rémi Coulom, CCC, November 15, 2009
- ↑ Error function from Wikipedia
- ↑ The Open Group Base Specifications Issue 6IEEE Std 1003.1, 2004 Edition: erf
- ↑ erf(x) and math.h by user76293, Stack Overflow, March 10, 2009
- ↑ Re: Calculating the LOS (likelihood of superiority) from results by Álvaro Begué, CCC, January 22, 2014
- ↑ Error margins via resampling (jackknifing) by Kai Laskos, CCC, August 12, 2016
- ↑ Properties of unbalanced openings using Bayeselo model by Kai Laskos, CCC, August 27, 2016
- ↑ Abraham Wald (
**1945**).*Sequential Tests of Statistical Hypotheses*. Annals of Mathematical Statistics, Vol. 16, No. 2, doi: 10.1214/aoms/1177731118 - ↑ Sequential probability ratio test from Wikipedia
- ↑ Arthur Guez (
**2015**).*Sample-based Search Methods for Bayes-Adaptive Planning*. Ph.D. thesis, Gatsby Computational Neuroscience Unit, University College London, pdf - ↑ Jack Good (
**1979**).*Studies in the history of probability and statistics. XXXVII AM Turing’s statistical work in World War II*. Biometrika, Vol. 66, No. 2 - ↑ How (not) to use SPRT ? by BB+, OpenChess Forum, October 19, 2013
- ↑ Re: The SPRT without draw model, elo model or whatever.. by Michel Van den Bergh, CCC, August 18, 2016
- ↑ GSPRT approximation (pdf) by Michel Van den Bergh
- ↑ Re: Stockfish Reverts 5 Recent Patches by Michel Van den Bergh, CCC, February 02, 2020
- ↑ GitHub - vdbergh/pentanomial: SPRT for pentanomial frequencies and simulation tools by Michel Van den Bergh
- ↑ Re: Stockfish Reverts 5 Recent Patches by Michel Van den Bergh, CCC, February 02, 2020
- ↑ GitHub - vdbergh/simul: A multi-threaded pentanomial simulator by Michel Van den Bergh
- ↑ Law of comparative judgment - Wikipedia
- ↑ Elo's Book: The Rating of Chess Players by Sam Sloan
- ↑ The Master Game from Wikipedia
- ↑ Handwritten Notes on the 2004 David R. Hunter Paper 'MM Algorithms for Generalized Bradley-Terry Models' by Rémi Coulom
- ↑ Derivation of bayeselo formula by Rémi Coulom, CCC, August 07, 2012
- ↑ Mm algorithm from Wikipedia
- ↑ Pairwise comparison from Wikipedia
- ↑ Bayesian inference from Wikipedia
- ↑ How I did it: Diogo Ferreira on 4th place in Elo chess ratings competition | no free hunch
- ↑ "Intrinsic Chess Ratings" by Regan, Haworth -- seq by Kai Middleton, CCC, November 19, 2017
- ↑ Re: EloStat, Bayeselo and Ordo by Rémi Coulom, CCC, June 25, 2012
- ↑ Re: Understanding and Pushing the Limits of the Elo Rating Algorithm by Daniel Shawul, CCC, October 15, 2019
- ↑ Ply versus ELO by Greg, HIARCS Forum, May 30, 2020 » Diogo R. Ferreira - Impact of the Search Depth ...
- ↑ Ordo by Miguel A. Ballicora
- ↑ Pairwise Analysis of Chess Engine Move Selections by Adam Hair, CCC, April 17, 2011
- ↑ Questions regarding rating systems of humans and engines by Erik Varend, CCC, December 06, 2014
- ↑ chess statistics scientific article by Nuno Sousa, CCC, July 06, 2016
- ↑ Understanding and Pushing the Limits of the Elo Rating Algorithm by Michel Van den Bergh, CCC, October 15, 2019
- ↑ LOS Table by Joseph Ciarrochi from CEGT
- ↑ Arpad Elo and the Elo Rating System by Dan Ross, ChessBase News, December 16, 2007
- ↑ David R. Hunter (
**2004**).*MM Algorithms for Generalized Bradley-Terry Models*. The Annals of Statistics, Vol. 32, No. 1, 384–406, pdf - ↑ Type I and type II errors from Wikipedia
- ↑ Arpad Elo - Wikipedia
- ↑ Regan's latest: Depth of Satisficing by Carl Lumma, CCC, October 09, 2015
- ↑ Resampling (statistics) from Wikipedia
- ↑ Jackknife resampling from WIkipedia
- ↑ Delphil 3.3b2 (2334) - Stockfish 030916 (3228), TCEC Season 9 - Rapid, Round 11, September 16, 2016
- ↑ World Chess Championship 2016 from Wikipedia
- ↑ Normalized Elo (pdf) by Michel Van den Bergh
- ↑ Most accurate K-factor - Elo rating system from Wikipedia
- ↑ FIDE Chess Rating calculators: Chess Rating change calculator
- ↑ table for detecting significant difference between two engines by Joseph Ciarrochi, CCC, February 03, 2006
- ↑ an interesting study from Erik Varend by scandien, Hiarcs Forum, August 13, 2017
- ↑ A Visual Look at 2 Million Chess Games by Brahim Hamadicharef, CCC, November 02, 2017