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Given that we are experimenting here, probably let's restrict Black from making the extra move with the same pawn. all we want to do here is give Black nearly equal chances of beating White. Re your second point, fine (except that I believe that Black will always have space for an extra move). I suppose in the last stages, Black may be trying for Stalemate.
That's my thinking. I honestly doubt that it would give Black too great an advantage; in fact, I'm not sure it goes far enough to repair the imbalance.
Could we improve Black's chances still further by giving Blkack an extra move, whenever he takes a piece (and not a pawn). So black might get, say, 6 extra moves, which might make all the difference.
(Cacher) Si tout à coup le site apparaît dans une langue étrangère, cliquez simplement sur le drapeau correspondant à votre langue pour rétablir la situation. (pauloaguia) (Montrer toutes les astuces)