A chess question of W.T. Tutte [closed]

Question

In "Graph theory as I have known it", p.12, Knights Errant, the late Tutte mentions as an aside the chess question "Does either Black or White have a certain win from the initial position, given perfect play by both sides".

Is there any literature on that possibility of a black win, that is the possibility of the initial chess position being mutual zugzwang? What is the earliest reference to the question?

EDIT: If posed in chess.stackexchange, the community might give an answer based on chess strategy. However the question of evaluating chess has a venerable mathematical tradition, including non-trivial work by Zermelo, Shannon & Elkies. So the intention of posting here in math.overflow is to ask from a purely mathematical perspective, e.g. what has combinatorial game theory to say?

Answer 1

The question "does either Black or White have a certain win from the initial position, given perfect play by both sides" was first addressed by Wilhelm Steinitz in his 1896 "Theory of Perfect Play" (Chapter 6 of Modern Chess Instructor). He concluded that "by proper play on both sides the legitimate issue of a game ought to be a draw".

You can find a quite detailed overview of the literature since Steinitz in Wikipedia. The advantages of Black over White seem to be largely psychological ("underdog").

Answer 2

In the German chess problem magazine feenschach, Heft 214 Band XXIV, July-August 2015 pp 178-180, I pose & solve the following mathematical puzzle: "Consider the 25 chess positions deriving from the 25 sequences of moves of the following form: 1. e4 e5 2. B~ B~. (I.e. two bishops are developed.) Show that at most six of these positions are wins for Black."

E.g. one position out of the 25 is:

[]

This is solved non-constructively with a simple strategy-stealing argument, which places an upper bound on the possible number of Black wins as 25/4. However, I don't think this fragile approach can be generalized to the Holy Grail of showing that chess is not a win for Black. The population of 5x5 (or more generally nxn) positions examined here is kind of a best case, where there is maximal stealing possible. If one considers e.g. the population of 400 positions after Black's first move, then then we can show that at most 304/400 can be wins for Black. This is a much higher proportion, and shows the limitations of the approach.

27/12/2015: After the closing of this question for allegedly not being mathematical, I added the following to emphasize that it can very well be interpreted mathematically.

Thanks very much, Mr Calvert, for your support. I was a bit bemused that the question was suddenly closed for allegedly not being mathematical immediately after I posted an entirely mathematical response. The underlying article has simple lemmas and everything :-) Whether Black wins chess may be addressed by chess players in terms of chess strategy. Indeed the accepted answer here is such an opinion! However my interest is only mathematical.

Claude Shannon wrote in his well-known 1950 paper (Philosophical Magazine, Ser.7, Vol. 41, No. 314 - March 1950. XXII. Programming a Computer for Playing Chess)

"It is interesting that a slight change in the rules of chess gives a game for which it is provable that White has at least a draw in the initial position. Suppose the rules the same as those of chess except that a player is not forced to move a piece at his turn to play, but may, if he chooses, 'pass'."

I was really looking for partial strategy-stealing solutions, in the spirit of Shannon, but in ordinary chess. So for White to lose a tempo takes at least two moves. Interestingly, parity concerns are particularly significant in the chess starting array.

In "More Games of No Chance" by MSRI (= "Mathematical Sciences Research Institute") Publications, Volume 42, 2002, Unsolved Problems in Combinatorial Games #29, p 465, the mathematician Richard Guy also raised the question whether chess could be a Black win. However he misstated the result that I had emailed him. The correct result has only now been published (feenschach, above).

Another possible angle of attack, given that chess is so hard, is to ask what properties a simpler symmetric positional combinatorial game would have to satisfy to offer non-trivial strategy-stealing. Intuitively, Zugzwang is a feature in cold positions, but the chess in the opening appears to be hot. What other games can one set up so that e.g. a two-move tempo loss can exist, without being apparently as (literally!) self-defeating as in chess.

As this mathematical question remains incorrectly closed - I'm happy to discuss this with anyone offline: there is an email contact given at the bottom of http://anselan.com/chess.html.

Answer 3

In my decade of chess-playing, I have never come across anything remotely resembling an answer to this question. If the perfect play question had already been answered by example, chess would be an exercise in memorization -- the absolute perfect path(s) of the game could all be played out down to the endgame, and whoever deviates first loses material. In the event that one side has a forced win, the other side would always be the one forced to lose material. Brute-forcing a solution to chess is nowhere near possible at the moment, given the amount of possible game positions (~ $10^{43}$, according to Claude Shannon). I've never heard of any sound way to argue this question other than brute-force.

Answer 4

Virtually no one in the Chess world believes Black wins, hence no literature on what you highlight.

Of course, that isn't mathematical proof, but if we were to ever get an answer and I had to gamble on it, I'd say it's a draw with optimal play.

Now if we look at stats, certainly Black loses more often than white loses, with draws becoming more common at higher levels of play.

I'd even say if White got 2 moves to start, and the moves were one for one thereafter, the game is a draw with optimal play. Granted, Black would be cramped by White's first two moves.