Folks, I got Gil's argument at last. Woohoo!
A chess tournament produces bits of information: who won which game. If most of those bits can be causally linked to skill differentials between players, rather than chance, Gil calls it a game of skill. Otherwise, luck. If stakes are high and zero-sum, players start trying to find weaker opponents. This means each game will be between players of roughly equal ability, so skill differentials cease to contribute bits to the bottom line.
Gil, is that right? If so, I apologize for misunderstanding at first. This sounds interesting.
There is no reason for bets to be of zero expected money. Just bet around zero expected utility of money. The whole argument might be a characterization of this bizarre betting rule, but not of the games themselves.
Gil Kalai, a well known mathematician, has this to say on the topic of chess and luck:
http://gilkalai.wordpress.com/2009/07/05/chess-can-be-a-game-of-luck/
I didn't follow his argument at all, but it seems like something other LW posters may understand, so I decided to post it here. Do comment on his arguments if you agree or disagree with him.