The Dismal Science...

Today's golden-oldie re-run I first saw on Greg Ross's delightful "Futility Closet" site:

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....why rational choices ain't always so rational:

Another lovely puzzle/paradox today from Greg Ross's "Futility Closet" volume. It's known as the "dollar auction" paradox created by economist Martin Schubik. The setup (I've adapted from Wikipedia):

An auctioneer is to auction off a single dollar bill with the following rule: the bill goes to the highest bidder, AND the second-highest bidder LOSES the amount that they bid (to the auctioneer). The winner could gain a dollar for say 20 cents, for example, but only if no one else bids higher. The second-highest bidder is the biggest loser since they pay out their bid and get nothing in return.

The opening, minimum bid is 5 cents (with 5-cent increments thereafter) from one player, who would make a 95-cent profit if no one else bid. But it's sensible for another player to bid, say 10 cents, and still make a 90-cent profit. Then similarly, another bidder may now bid 15 cents, making 85-cents profit.

Whoever is the second-highest bidder at any point in time will wish to convert his potential loss to a gain by bidding higher than the highest-bidder, and so on. Obviously, if this keeps up, at some point, the dollar will COST someone a dollar to purchase -- but at least they will suffer no loss, while the 2nd highest bidder will lose 95 cents, giving them an incentive to bid $1.05 and thus decrease their loss to a nickel... at which point, the other bidder loses a whole dollar... and on and on. Bids beyond $1.00 mean that both top bidders lose money, thus minimizing the amount of loss then becomes the focus. A series of rational bids will reach and ultimately surpass the one dollar point, as the bidders seek to minimize their losses. Thus, "rational" bidding leads inevitably to both the two highest bidders losing money (while the auctioneer makes out well).

No wonder some call economics "the dismal science." ;-)

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Hahhh! I've pre-timed these re-run posts to run on forthcoming days, and on Monday (4/13) noticed that Kyle Evans entered The Aperiodical's "Big Math-Off" contest with a British version of this very same puzzle:
https://aperiodical.com/2020/04/the-big-lockdown-math-off-match-2/