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> This also deals with the discovered check problem in multiplayer

Not to mention variants where both (or many) sides move simultaneously and
then the moves must be resolved. For example, the multiplayer game
Diplomacy, which IM Bela Toth introduced me to, has everybody move
independently and then conflicts are resolved by complicated rules; and for
contrast, the multiplayer game Risk (and thanks to Bela I can win this
game) is completely turn-based.

These things are complex, and this is unfortunate in a way because we
susally seek simplicity; but it may be necessary.

Synchronous Chess is an ancient chess variant where both sides move at once
and the conflicts are somehow resolved. 

Realtime Chess is a first-draft inspiration of the moment by me, and the
idea is that you just move your pieces without waiting for the other guy to
move. Your clock is *always* running. ((I see no way to make rules that
keep it from bing a simple stupid clickfest.))

Isn't that obvious from the symmetry? The rules make no distinction anymore between black and white, so they must have equal opportunity.

Of course you could simply acheive that by only flipping a coin to decide who starts.

Fairness is of curse not a mathematically defined concept; one could argue that a game that involves chance elements is intrinsically less fair than one that doesn't. If under the coin-flip rule I play two games against the same opponent, he might very well be lucky, and get the first-move advantage twice. Is that fair, just because I _could have_ gotten that advantage as well if I had been more lucky? Or is it just that it _could have been fair_ if I had been less unlucky (but is quite unfair in practice). Personally, I think it is much more fair to simply play two games with alternating colors, where white starts. Or in a tournament with only a single game per pairing, alternately give a player white and black.

What you are asking for is not fairness, but stability. The two have nothing to do with each ther. And indeed, Chess is by nature an unstable game, where advantages tend to grow with the move number. The quality of earlier moves has a higher weight in determining the outcome than the quality of later moves. Take away someone's Queen on move 3, and 'restoring' the balance by taking away the opponent's Queen on move 33 will almost certainly leave him in a lost position, as the opponent hs been hammering on him with an extra Queen for 30 moves.

So in practice the game will be decided in favor of the first player who has a stretch of luck in the first-move lottery. But even if the first coin flip would fully determine the outcome, it would still be a far game. Just a less interesting one.