In a previous blog post I used logistic regression to estimate the values of pieces in Atomic chess. In that study I computed material differences between the two players using a snapshot 8 plies before the end of the match. (A "ply" is a move by a single player.) That choice of snapshot was arbitrary, but it is typically late enough in the match so there is some material difference to measure, and also near enough to the end to estimate the "power" of each piece to bring victory. However, this valuation is rather late in the game, and is probably not representative of the average value of the pieces. That is, a knight advantage early in the game could be parlayed into a queen advantage later, which could then prove decisive.

To fix that issue, I will re-perform that analysis on other snapshots. Recall that I am working from 9 million rated Atomic games that I downloaded from Lichess. For each match I selected a pseudo-random ply after the second and before the last ply of each game, uniformly. (There is no material difference before the third ply.) I also selected pseudo-random snapshots in the first third, the second third, and the last third of each match. I compute the difference in material as well as differences in passed pawn counts for each snapshot. You can download v2 of the data, and the code.

Recall that I am using logistic regression to estimate coefficients in the model

$$\operatorname{log}\left(\frac{p}{1-p}\right) = \frac{\operatorname{log}(10)}{400}\left[\Delta e + c_P \Delta P + c_K \Delta K + c_B \Delta B + c_R \Delta R + c_Q \Delta Q \right],$$

where $$\Delta e$$ is the difference in Elo, and $$\Delta P, \Delta K, \Delta B, \Delta R, \Delta Q$$ are the differences in pawn, knight, bishop, rook and queen counts. Here $$p$$ is the probability that White wins the match. By putting the weird constant $$\operatorname{log}(10)/400$$ in front of the expression, the constants $$c_P, c_K$$ etc. are denominated in Elo equivalent units. Similarly, I fit a model with terms for passed pawn counts in various ranks.

As previously, I subselect to games where each player has already recorded at least 50 games in the database, where both players have pre-game Elo at least 1500, and games which are at least 10 ply in length.

Here is a table of the estimated coefficients for the four regressions, with coefficients and standard errors in Elo equivalents, as well as Wald statistics. The p-values all underflow to zero. The intercept term can be interpreted as White's tempo advantage.

snapshot term Estimate Std.Error Statistic
random Elo 0.957 0.001 665.0
random White Tempo 53.136 0.281 188.9
random Pawn 31.301 0.329 95.0
random Knight 47.058 0.471 99.9
random Bishop 57.368 0.484 118.4
random Rook 105.767 0.718 147.4
random Queen 244.058 0.890 274.2
first third Elo 1.006 0.001 710.2
first third White Tempo 62.620 0.270 232.1
first third Pawn 58.935 0.697 84.6
first third Knight 83.380 0.878 95.0
first third Bishop 76.827 0.986 77.9
first third Rook 80.067 1.754 45.7
first third Queen 148.910 1.768 84.2
second third Elo 0.955 0.001 666.7
second third White Tempo 50.868 0.283 179.5
second third Pawn 38.063 0.322 118.3
second third Knight 57.589 0.449 128.2
second third Bishop 59.246 0.462 128.1
second third Rook 107.305 0.742 144.7
second third Queen 210.759 0.917 229.8
last third Elo 0.915 0.001 619.6
last third White Tempo 45.537 0.292 155.9
last third Pawn 26.008 0.264 98.5
last third Knight 32.499 0.397 81.8
last third Bishop 55.957 0.396 141.2
last third Rook 109.550 0.545 201.2
last third Queen 283.669 0.716 396.1

The data are hard to digest in table form, so below I plot the coefficients for the four different snapshots, with standard error bars. We see that a queen is generally worth around 150-250 Elo points, a rook around 100 (though somewhat less early in the match), a bishop around 60 (more early in the match), a knight 30-80, and a pawn 15-60. As one progresses along the match (from first to second to last third), the queen and rook gain value, while the bishop, knight and pawn lose value.

The top axis is denominated in 'pawn' units, where I eyeballed a pawn as worth around 30 Elo, but this is so variable over the different match snapshots it is hard to quote a consistent valuation scheme in pawn units. This is in contrast with the previous blog post where we suggested a 1:2.5:4:8:22 valuation for pawn, knight, bishop, rook, queen; that scheme is only appropriate for the very end of the match (and it can be hard to tell you are at the end of the match while playing). Below I denominate piece values relative to the estimated pawn values. (I drop the error bars because I am too lazy to code up the delta method.) For the random snapshot the pawn value estimates are as below

Piece Pawn value
Knight 1.5
Bishop 1.8
Rook 3.4
Queen 7.8

As in the previous blog post, I ran the regressions again with filters for minimum Elo. The reasoning is that better players will exhibit higher quality play, rather than average play. Here are plots by minimum Elo, snapshot and piece. We see that better players are better able to capitalize on bishops and perhaps knights, but otherwise the valuations are largely consistent across player ability.

