In commenting on the game of chess, Will Baude notes the following.
Professor Leitzel of Vice Squad writes in to remind me of the 1913 Zermelo’s Theorem, which establishes just that: given the game’s finiteness (established above), there exists some strategy s.t. either white always wins, black always wins, or nobody does.]
Not so, as it happens, although it’s been the conventional wisdom among game theorists until recently – the inestimable James Morrow (whose “Game Theory for Political Scientists” I’m using as a coursebook this semester) states it a little more formally when he says that
Zermelo showed that chess has a winning strategy: White can force a victory, Black can force a victory, or either can force a draw.
But, as discussed on my old blog last year, this very interesting paper by Ulrich Schwalbe and Paul Walker, shows that Zermelo said no such thing. Zermelo proved a much narrower result, and indeed explicitly states that he hasn’t proved that chess has a winning strategy.
The question as to whether the starting position … is a winning position is open. Would it be answered exactly, chess would of course lose the character of a game at all.
It would be very interesting to trace back how this error (and a variety of others) crept into the literature. Zermelo was never translated into English before Schwalbe and Walker’s paper, so I imagine that nobody much bothered to try to read him (especially since his article was published in 1913 and was quite likely printed in Fraktur). One person’s error was presumably picked up by others, and then disseminated until it became accepted dogma in the wider literature. Academic research sometimes resembles a game of Chinese whispers – because we all rely on the research of others, serious blunders can be perpetuated for generations before someone bothers to go back and recheck the work of their elders.
Update: Peter Northup has convinced me in comments and email that I’ve misunderstood Zermelo a little myself, and that the formulation that Morrow uses isn’t as offbase as I thought it was. What’s clear is that game theorists are incorrect in saying that Zermelo used backward induction as such, and that he doesn’t show that there is a winning strategy as such. I stand corrected.