The pleasures of chess

Garry Kasparov this month thinks about reviewing something-or-other in the New York Review of Books, becoming happily diverted into a discussion of what makes chess truly interesting. (I draw also from some recent conversations with S. O. Killmier.)

The big point: chess is not about who can see the most moves ahead. Computers (and humans) that win by doing this are simply winning by brute force, rather than by intelligence; in the article Kasparov memorably denigrates his result against Deep Blue as ‘losing to a $10 million alarm clock.’ If one insists that the only purpose of chess is to win, then brute force seems a very successful, though by no means infallible, way to do this. I’d like to spend a little time describing just why it isn’t fool-proof; and a lot of time showing why victory in chess is less than half the point.

Imagine you are a chess computer; in fact, imagine you are a chess computer with limitless computational power. Now here is a famous chess position—find the winning move:

White to play… and win!

1. Brute force is dumb force

From the probabilists’ viewpoint, some of the twenty moves available to White in this position must be more likely than others to lead to a victorious final position. After all, there are only a finite number of possible arrangements—how many is not known, though 10⁵⁰ is bandied about as an estimated upper bound—and only a tiny fraction of these are final positions. So, which move is best?

We have no idea. Claude Shannon thought a little about this (pdf) in 1950, but skirts around the issue of how it would actually be calculated:

By working backward from the end one can determine whether there is a forced win, the position is a draw or is lost.

Okay… maybe. As Shannon points out, because of the rule that a draw results after 50 moves in which no piece is captured and no pawn moved, every game will terminate, but threading backward from a position is extremely difficult. To evaluate which move is best at the start of the game, one needs to know how many possible threadings there would be. But how many games of chess are there? No-one knows. Even a simpler question such as: what is the maximum possible length for a game of chess?, goes unresolved. I don’t know of any sophisticated attempts to answer either question^[A].

But even though I’ve not seen any proof of the result, it feels as though the number of possible games has to be finite. So, armed with a way of counting them up, it would be possible to enumerate how many started with each of the first twenty moves. Then one would, at long last, know which of the twenty possible first moves is most likely to lead to victory.

But then, that wasn’t quite what we wanted to know.

2. Beautiful chess is good chess

Ask any chess player what the best starting move is and they’ll tell you, in roughly descending order of good-ness (and written in algebraic notation): d4, e4, maybe Nf3, anything else. How on Earth do they know? Well, one answer is purely numerical. The collection of all chess games played—indeed, the entire enterprise of chess itself—is a grand piece of experimental mathematics designed to answer just the question posed at the start of this post. The infamous Opening Explorer performs the enumeration I suggest above, though just with the much more tractable collection of all games recorded in its database^[B]. Perhaps one day, every possible game of chess will have been played and recorded there.

The other answer is empirical. And this distinction between the numerical and the empirical is close to the heart of the game, and to the heart of science. By playing chess, we have formulated hypotheses and passed knowledge about results and their generalisations along to successive generations, so that the modern chess player has a working knowledge of the kinds of positions that lend themselves to victory. The best opening moves are the ones that aspire to those kinds of positions.

Magnus Carlsen, currently the official world number one, refers to a ‘feeling’ of what the best move in a position will be. This is much less vague than it sounds. It reflects the lessons unconsciously learned and his ability to generalise them accurately to broad range of situations. There’s science in that. It might be that, one day, all the rules for best chess play will have been discovered. But it isn’t a case of probabilities. Chess, to a greater degree than the now-solved checkers and a lesser degree than poker, is definitely a game of temperament: the best move depends not only on the position but the mindsets of the players. If a successor to Shannon can generalise his experiment to incorporate this element too, perhaps we’ll be close to finally ‘solving’ chess.

Kasparov has poignant final words:

Perhaps chess is the wrong game for the times. Poker is now everywhere, as amateurs dream of winning millions and being on television for playing a card game whose complexities can be detailed on a single piece of paper. But while chess is a 100 percent information game—both players are aware of all the data all the time—and therefore directly susceptible to computing power, poker has hidden cards and variable stakes, creating critical roles for chance, bluffing, and risk management.

These might seem to be aspects of poker based entirely on human psychology and therefore invulnerable to computer incursion. A machine can trivially calculate the odds of every hand, but what to make of an opponent with poor odds making a large bet? And yet the computers are advancing here as well. Jonathan Schaeffer, the inventor of the checkers-solving program, has moved on to poker and his digital players are performing better and better against strong humans—with obvious implications for online gambling sites.

Perhaps the current trend of many chess professionals taking up the more lucrative pastime of poker is not a wholly negative one. It may not be too late for humans to relearn how to take risks in order to innovate and thereby maintain the advanced lifestyles we enjoy. And if it takes a poker-playing supercomputer to remind us that we can’t enjoy the rewards without taking the risks, so be it.

Micallef can go one better though:

A. Here are some back-of-the-envelope computations, indicating that the answers are about 10²⁹⁵⁰⁵ and 6000, respectively.

B. You’ll note that it evaluates e4 as slightly more likely to win than d4, yet virtually every top player would agree that d4 is the better move. The point is that both give excellent chances for White to win, and which one is better for a particular player can well depend on their style of play.

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