## Passed pawns

As in the previous blog post, I computed the difference in counts of passed pawns. I classified passed pawns as belonging to ranks 2, 3 or 4, to rank 5, to rank 6, or to rank 7; any pawn on rank 7 is automatically a passed pawn. When computing the material difference from White's point of view, the ranks are mirror image for Black in the obvious way. Again, I fit a model of the form

$$\operatorname{log}\left(\frac{p}{1-p}\right) = \frac{\operatorname{log}(10)}{400}\left[\Delta e + c_{234} \Delta PP_{234} + c_{5} \Delta PP_{5} + c_{6} \Delta PP_{6} + c_{7} \Delta PP_{7} \right].$$

Here are the estimated regression coefficients and standard errors denominated in Elo, along with the Wald statistics.
Below I plot the coefficients. It should seem odd to you that high rank passed pawns can sometimes have negative value. For example, a pawn on the seventh rank in the first third of a match appears to be worth around -15 Elo. The reason for this apparent contradiction is that this valuation is conditional on taking a snapshot in the first third of a match, but in most situations if you have a pawn on the 7th rank, it can often quickly lead to a victory. We are, however, looking at those cases where it does not. For this analysis it probably makes more sense to look at the random snapshot to get an "average" value.

snapshot term Estimate Std.Error Statistic
random Elo 1.01 0.001 715.29
random White Tempo 64.87 0.261 248.86
random P.P. Rank234 59.01 0.955 61.81
random P.P. Rank5 39.98 1.415 28.25
random P.P. Rank6 27.96 1.316 21.24
random P.P. Rank7 69.49 1.408 49.34
first third Elo 1.02 0.001 719.17
first third White Tempo 66.06 0.260 254.11
first third P.P. Rank234 147.94 5.674 26.07
first third P.P. Rank5 68.32 7.697 8.88
first third P.P. Rank6 -8.71 7.618 -1.14
first third P.P. Rank7 -15.68 11.592 -1.35
second third Elo 1.01 0.001 715.28
second third White Tempo 64.50 0.261 247.28
second third P.P. Rank234 106.15 1.174 90.42
second third P.P. Rank5 73.42 1.692 43.39
second third P.P. Rank6 46.71 1.655 28.22
second third P.P. Rank7 51.19 1.955 26.19
last third Elo 1.01 0.001 711.41
last third White Tempo 63.95 0.261 244.61
last third P.P. Rank234 44.47 0.651 68.36
last third P.P. Rank5 33.59 0.972 34.55
last third P.P. Rank6 25.85 0.888 29.10
last third P.P. Rank7 77.31 0.926 83.50

## Spline Regressions

It is hard to interpret the coefficients because the value of pieces appears to depend on the phase of play. To remedy this, I interacted the material difference with some spline terms. That is, I compute some spline functions of the ply at which the snapshot is taken, the multiply those by the material differences. I then estimate the coefficients of the interaction terms, and combine them with the spline functions. This gives material value as a function of the ply, which I plot below. I did this two different ways: once computing splines over the raw ply, and then using the ply divided by the total ply. In the latter formulation you can view the valuation as percent progress in the match. The problem with this formulation is that you typically do not know how far along in the match you are. On the other hand, because matches progress at different speeds, basing value on the raw ply also seems flawed.

For the raw ply plot we plot in a single facet. For the percentage regression, this results in a visually unreadable plot, so we use separate facets for the different pieces. For the raw ply regression we see near equal values of the bishop and knight through most of the match; rooks increase in value after the 20th ply; queens are valuable from early in the match, and increase in value as the match progresses. This phenomenon is intuitive, as the queen is less likely to be captured later in the match when there are fewer pieces on the board.

Below I express those estimates relative to the estimated value of a pawn. Again we lose the standard error bars. For the raw ply regression, we see a bulge at around 25 ply where pawns have very low value, and queens peak. A different pattern emerges in the percent ply regressions, where queens increase in value steadily over the course of the match.

## Future work

The analysis here indicates we need a better measure of match progress, one which can be computed in real time, but which matches the tempo of the particular match. It would seem that something like total material on the board would be a good measure. This is intuitive, as crowded positions are dangerous in Atomic and can quickly lead to large changes in the material difference. I also want to perform an analysis using survival analysis.