What Is a Centipawn Advantage?

What Is a Centipawn Advantage?(win-vector.com)

69 points by jmount 87 days ago | 32 comments

I'm not a chess engine guy, but I've talked to some, and, from what I recall, there is a very interesting difference between an engine like Leela Chess Zero (lc0) and Stockfish. Stockfish internally calculates in centipawns while lc0 calculates in WDL's. Stockfish has a model they use that converts their centipawn calculation to WDL's, but it's not _really_ WDL of the position, it's just their estimate of it according to a probabilistic model. Same in reverse applies to lc0. Why I find this interesting is that it shows how they come from different generations, with Stockfish representing the old deterministic style with deep search, and lc0 being directly inspired by Alpha Zero and the new generation of engines based on neural nets. Stockfish has by now adopted the best of both worlds (deep search with a small neural net) and is the better for it, but I still think the developers of both engines banter over who is really producing the True WDL numbers for a given position.

For my part, I find that WDL is more amendable to interpretation. Being up 5 pawns worth of material sort of makes sense, but being told you have a 95% chance of winning makes more sense to me at first blush.

n_e 83 days ago | |

> but I still think the developers of both engines banter over who is really producing the True WDL numbers for a given position

In fact, stockfish's WDL is very rudimentary: it is a function of the centipawn evaluation of the position and the value of the remaining material.

See https://github.com/official-stockfish/Stockfish/blob/a6d055d...

knuckleheads 83 days ago | | |

Yeah, it's still unclear to me why Stockfish produces WDL numbers beyond sometimes people ask for them.

tarentel 83 days ago | |

To your last point, the centipawns thing doesn't make a whole lot of sense from an interpretation perspective because it is so shallow. WDL can give you much more insight into how tame or chaotic things are. A 1 pawn evaluated advantage with a 95% chance to win is wildly different from a similar evaluation and a 50% chance to win. The first position likely has an obvious tactic that leads to a win, the latter may require perfect play for 15 moves that only a computer can calculate.

Also, from a computer perspective, a >= 1 pawn is usually sufficient for a computer to win 100% of the time so it's not really interesting and says very little about whether a person could win 100% of the time.

knuckleheads 83 days ago | | |

Yep, exactly. I spent a lot of time trying to figure out better ways for interpreting the evaluations of engines for https://www.schachzeit.com/en/openings/barnes-opening-with-d... and I ended up liking WDL much better than centipawns. A blunder defined in terms of decreasing your chance of winning by such and such percentage is, to me, a much better definition than a blunder losing such and such material. What does that mean? It makes sense to me now, but it took a long while.

Relatedly, there is an interesting thing that lc0 has been doing as well, where it takes the contempt concept even further, and can beat you with queens odd. https://lczero.org/blog/2024/12/the-leela-piece-odds-challen... It assumes it is better than you and that it shouldn't just give up because you might be up a knight, rook or even queen.

ramblerman 82 days ago | | |

If there is a 15 move sequence that leads to a guaranteed win, stockfish would not call it a 1 pawn advantage (given the sufficient calculation time) instead calling it a won position.

I think you may be mistaking your understanding of stockfish as shallow in that regard.

Where the big differences might emerge is in strategic mid game positions without any clear tactics or forcing moves. There lc0 can somehow "feel" that a position seems better.

deklesen 83 days ago | |

FWIW, as an avid chess player, I find the "up 5 pawns" has more intuitive signal.

knuckleheads 83 days ago | | |

Was it intuitive when you were first learning to play? Or have you gotten used to understanding positions via centipawns?

paulddraper 83 days ago |

Surprised it didn’t mention until the very end, but since chess is deterministic, there is no objective probability.

Every position is objectively plus infinity, minus infinity, or zero.

The “advantage” is an engine-specific notion that helps prune search paths.

Some chess engines don’t even evaluate an advantage.

kuboble 83 days ago | |

There are also objective measures for more fine position evaluation.

