Playing Strength

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Playing Strength, (Performance, Skill Level) of a chess player, or chess playing entity, program or engine, reflects the ability to win against other players, given by a number or other element of an ordered set such as an Elo number.

The ability to solve test-positions, that is, finding the specified, likely one and only best move, might be an indicator for various particular engine skills, but does not necessarily correlate with playing strength. In his Parallelism and Selectivity in Game Tree Search lecture, Tord Romstad introduced the Worst Moves Observation (WMO), which states the practical playing strength is not primarily determined by the quality of the players best moves nor average moves, but by the quality of the players worst moves.

=Measuring= A statistical valid method to measure playing strength within a defined confidence interval is to play an appropriate huge number of games with both sides versus a wide range of different opponents with symmetric time constraints, and to apply match statistics. Performance isn't measured absolutely; it is inferred from wins, losses, and draws against other players or engines. Players' rating depend on the ratings of their opponents, and the results scored against them. While relative playing strength of chess engines is not strictly transmissive over various time controls, the number of games played is more relevant than their duration, the todays de facto standard in measuring playing strength is parallel playing fast chess with (ultra) short time control, such as blitz, bullet or even lightning chess, as for instance used in the Fishtest framework of Stockfish.

=Strength= The strength of a chess program depends on many things, the quality and efficiency of the algorithms involved to determine the best move of a position, the balance of the so called search versus knowledge tradeoff to evaluate or compare leaf nodes of a search tree, how to shape that tree and to propagate a score up to the root, and time management, that is how to allocate time for searching a move under time control requirements. Time used is roughly proportional to the number of visited nodes of the common depth-first search inside an iterative deepening frame, which grows exponentially by its effective branching factor raised to the power of search depth. Playing strength might be improved over the (playing) time due to learning algorithms.

=See also=
 * Automated Tuning
 * Depth
 * Depth | Diminishing Returns
 * Engine Rating Lists
 * Engine Testing
 * Knowledge
 * Learning
 * Match Statistics
 * Opponent Model Search
 * Pawn Advantage, Win Percentage, and Elo
 * Pondering
 * Search Pathology
 * Search with Random Leaf Values
 * Test-Positions
 * Time Management
 * Who is the Master?

=Publications=

1970 ...

 * 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
 * Peter W. Frey (ed.) (1977). Chess Skill in Man and Machine. Springer, New York, N.Y. 2nd ed. 1983
 * 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
 * Don Beal (1978). Handicapping Computer Chess Programs. ICCA Newsletter, Vol. 1, No. 2
 * 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
 * Editor (1979). Chess Ratings from Problem Solving. Personal Computing, Vol. 3, No. 12, pp. 77 » CAPS II, Chess Problems, Compositions and Studies

1980 ...

 * Floyd R. Kirk (1980). Bloss Flunks Test. Personal Computing, Vol. 4, No. 8, pp. 72 » Ratliff, Bloss
 * David E. Welsh (1980). Rating Boris 2.5. Personal Computing, Vol. 4, No. 10, pp. 78 » Boris 2.5
 * W. W. Foster (1980). Six Test Problems for Chess Playing Machines. Personal Computing, Vol. 4, No. 10, pp. 81
 * David E. Welsh (1980). Rating Boris 2.5 - Part II. Personal Computing, Vol. 4, No. 11, pp. 87
 * Ken Thompson (1982). Computer Chess Strength. Advances in Computer Chess 3
 * Ivan Bratko, Danny Kopec (1982). A Test for Comparison of Human and Computer Performance in Chess. Advances in Computer Chess 3
 * Danny Kopec, Ivan Bratko (1982). The Bratko-Kopec experiment: a comparison of human and computer performance in chess. Advances in Computer Chess 3
 * Danny Kopec, Enrique Irazoqui, Ivan Bratko (1983). The Updated Bratko-Kopec Test. Computer Chess Digest Annual pp. 45-63
 * Alexander Szabo (1984). Computer-Chess Tactics and Strategy. M.Sc. Thesis, University of British Columbia, pdf
 * Göran Grottling (1984). The Truth about their Strength. ICCA Journal, Vol. 7, No. 4
 * Göran Grottling (1985). How to Bracket the Truth in Playing Strength. ICCA Journal, Vol. 8, No. 1
 * Monroe Newborn (1985). A Hypothesis Concerning the Strength of Chess Programs. ICCA Journal, Vol. 8, No. 4
 * Peter W. Frey (1986). Algorithmic Strategies for Improving the Performance of Game-Playing Programs. In Doyne Farmer, Alan Lapedes, Norman Packard, Burton Wendroff (Ed.) (1986). Evolution, Games and Learning: Models for Adaptation in Machines and Nature. Proceedings of the Fifth Annual International Conference of the Center, Elsevier
 * 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
 * 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
 * Luis Antunes, Luis Moniz, Carlos Azevedo (1993). Rb+: the Dynamic Estimation of the Opponent's Strength. INESC-ID, Lisbon, Portugal » RB
 * Robert Hyatt, Monroe Newborn (1997). CRAFTY Goes Deep. ICCA Journal, Vol. 20, No. 2 » Crafty

