May 14, 2003

Checking facts

Just one final word on games, game theory and Survivor, and I'll shut up for a while, I promise. Stephen Karlson makes two germane points in a recent post on Survivor.

First, that it's a waste of time for an academic to look for "deep insights in a contrived conflict" like Survivor. Guilty as charged, m'lud - but there's a rather larger constituency out there of people who like Survivor than people who like game theory. My work is influenced by Russell Hardin at NYU/Stanford - he's a political theorist who likes to draw examples for theoretical points from opera and highbrow literature. My tastes are less refined, and thus my examples more mundane.

Second, Professor Karlson suggests that game theory can indeed explain what's going on. As I understand his argument, he's saying that all games with a final move, like chess or Survivor must have a sequence of strategies by which one player can force a victory. We can understand players' actions as luring your opponent into a variation where you then spring your pre-prepared move. Well, up to a point Lord Copper. There are two counter-arguments to this. The first has already been made by Kieran which is the rather simple observation that this isn't the way that experts actually -play- games. Even the best players tend to get rather dizzy if they try to go more than a couple of levels down the game tree; possible moves proliferate exponentially, and even mixed strategies aren't much help. This may be mathematically useful, and may help you program a computer to play chess, but it's a rotten model of how people play the game. Or, of how they play "Survivor." Highly complex games are more or less equivalent to indeterminate ones from the point of view of human beings, who have limited mental processing power.

But there may be an even more fundamental objection. When Professor Karlson says that in chess, " there is a sequence of moves by which White can force a win, Black can force a win, or Black can force a draw," he's referring to Zermelo's theorem. This is one of the staples of post-grad courses in game theory; it's thought of as one of the earliest (if not the earliest) game theoretic results in the literature. But it's based on a quite fundamental misinterpretation of what Zermelo actually said. Like Professor Karlson, I thought until a couple of months ago that Zermolo's theorem stated, in effect, that chess was solvable, and that either White or Black had winning strategies or could force the game to a draw. But then I came across this rather interesting paper which seems to show convincingly that Zermelo has been grossly misinterpreted by game theorists. His concerns were much narrower; he wanted to find out whether there was an upper bound on the number of moves that a player, who was already in a winning position, needed to make in order to force a win. It's a much narrower result; in Zermelo's own words

"The question as to whether the starting position ... is already a 'winning position' for one of the two parties is still open. Would it be answered exactly, chess would of course lose the character of a game at all."

There may be game theorists since who have come up with a more general proof that there is such a winning position at the start since then, but I'm not aware of them.

Posted by Henry at May 14, 2003 12:00 PM | TrackBack

Tiny nitpick: Zermelo seems to make a mistake about the rules of chess. Chess can’t, even in theory, go on forever, because of the 50-move rule. (If 50 moves pass without a pawn being moved, or a piece being taken, it’s a draw; in some situations there’s an extension.) So there are only a finite (though impossibly huge) number of possible chess games, which I think entails that best strategies exist for both sides.

Posted by: Matt Weiner at May 16, 2003 01:30 AM

Very interesting post

Posted by: Fred at October 21, 2003 06:53 AM
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