For simplicity, define the object of the game of chess to be to capture the opponent's king. (In contrast to the conventional rules which define the end of the game at checkmate where king capture is inevitable but hasn't actually happened.)
Modify the rules of chess so that both players are permitted the same number of moves.
By the modification, black is permitted one move after his or her king is captured. White is not, because he or she moved first. If Black's "extra" last move captures White's king, then the game is drawn by king exchange.
White's first move advantage is greatly reduced, perhaps too much. Black may respond to white giving check by protecting the Black king or by attacking (giving check to) the White king.
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