Current computer chess engines include a neat feature: the analysis bar. For example here is the analysis bar from chess.com.
White to move, FEN:
rnb2rk1/ppp2ppp/3p1q1n/4p3/7P/b1PPP3/PP1BBPP1/R2QK1NR w KQ – 0 1
The analysis bar is on the left, and the white portion is 74 “centipawn units” below the midpoint (matching the -0.74 “pawn units” in the top right analysis tab). This means the analysis feels white is at a 74 centipawn disadvantage to black in this position. The question is, what is a centipawn advantage?
Lichess gives a similar analysis and score (in this case -50 centipawns).
As probabilities
From the Stockfish Interpretation of the Stockfish evaluation document we have: “a 1.0 pawn advantage being a 0.5 (that is 50%) win probability.” A player with a 100 centipawn advantage is thought to have a 50% chance of winning (probability taken over chess engine near-optimal variations of play). Notice this is not necessarily the traditional material value of a pawn on the board. The intent is given in a reference graph that we reproduce here.
To use this graph we lookup our centipawn advantage (or disadvantage) on the x-axis and then measure the heights we cross the win, draw, and lose curves at. This reads off estimated probabilities of each outcome. Notice at a 0 centipawn advantage a draw is considered almost inevitable under optimal play. In our case the 74 centipawn disadvantage is shorthand for claiming a 0.23 probability of a loss, a 0.77 probability of a draw, and little chance of a win.
The way the engine likely uses the graph is it estimates probabilities (W, D, L) and then finds the point on the x-axis minimizing the (possibly square) distance of these from all three curves. The resulting centipawn advantage is then an approximate summary of the engine calculation.
Some engines, such as the OSX Stockfish UI show below, report both the centipawns (in this case -98 centipawns) and probabilities together.
As Elo
Chess players tend to use differences in Elo rating to measure advantage between players. So they may be more comfortable with Elo differences than with probabilities. One can convert centipawns to Elo difference using the following graph.
We see the relation is not linear. The flattish portion in the middle is due to the fact that the centipawn system models draws separate from wins and losses, whereas the Elo system treats draws as half-wins. A 100 centipawn advantage corresponds to an Elo difference of about 190. An Elo difference of 100 is about 80 centipawns.
Why pawns?
Chess does not have an in-game point system (unlike games like Go). We are left to choose our own domain of advantage measure. Common choices include: probability, Elo difference, material (pawn and piece counts), position, or time (moves or tempi). The common evaluation taught is an “all things being equal” chess piece relative value with the following simplified values.
Converting probabilities into centipawns is an attempt to use a unit familiar with chess players. In math terms: it is using a logistic link instead of a probability or likelihood ratio.
Positional concerns
In chess one is usually not in “an all things being equal” situation. A given pawn may be protecting a piece, so may be worth much more than a point. For example here is the Stockfish estimated change in centipawn advantage for each single piece removal from the standard initial chess board.
Notice not all the pawns are priced at the same number of centipawns. The average value for white pawns is 128 (for white to move) and the average value for black pawns is 154 (again for white to move). So the pawns are averaging to a bit more than 100. It makes sense that the centipawn score can not simultaneously be completely calibrated to the claimed 100 centipawns is a 50% chance of winning and to an actual pawn on the board always being worth 100 centipawn advantage.
More on tempi
Beyond the usual “a pawn is worth 2 or 3 tempi” rule it would be interesting to work out a directly theory of “fractional tempi” for chess in the spirit of “Winning Ways for Your Mathematical Plays” by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. One idea would be to play over an ensemble of boards, where players alternate over choosing which board to move on (ignoring if it is the white or black turn on each board). Another idea is adding a small random chance of getting an additional move. Some concept like this might allow expressing positional advantage in a direct fractional tempo unit.
Conclusion
Shannon pointed out (Shannon, C. E. (1950). XXII. Programming a computer for playing chess. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(314), 256–275. https://doi.org/10.1080/14786445008521796) 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.
As we don’t currently have access to this perfect evaluation function we settle for other evaluation domains such as piece/material values (augmented with positional features), tempi, estimated probabilities, and differences in Elo or matchmaking rating. Total material value plus position came first, as it is easiest to evaluate. Probabilities came quite late as they could not be mechanically estimated in a satisfactorily manner until the 1990s.
Categories: Tutorials