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 (2002). Reference Fallible Endgame Play. 7th Computer Olympiad Workshop, pdf
 * 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

2005 ...

 * Jan Renze Steenhuisen (2005). New Results in Deep-Search Behaviour. ICGA Journal, Vol. 28, No. 4, pdf
 * Matej Guid, Ivan Bratko (2006). Computer Analysis of World Chess Champions. ICGA Journal, Vol. 29, No. 2, pdf
 * Matej Guid, Ivan Bratko (2006). Computer Analysis of Chess Champions. CG 2006
 * Matej Guid, Ivan Bratko (2007). Factors affecting diminishing returns for searching deeper. CGW 2007 » Crafty, Rybka, Shredder, Diminishing Returns
 * 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
 * Matej Guid, Ivan Bratko (2007). Factors affecting diminishing returns for searching deeper. ICGA Journal, Vol. 30, No. 2, pdf
 * Matej Guid, Aritz Pérez, Ivan Bratko (2007). How trustworthy is Crafty's analysis of world chess champions? CGW 2007 » Crafty
 * Matej Guid, Aritz Pérez, Ivan Bratko (2008). How trustworthy is Crafty's analysis of world chess champions? ICGA Journal, Vol. 31, No. 3, pdf
 * Rémi Coulom (2008). Whole-History Rating: A Bayesian Rating System for Players of Time-Varying Strength. CG 2008
 * Giuseppe Di Fatta, Guy McCrossan Haworth, Kenneth W. Regan (2009). Skill Rating by Bayesian Inference. CIDM 2009, pdf
 * 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
 * Matej Guid (2010). Search and Knowledge for Human and Machine Problem Solving. Ph.D. thesis, University of Ljubljana
 * Kenneth W. Regan, Guy Haworth (2011). Intrinsic Chess Ratings. AAAI 2011, pdf, slides as pdf
 * 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. Physics and Society, arXiv:1103.1530v2
 * Matej Guid, Ivan Bratko (2011). Using Heuristic-Search Based Engines for Estimating Human Skill at Chess. ICGA Journal, Vol. 34, No. 2
 * Rémi Coulom (2012). Paired Comparisons with Ties: Modeling Game Outcomes in Chess. pdf preprint
 * Diogo R. Ferreira (2012). Determining the Strength of Chess Players based on actual Play. ICGA Journal, Vol. 35, No. 1
 * Diogo R. Ferreira (2013). The Impact of the Search Depth on Chess Playing Strength. ICGA Journal, Vol. 36, No. 2
 * Yoshikuni Sato, Makoto Miwa, Shogo Takeuchi, Daisuke Takahashi (2013). Optimizing Objective Function Parameters for Strength in Computer Game-Playing. AAAI 2013
 * Erik Varend (2014). Quality of play in chess and methods for measuring. pdf

2015 ...

