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A variation to fix first-move advantage

We all know how the white player has a higher chance to win at chess. But I think the gap could be almost entirely filled with a simple variant:

• One wins the game when the enemy king is captured and, at the end of the turn, her king isn't. If both kings are captured at the end of the turn then the game is a draw. No such a "check" thing exists nor stalemate.

Let's analyze the position after the (unrealistic) game {1. e4 e5 2. Qf3 Qh4 3. Bc4 Bc5 4. Kf1 Kf8 5. Qxf7}: {rnb2knr/pppp1ppp/8/2b1p3/2B1P2q/5Q2/PPPP1PPP/RNB2KNR w - - 1 5}.
In a normal chess game it would be mate. But this is not the case: the turn is not over since the black hasn't moved yet.
In fact, black could block the mate with {5... Qxf2 6. Qxf8 Qxf1 -> draw (or 6. Qxf2 Bxf2 7. Kxf2 -> white +3.21)}.

What do you think? I hadn't tried it out (not yet) but seems interesting.
I think white would still have a higher chance to win.
If you want to eliminate the first move advantage, then allow both sides to move simultaneously. (This, of course, will have its own set of problems, starting with how this scheme will be implemented.)

Back to your idea, @SineRequie, is this half-move leniency really enough to create a balance? Like ChessWhiz, I think white will still have a slight advantage.

On the other hand, what if all "games" are required to be played in two rounds? Both players will have a turn at being white.
#4 ...but then, would there be an advantage with being white FIRST?
Yes, that is the "fix" to the problem of white's advantage. Both players write down their moves, then reveal simultaneously. It's a strange game. :-)
@Toutatis, I presented 3 different ideas, all of which centered around the idea of eliminating the "first move advantage", the OP's topic.

I assume you were not referring to the first since the colors become irrelevant when both players are making moves simultaneously. (Like I wrote earlier, various implementations will have different problems so I don't really want to delve on it a.t.m., haha.)

The second one is just a supposition based on how white has the advantage in that black still can't just mirror the moves (to force a draw) ... even with a half-move leniency (among other factors that I'm not really smart enough to think of and list to exhaustion).

With the third, both players play two games, both having a turn at white, with the usual white-first-move advantage still intact.
Tbh, I don't think giving black an extra move at the end would change anything. Almost 0 games finish with mutual checkmating attacks where black gets mated, but could deliver checkmate as well if he'd get another move.

The reason why white has an advantage in chess is simply that black can't copy his opening for more than a few moves. E. g. after 1. e4 e5 2. Nf3 Nf6 3. Nxe5 Nxe4? 4. Qe2 Qe7?? 5. Qxe4 Qxe5?? 6. Qxe5 or 1. d4 d5 2. c4 c5?! 3. cxd5 cxd4?! 4. Qxd4 Qxd5?? 5. Qxd5, black would just lose the queen (without being able to capture white's queen).
So, instead of copying white's moves, black will always have to play a more passive move at some point. And this is how white can convert his single extra tempo into something more very quickly.
@Toadofsky, yes, simultaneous should be okay if both are up to it. Most of us mortals (average and worse players) would likely prefer series over parallel games though.

@ChessWhiz, not surprising that there's an existing method for it considering there are oddities like boxing-chess (where two people box for 3 minutes and play chess in between rounds).

@Dr_King_Schultz, in a manga called Historie that I'm following, there is a chess game where they have a prince instead of a queen. At some point, the player has the one-time option of abdicating the crown of the king to the prince to avoid checkmate. (This randomly came to mind when you mentioned the unlikely scenario of mutual annihilation in most games.)

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