For winning/drawn positions: "What is the smallest program that can guarantee your side to win/draw" probably adding some time constraint.

im3w1l 83 days ago | | |

I think program size is probably not a good measure since any heuristic you can put in could be discovered at runtime with a metaheuristic that searches for good heuristics. Time and memory make more sense.

paulddraper 83 days ago | | |

Measuring the size of a model that produces a win?

Theoretically valid, but that's not going to be a very useful/diable.

jmount 83 days ago | | |

That is a neat variation.

TZubiri 83 days ago | |

Not only it is mentioned, but it's mentioned that it was mentioned as early as 1950, by none other than Claude Shannon:

>""under perfect play all chess games be a the same single one outcome of the following (we just currently don’t know which one, “A” playing the white pieces): Mr. A says, “I resign” or Mr. B says, “I resign” or Mr. A says, “I offer a draw,” and Mr. B replies, “I accept"

janalsncm 83 days ago | |

Yeah it’s confusing because there are really three “evaluations” you could have for a position

1) god-mode 1/0/-1 which you could argue is the “true” position 2) engine centipawns which help the search algorithm 3) human evaluation which would distinguish between two positions in terms of a subjective difficulty

For example, two positions might be 0.0 on the eval bar but one position is an obvious draw and in the other position one player has to walk a tightrope of precise moves to draw. Just because that’s obvious to a computer doesn’t mean a human can easily draw the second position.

salamo 83 days ago |

You'll also have some fun pinning down the difference between an "inaccuracy", a "mistake", and a "blunder". These are meaningful delineations for humans but not for a chess algorithm. Objectively, any amount of centipawn loss either changes the best possible outcome for the player or it does not.

So in practice, a drop in win probability greater than 14% is considered a blunder on Lichess.

For reference, lichess uses the following function to map centipawn advantage to the probability bar, derived from observed outcomes: https://github.com/lichess-org/lila/pull/11148

From an ML perspective, this is basically logistic regression with a single feature. However, once we leave the realm of theoretical centipawn value and begin to optimize predictive power, we could imagine adding in other things like the players' ELOs or time remaining per player, etc.

I think there are some interesting theoretical differences between predicted win probability derived from Stockfish CP and actual outcomes. As in, you could even imagine predicting positions where certain players struggle and steering them towards those positions. [0]

[0] https://www.youtube.com/watch?v=KgOC1D8wkyE

ramses0 83 days ago |

"""under perfect play all chess games be a the same single one outcome of the following (we just currently don’t know which one, “A” playing the white pieces):

Mr. A says, “I resign” or Mr. B says, “I resign” or Mr. A says, “I offer a draw,” and Mr. B replies, “I accept.” That is, under perfect play, each chess position is either a forced win, forced draw, or forced loss. The domain of a perfect chess position evaluation function is these three cases as symbols."""

There's an interesting point I've heard of in Backgammon, somewhat related to this statement. Modern Backgammon offers "the doubling cube" as a play option. https://en.wikipedia.org/wiki/Backgammon#Doubling_cube

...basically if you think you're going to win (aka: you have a 200 centi-pawn advantage), you can offer the doubling cube to your opponent (doubling the stakes of losing). If you're playing to win $5, and halfway through you think "yep, 90% chance I'm going to win this one...", you push the doubling cube to 2x (aka: $10 consequence), and kindof like poker your opponent has to evaluate whether it's "worth it" for them to stay in the game.

You might imagine a "2xELO penalty" where White takes a Queen with a Pawn, and then offers "2x, or I'm gonna beat 'ya!". If Black say "Naaah, you just activated my trap card!" and then either accepts "2x" or pushes back at "4x", then it becomes a little more like poker... you think you can beat me, then prove it!

Not that I'm suggesting changing the rules of Chess, but overall I'm really fascinated by the concept of formalized semi-out-of-band risk-taking to potentially end games early.

throwaway198846 83 days ago |

fractional tempi sounds fascinating