 * Shogo Takeuchi, Tomoyuki Kaneko (2015). Estimating Ratings of Computer Players by the Evaluation Scores and Principal Variations in Shogi. ACIT-CSI
 * Guy Haworth, Tamal T. Biswas, Kenneth W. Regan (2015). A Comparative Review of Skill Assessment: Performance, Prediction and Profiling. Advances in Computer Games 14
 * Chu-Hsuan Hsueh, I-Chen Wu, Wen-Jie Tseng, Shi-Jim Yen, Jr-Chang Chen (2015). Strength Improvement and Analysis for an MCTS-Based Chinese Dark Chess Program. Advances in Computer Games 14
 * Jean-Marc Alliot (2017). Who is the Master? ICGA Journal, Vol. 39, No. 1, draft as pdf » Stockfish, Who is the Master?
 * Takafumi Nakamichi, Takeshi Ito (2018). Adjusting the evaluation function for weakening the competency level of a computer shogi program. ICGA Journal, Vol. 40, No. 1
 * Chu-Hsuan Hsueh, I-Chen Wu, Tsan-sheng Hsu, Jr-Chang Chen (2018). An investigation of strength analysis metrics for game-playing programs: A case study in Chinese dark chess. ICGA Journal, Vol. 40, No. 2 » Chinese Dark Chess

=Forum Posts=

1990 ...

 * Weakest Chess Program needed by Jones Maxime Murphy, rgc, November 12, 1991
 * Estimating Playing Strength of Computers (was: Re: Rating of Radio Shack 1650) by Peter Marko, rgc, September 15, 1992 » SciSys Explorer
 * Rating boundaries by Kevin Stokker, rgc, January 14, 1993
 * Cognitive Psychology and Chess Skill by John Rummel, rgc, June 07, 1994 » Cognition

1995 ...

 * Primitive Chess Program by David Ewart, rgc, June 09, 1995
 * Rating of perfect chess by an18...@anon.penet.fi, rgc, July 17, 1995
 * 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
 * ELO performance? by Stefan Meyer-Kahlen, CCC, May 22, 1999 » Match Statistics, Pawn Advantage, Win Percentage, and Elo
 * from "Scacco": No steps forward in Computer Chess in the Last years !? by Gianluigi Masciulli, CCC, September 18, 1999

2000 ...

 * Strength of the engine in chess programs by Rolf Tueschen, CCC, May 20, 2002
 * Value of playing different versions of a program against each other by Tom King, CCC, January 06, 2003 » Engine Testing
 * Rules of thumb for strength estimation? by Albert Bertilsson, CCC, April 25, 2003

2005 ...

 * bayeselo: new Elo-rating tool, applied to CCT7 by Rémi Coulom, CCC, February 13, 2005 » CCT7
 * a beat b,b beat c,c beat a question by Uri Blass, CCC, May 16, 2007
 * A thought about ratings by Dave Dyer, Computer Go Archive, December 10, 2007 » Go
 * Re: A thought about ratings by Don Dailey, Computer Go Archive, December 10, 2007
 * Re: A thought about ratings by Edward de Grijs, Computer Go Archive, December 10, 2007
 * Re: A thought about ratings by Don Dailey, Computer Go Archive, December 10, 2007

2008 2009
 * Glaurung Mac OS X: New GUI, now with adjustable strength. by Tord Romstad, CCC, May 18, 2008 » Glaurung
 * Artificial stupidity - making a program play badly by Tord Romstad, CCC, May 20, 2008
 * New testing thread by Robert Hyatt, CCC, August 07, 2008 » Engine Testing
 * Hardware vs Software by Charles Roberson, CCC, December 02, 2008
 * Speed/ELO relationship, and question to engine programmers by Russell Reagan, CCC, March 02, 2009
 * Elo Calcuation by Edmund Moshammer, CCC, April 19, 2009
 * different kinds of testing by Don Dailey, CCC, November 09, 2009

