You and I spot a $20 bill on the street. To divide it, we agree to an auction: Each of us will write down a bid, and the high bidder will keep the $20 but pay the amount of his own bid to the other player. If we submit the same bid then we’ll split the $20. What should you bid?
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If you bid too low (say, $1), then in the worst case I’ll bid a penny higher (leaving you with $1.01).
If you bid too high (say, $19), then in the worst case you’ll win the auction but have to give me most of your winnings.
Bidding $10 guarantees you at least $10 profit: If I bid higher than I’ll have to pay you my bid, and if I bid lower then you’ll win the $20 and have to pay me only $10.
But bidding $9.99 also guarantees you $10 if I bid higher — and it nets you an extra penny if I bid lower. That’s your best bet.
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