What is the advantage with the white pieces?

We saw last time that, everything else equal, playing with the white pieces gives you an advantage of a few percentage points. Although I expected to see an advantage to white in master play, I was surprised to see the advantage show up so strongly in the two million games I looked at.

To state the obvious, white goes first and has the ability to dictate the opening. In most cases, you’d expect people to choose an opening that they know well and can play strongly. However, there are many openings, and black certainly has every opportunity to change the direction of the opening. Something like Alekhine's Defence (which I used to love playing until I kept getting demolished because I found it difficult to get my pieces out) is a good example of a way to turn e4 into a very different game from the usual Ruy Lopez and Italian Games.

So, on the one hand, at lower levels of the game, where opening knowledge is lacking and blunders tend to predominate, I would've thought that white's advantage would be minimal. On the other hand, imagine at a random point during a game suddenly giving a particular player two moves in a row. Surely that would be a decisive advantage, which suggests that having the first move should really be significant.

Let's quantify the advantage of playing with the white pieces, explore exactly how long the advantage lasts, and see if it differs across different rating levels. I revisited the two million games we looked at last time, where white won approximately half the time, draws occurred 3.5% of the time, and black won about 46.5% of the games. In this dataset, the median number of full moves (i.e., white and black both moving) is 35 and 90% of all games end before move 54.

To look at how the advantage for white changes as the game goes on, remove all the games that end before the first full move and recalculate the number of wins for white, draws and wins for black. These new percentages give us estimates of the win/draw probabilities of all games still being played at the end of move one. Next, remove games that finish by the second full move and so on, and we obtain conditional probabilities for white wins, draws and black wins. The graph below shows the outcomes of games lasting at least the number of moves in the x-axis.

From this we see that the initial probabilities of different outcomes (around 50% / 3.5% / 46.5%) remain fairly constant until about move 15. For games that last longer than 15 moves, the conditional probabilities of white as well as black winning both start to decrease. A game that lasts more than 100 moves is more than 50% likely to end in a draw. In other words, the longer a game lasts, the more likely it is to end in a draw.

Next, let's look at differences across rating levels. To do this, we calculate the expected score (for white), which is calculated as the number of white wins plus half of the number of draws divided by the total number of games. The expected score out of all two million games is thus approximately equal to 50% + 3.5%/2 or 0.5175. For each game, calculate the average rating of both players, and then divide the range of game ratings into four equal intervals.

As before, the expected score can be calculated conditional on the game still being played on increasing number of moves and plotted for each ratings band:

Interestingly enough, the lower rated games have the highest advantage for white (at least initially). Across all rating levels, the advantage with the white pieces steadily decreases as the game continues, although we saw that this reduction in white's advantage manifests itself almost entirely through a higher draw probability rather than an increase in black's chances of winning.

The shaded areas are approximate 95% confidence intervals for the expected score with the idea being that once the shaded area crosses the 0.5 line then black has for all intents and purposes (i.e. statistically) eliminated white's advantage. The only rating level that approaches even odds for black and white by move 30 is the highest band (average Elo rating of 2231-2712). The middle two rating bands look like they are on the way to nullifying the advantage by move 50, but there are too few games remaining in play by this point to really tell.

Finally, you may have noticed the kink in the curves above for the first few moves. Since the fastest checkmate can occur only after two full moves, these results are either resignations, time-outs or draws by agreement. Obviously, a definitive result in the first few moves is not a true chessic outcome, and, apart from cheating and time-outs, I can only think of a few different reasons why the game might end so quickly, namely quitting due to not wanting to play black, and quitting (by white or black) due to an unexpected opening move.

The following table gives the count of results for games ending on or before full move number two. The top row shows that quite a few games end before any move has been made, and these are mostly wins to black. The proportion of games won alternates between black and white according to who made the last move. This suggests either time-outs or resignations due to unexpected moves. Actually, this see-sawing of results happens throughout the dataset, which is not evident in the previous results since it is smoothed out by looking at full moves rather than half moves.

I was quite surprised to see such a strong and persistent advantage to playing with the white pieces, especially as it persists for almost all of the game and is evident across all ratings. Black's consolation is perhaps that, being only a few percentage points, the advantage is statistical rather than decisive.