2010 ...
2011 2012 2013 2014
 * How to reduce playing strenght to play against humans by Fermin Serrano, CCC, March 05, 2010
 * Current skill command (Crafty) results by Robert Hyatt, CCC, July 21, 2010 » Crafty
 * harware vs software advances by Don Dailey, CCC, September 08, 2010
 * hardware advances - a different perspective by Robert Hyatt, CCC, September 09, 2010
 * old crafty vs new crafty on new hardware by Robert Hyatt, CCC, September 11, 2010
 * Crafty tests show that Software has advanced more by Don Dailey, CCC, September 12, 2010
 * Final results - Crafty - hardware vs software by Robert Hyatt, CCC, September 13, 2010
 * Deep Blue vs Rybka by Don Dailey, CCC, September 13, 2010 » Deep Blue, Rybka
 * Ply vs ELO by Andriy Dzyben, CCC, June 28, 2011
 * Increase in Elo ..Question For The Experts by Steve B, CCC, December 05, 2011
 * StockFish LS with LimitStrength feature by Alexander Schmidt, CCC, January 01, 2012 » Stockfish
 * Depth vs playing strength by David Whitten, CCC, January 09, 2012
 * Reducing Strength by Ted Wong, CCC, March 15, 2012
 * Human Elo ratings: averages and standard deviations by Jesús Muñoz, CCC, March 18, 2012
 * Elo versus speed by Peter Österlund, CCC, April 02, 2012
 * Rybka odds matches and the strength of engines by Kai Laskos, CCC, June 09, 2012 » Rybka
 * Score Inaccuracy: An Engine Weakening Algorithm by Erik Madsen, CCC, October 31, 2012 » MadChess
 * 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
 * Noise in ELO estimators: a quantitative approach by Marco Costalba, CCC, January 06, 2013
 * Scaling at 2x nodes (or doubling time control) by Kai Laskos, CCC, July 23, 2013 » Doubling TC, Diminishing Returns, Playing Strength, Houdini
 * 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
 * ELO rating and thinking time by Henk van den Belt, CCC, November 22, 2013
 * MadChess UCI_LimitStrength Algorithm by Erik Madsen, CCC, April 12, 2014 » MadChess, UCI
 * Playing strength development - increasing time control by Andreas Strangmüller, CCC, April 17, 2014
 * How many Elo points change for a doubling of uP time? by Jay Rodrequez, CCC, August 14, 2014
 * 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
 * Questions regarding rating systems of humans and engines by Erik Varend, CCC, December 06, 2014

2015 ...
2016
 * How to dumb down/weaken/humanize an engine algorithmically? by Dominik Klein, CCC, January 18, 2015
 * computing elo of multiple chess engines by Alexandru Mosoi, CCC, August 09, 2015
 * 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
 * ELO error margin by Fabio Gobbato, CCC, October 17, 2015
 * testing multiple versions & elo calculation by Folkert van Heusden, CCC, October 27, 2015
 * a direct comparison of FIDE and CCRL rating systems by Erik Varend, CCC, February 22, 2016 » FIDE, CCRL
 * Computer Chess Strength by John Fishburn, CCC, February 28, 2016
 * How much benefit from opening book? by John Fishburn, CCC, March 06, 2016 » Opening Book
 * Computer chess progress over say the last 20 years? by Martin Fierz, CCC, March 10, 2016
 * Computer chess progress over the last 20 years! by Martin Fierz, CCC, March 13, 2016
 * Computer Chess Progress: Stockfish 7 vs Ruffian 1.0.5 by Martin Fierz, CCC, March 17, 2016 » Stockfish, Ruffian
 * skill levels by Alexandru Mosoi, CCC, April 28, 2016
 * Strategies for weaker play levels by Evert Glebbeek, CCC, June 28, 2016
 * ELO inflation ha ha ha by Henk van den Belt, CCC, September 16, 2016 » Delphil, Stockfish, Match Statistics, TCEC Season 9
 * About expected scores and draw ratios by Jesús Muñoz, CCC, September 17, 2016

