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dreeves comments on Decision Auctions aka "How to fairly assign chores, or decide who gets the last cookie" - Less Wrong Discussion
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In practice the sealed-bid version seems to be ungameable, at least for us! None of the problems you mentioned have arisen. My parents have tried this and had more problems but as far as I could tell it always involved contention about what to consider to be joint 50/50 decisions. Bethany and I seem to have no problem with that, using the heuristic of "when in doubt, just call it a 50/50 decision and yootle for it".