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Yobi comments on Rationality Quotes March 2012 - Less Wrong
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I read this as implying that the loser is the one who makes the last mistake — the mistake that allows his opponent to win.
But yeah, I think the quote is kinda sloppy — it assumes that the opponents take turns in making mistakes.
This is true if you only count as mistakes moves which turn a winning position into a losing position, as gRR said elsethread. (I think I picked up this meaning from Chessmaster 10's automatic analyses, and was implicitly assuming it when reading the Tartakower quote.)