Wikipedia:Reference desk/Archives/Mathematics/2016 December 27

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December 27

Could chess be a win for Black?

By the Gale–Stewart theorem, there are exactly three exhaustive and mutually exclusive possibilities:

• Chess with perfect play is a win for White
• Chess with perfect play is a draw
• Chess with perfect play is a win for Black

Which of these obtains, as far as I know, is still unknown, or at least unproven. However our article first-move advantage in chess does not mention any authorities who explicitly contemplate the third possibility. The possibility that "Black is OK!" is mentioned, but the arguments do not seem to be about perfect play.

I understand that experts, presumably with good reason, consider it to be very unlikely. But I'm curious whether there are any who explicitly consider it a genuine possibility, even if one that would be very surprising if true. --Trovatore (talk) 09:01, 27 December 2016 (UTC)

That's a very intesting question. I'm no expert at all, but I think mainstream considers that if there is a win for Black, it may live inside the Sicilian Defence. Databases put the winning probability for the Black side at around 46-48%, which is the best Black can get.

It is aparent to me that under perfect play, symmetrical opening positions are very unlikely to host the winning Black strategy, if you want, for strategy-stealing arguments. The quoted article on the Sicilian defence explains the imbalance the opening gives to Black breaking the symmetry. Pallida  Mors 14:08, 27 December 2016 (UTC)

The strategy-stealing argument applies to any symmetric game (...) in which an extra move can never be a disadvantage. - but chess is known to not be such a game, see zugzwang. Of course, you would be hard-pressed to find a good chess player who thinks the starting position is a zugzwang, but a reasonable and widely-held feeling is not the same as a proof. Tigraan^{Click here to contact me} 20:31, 27 December 2016 (UTC)

Since the OP was not asking for a proof, but rather, arguments of plausability, I think my remarks are of value.Pallida  Mors 12:45, 29 December 2016 (UTC)

If there is a win for Black in the Sicilian Defense, it doesn't matter, because White can avoid it. Bubba73 ^{You talkin' to me?} 04:10, 31 December 2016 (UTC)

I don't really have any knowledge about this above what can be found by a Google search, however, a few comments:

• The result you mention is more commonly known as a consequence of Zermelo's theorem (which seems to be a special case of Gale-Stewart). People will more quickly understand what you mean if you reference that.
• First-move advantage in chess isn't the only Wikipedia article to search for information on this, for example Solving chess has some more discussion and does explicitly state black win as a possibility.
• AFAIK most professional games end in draws, and more so the higher the ratings. This strongly indicates that perfect play results in a draw as well.
• If you take an arbitrary chess position, most likely it is advantageous to be the side to play. Zugzwangs are the exception. If there was some option to pass in Chess, it would rarely be used. It stands to reason that this holds for the initial position as well, so if there is any advantage, it is white's.
• I'm not even sure what "explicitly considering black win a genuine possibility" would even mean. When discussed in a purely formal setting, it will of course be mentioned because we haven't proven otherwise. When discussed in a chess-centric context, you'll be hard-pressed to find someone championing this possibility because there isn't really any evidence to support it. You could explicitly state it is a possibility but you won't have any arguments in its favor, so it's kind of pointless.
• As a counterpoint, if you consider my own record of 1000 blitz games on Lichess, you'll see that I win as black more often than white ([1]). While this is anecdotal and not statistically significant (the difference is less than half a standard deviation), it does seem to suggest that the white advantage could depend on play style, speed and rating. So I am a bit suspicious whenever I see discussions of white advantage.

-- Meni Rosenfeld (talk) 14:28, 27 December 2016 (UTC)

The first move advantage in chess article deals with the observed fact that in games among top players in the last two centuries or so, White has consistently won more often than Black. This is a different question to that of the evaluation as "1-0, 0-1 or 1/2-1/2" of the starting position, and arguably an almost unrelated one; but if you feel like adding the third option to the article, go ahead.

For further reading, losing chess is the closest game that I know which is "simple" enough that it has been solved by computers (see the article).

My speculation on the subject:

"the better the players, the higher the chance to draw" is an often-repeated argument in favor of the view that the starting position is a draw. I never found it convincing because the underlying reasoning is to extrapolate the percentage of draws between two players of equal caliber as a function of the "playing strength" (loosely defined), for playing strength going to "perfect play". The problem is, we do not know whether today's top players are really close to "best play"; the extrapolation is only reasonable if they are. That argument can also be improved, with information I got from one of the top US Advanced Chess players (in a private forum). (Preliminary note: sometimes, players "play for the draw", that is, will play moves that entail a smaller risk to lose though they give less chances to win as well, because a draw is enough to achieve a desired tournament result - for instance, if you are leading the tournament and a draw in the last round guarantees first place, or if you are playing against someone considerably stronger so that a draw at that round would satisfy you.) In that game format, he said, it is empirically observed that the ~100 best players can force a draw with either color if they so desire (of course, if both players do so, neither will win the match, so both will usually agree to take some risks at some point), which is not the case in regular chess (there are multiple examples of players at the top level losing when playing for a draw). While this is not yet a proof (maybe when aliens visit us, they will crush today's top players even when those aim for the draw), at least going from a draw is probable, if wanted by either player to a draw can be forced is a significant step.

Tigraan^{Click here to contact me} 20:31, 27 December 2016 (UTC)

There is zero evidence that the results from top human players are a good predictor of the results from perfect play. There are an enormous number of partly played games where a computer using a tablebase can play perfectly and win as black but humans cannot find the win. --Guy Macon (talk) 00:23, 28 December 2016 (UTC)

Who said anything about humans? My understanding is that matches between top computers also mostly end in draws, maybe even more so than humans. Your second statement really says nothing other than that computers are stronger than humans. This would support the suggestion that playing strength correlates with drawishness, and thus, the idea that perfect play draws. -- Meni Rosenfeld (talk) 00:40, 28 December 2016 (UTC)