Quantifying Subjective Advantage
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Stockfish and other engines are well known to quantitizate* position and move evaluation in the shape of a number. However, this number, counted in centipawns, only encompasses the situation of the board, and is insufficient to predict accurately the course of the game, or even to tell whether one side has an advantage. In this thread, I open a discussion about subjective advantage and how to put a centipawn number on them, thus allowing to bring much more accurate evaluation of a position and analysis of a game.
Defining subjective advantages
Subjective advantages include, but are not limited to :
Psychological advantage.
One could say that all subjective advantage is psychological, but in this precise context, psychological advantage only encompass what happens between the two players and the board, for example :
Time advantage
This one is pretty straightforward, it is how much time is left on the clock, eg. is one side in time trouble. However, it is one of the hardest to quantify, as it combines with the difficulty of the position.
Environmental advantage
In OTB, it can be the quality of the chair, the brightness of the lights, the quality of the pieces... and of course, it takes into account the respective sensibility of players to given factor.
In on-line chess, it can also be the comfort of the playing site, as well as the quality of the mouse, the quality of the connection, the quality of the platform, sneaky breadcrumbs lowering the performance of a laptop, et caetera.
Eruditional advantage
It still is up to debate whether 'eruditional' should be written with one or two n. For the sake of consistency with 'intimidational', 'deludational' and 'psychological' which have only one consonant before the 'al', I'll write 'eruditional'. But I guess the two orthographs would make sense, as, for example, 'annal' can be written with either one or two 'n's.
Eruditional advantage is the advantage which comes from chess knowledges, in general, whether it is about knowing the exact en passant and castling rules, the rules under which the game is played (FIDE/USCF/Chess.com), knowledge and preparation of openings, knowledge of endgame drills...
In theory it can be extended to any chess skill, even one that are not literally linked to erudition, but can't be summed up into a rating number as the analysis of eruditional advantage aims to provide a finer analysis taking into account the current position of the board.
An approach to quantification.
Introduction and Expected scoring
The main approach I have considered to quantify** subjective advantage is considering expected results of a game. For example, Stockfish gives to the Soller Gambit an objective evaluation of about 136 centipawns (depending on how deep she/it/he(in no intended specific order) digs) yet according to chess.com database, black scores 0.5 on average, meaning that the position is overall equal, meaning that in the Englund Soller Gambit, Black has, on average, a subjective advantage of 136 centipawns.
https://www.chess.com/explorer?moveList=d4+e5+dxe5+f6&ply=4&origPly=2&origMoves=d4+e5
Of course calculation of such advantage has to be adjusted according to rating difference, for example, still according to chess.com data base, White has usually a higher rating than black in the Soller Gambit, suggesting black's subjective advantage is bigger than 136 centipawns. Sadly, chess.com data base has very few soller gambit for which we know the ratings of the players, but ignoring theese and keeping the game for which we don't know the ratings and assuming they are about equal, Black score an astounding 0.625, suggesting that black subjective advantage is actually much higher than 136 centipawns.
https://www.chess.com/games/search?f=3255786
Rating vs centipawns
The natural way to quantify winning odds is to apply the formula for expected scores using elo ratings :
1/(1+10^((RB-RA)/400)), which gives the expected score of a player with rating RA against a player of rating RB.
https://en.wikipedia.org/wiki/Elo_rating_system#Theory
This gives the expected score prior to a match. We already see the flaw of such formula, as it does not takes into account subjective factors like environmental advantages, or matchup advantage, which can be deduced from psychological and eruditional advantages.
We can understand then that the right way to quantify subjective advantage that apply before the game or independently to the position of the board are to integrate in the shape of a rating alteration, giving an effective rating. This rating is exactly the a priori converse of the a posteriori performance rating and an attempt to predict it. It can be approximated with recent performance rating and with knowledge of performance rating and performance rating anomalies of past similar events, and corrected during a tournament given the current rating performance, and even maybe during a game given the strength of the moves played.
On the other hand, subjective advantages directly related to the position on the board are to be incorporated into the centipawn evaluation of the position.
Corrected Scoring Formula
The formula 1/(1+10^((RB-RA)/400)
can be rewritten :
QA/(QA+QB)
Where QA = 10^(RA/400) is a measure of the strength of player A. To fix this formula, my SWAG*** tells me that we need to change the expression of Q.
Q = S.T.R.E.P
Where T is the Theoretical strength of a player, based on his rating R : T = 10^(R/400).
E is the Environmental strength modification of a player, that is E = 10^(dR/400) where dR quantifies how environmental factors alters a player's performance, id est R+dR is the efficient rating of the player during a tournament, thus E.T is the strength a priori of a player (reguardless of his opponent, and thus of players specific preparation, who are accounted for in S).
Likewise P is the Psychological strength modification of a player, that is E = 10^(dR'/400), where dR quantifies how psychological factors alters a player's performance, id est R+dR + dR' is the efficient rating of the player during a specific game thus P.E.T is the strength of a player at a given point of the game. While E is generally constant during the course of a whole game, P usually varies a lot. Conveniently, P may also encompass factors like whether you score above or under the traditional expectation when facing a strong or a weak opposition, even when it is not psychologically grounded, as well as time trouble.
R (not the same R as rating) is the Realistic, objective strength of the board position of a player, expressed as follow : R = 10^(P/A) where P is the engine evaluation of the position in centipawn (or its opposite for black), and A is a number of the same order of magnitude that a winning advantage, id est between one or two pawns at GM level, and closer to 10 pawns at three digits level. The precise value of A and its dependencies.
S is the Subjective strength of the board position of a player, expressed as follow : R = 10^(SA/A) where A is the same A as in R, and SA is the subjective advantage of the board, so that P+SA is the accurate (as opposed to objective) evaluation of a position. In practice, as we will show in the next section, P+SA is much easier to estimate than SA separately, so SA is found by calculating P+SA, and is only useful for human analysis. As a consequence, R.S is usually calculated all at once.
accurate (as opposed to objective) position evaluation
The immense majority of chess engines works by :
For the purpose of evaluating winning odds in a match between humans, this algorithm can be fixed by running two instances, of corrected algorithm, one simulating player A, the other player B :
Once you got this algorithms you can predict the moves that will likely be played, the position that will likely be reached, and you can apply a third evaluation function to it, which this time takes both player strength and weakness into account (for example if one player is clueless about pawn structure, we can ignore almost any structural advantage he might have and almost any structural weakness his opponent has).
Examples
Coming soon, in the posts below.
Brought to you by an humble member of the Intellectual Chessists Society
*Stockfish evaluations are, in essence, discrete, and so should be a perfect objective analysis of a game, hence the preference of quantization over quantification, even though quantification is acceptable as it is broader than quantization, and one can even argue against the use of the word quantization as an engine evaluation is usually much less quantic (discrete) that the objectly perfect assessment of a position consisting of a result and the number of turns needed to reach it, assuming perfect play.
** Subjective advantages, on the other hand are, in essence, continuous, thus quantizating them would be improper, although possible as an approximation. Even if you take that the world is quantic, that is discrete, the time and space scale involved in chess generally imply measurement, thus likely randomness, thus the approach used should be statistic, probabilistic at best, and in any case our model continuous.
***Scientifical Well Appreciated Guess.