Suppose we have a game in which one of the players has an intrinsic advantage over the others. In some games, a point handicap can be used to even things out. All players in the tournament could propose a handicap, and the average handicap could be used.

However, having all the games use the same handicap might not be right. Certainly it could leave the players in some of the games feeling that they've been screwed by the coin toss.

What ought to work for a two player game is, for each game, the two players each secretly bid what they think it's worth to be able to choose their starting position. I.e., "I'd be willing to give the other guy 8 extra victory points if I get to choose between going first and going second." Whoever bids higher gets to choose, and the other player gets the arithmetic mean of the two bids.

So, if Joe thinks that choosing starting position is worth only 4 points, and Bob thinks it's worth 8, then Joe should be happy to get 6 points for letting Bob choose, and Bob should be happy to get to choose while giving Joe only 6 points.

But it's important that the players be allowed to bid fresh for each game, since their evaluation of what the choice of position is worth might depend on the opponent. (Even without knowing the individual opponents, you might expect that early rounds you'll face weaker competition than in later rounds, and thus the choice of position might not be worth as much as in the final round, for instance.)

I'm not sure how to extend this approach to a multiplayer start. And as someone else pointed out, of course, making it too complex makes it overshadow the rest of the game.