2017
 * Perfect play by Patrik Karlsson, CCC, September 28, 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
 * L3 cache, RAM and other performance factors by Nimzy, Rybka Forum, December 04, 2016 » Memory
 * Absolute ELO scale by Nicu Ionita, CCC, December 17, 2016
 * Diminishing returns and hyperthreading by Kai Laskos, CCC, December 27, 2016 » Diminishing Returns, Match Statistics, Thread
 * Progress in 30 years by four intervals of 7-8 years by Kai Laskos, CCC, January 19, 2017 » Match Statistics
 * Crafty Play By Elo ( Crafty v25.3) by Michael B, CCC, January 23, 2017 » Crafty
 * 6-men Syzygy from HDD and USB 3.0 by Kai Laskos, CCC, April 04, 2017 » Komodo, 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
 * RAM speed and engine strength by John Hartmann, CCC, May 03, 2017 » RAM
 * 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
 * Another attempt at comparing Evals ELO-wise by Kai Laskos, CCC, May 22, 2017 » Evaluation
 * Testing endgame strength by Álvaro Begué, CCC, June 21, 2017 » Endgame, Engine Testing, RuyDos
 * Invariance with time control of rating schemes by Kai Laskos, CCC, July 22, 2017
 * Ways to avoid "Draw Death" in Computer Chess by Kai Laskos, CCC, July 25, 2017
 * ELO measurements by Peter Österlund, CCC, August 06, 2017 » Lazy SMP, Parallel Search
 * an interesting study from Erik Varend by scandien, Hiarcs Forum, August 13, 2017
 * Wall and Regression by Srdja Matovic, CCC, August 31, 2017
 * Scaling from FGRL results with top 3 engines by Kai Laskos, CCC, September 26, 2017 » FGRL, Houdini, Komodo, Stockfish
 * "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?

2018
 * ELO progression measured by year by Ed Schroder, CCC, December 13, 2017
 * Is modern chess software lossless or lossy? by Meni Rosenfeld, CCC, January 10, 2018 » Selectivity
 * 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, 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

=External Links=

Chess Player

 * Adolf Anderssen - Playing strength and style from Wikipedia
 * Magnus Carlsen - Rating from Wikipedia
 * Garry Kasparov Chess ratings achievements from Wikipedia
 * Fred Reinfeld | Playing strength from Wikipedia
 * József Szén | Playing strength from Wikipedia

Chess Engines

 * Chess engine - Increasing strength from Wikipedia
 * ChessGenius - Playing strength from Wikipedia
 * Chess Tiger - Playing strength from Wikipedia
 * Komodo (chess) - Playing strength and style from Wikipedia
 * MChess Pro - Playing strength from Wikipedia

Analysis

 * Computers choose: who was the strongest player?, ChessBase News, October 30, 2006
 * Computer analysis of world champions by Søren Riis, ChessBase News, November 02, 2006
 * Chessanalysis homepage by Erik Varend

Rating Systems

 * Chess rating system from Wikipedia
 * Chessmetrics from Wikipedia
 * Elo rating system from Wikipedia
 * Elo Win Probability Calculator by François Labelle
 * Arpad Elo and the Elo Rating System by Dan Ross, ChessBase News, December 16, 2007
 * Kirr's Chess Engine Comparison KCEC - Draw rate » KCEC
 * 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, Match Statistics

Misc

 * Strength from Wikipedia
 * Strength of schedule from Wikipedia
 * go from strength to strength - Wiktionary
 * Spielstärke | Wikipedia.de (German)
 * Sports rating system from Wikipedia
 * Performance (disambiguation) from Wikipedia
 * performance - Wiktionary
 * Playing Strength and Gender at Sensei's Library
 * Ranking from Wikipedia
 * regression - Wiktionary
 * Skill from Wikipedia
 * Chicago - Feelin' Stronger Every Day (1973), YouTube Video

=References= Up one Level