The holiday season is approaching, bringing with it the famous prize games, either homemade or organized by associations, which vary across the world. Even in games involving small amounts of money, profound mathematical (and sociological) problems can arise.
One such problem is the famous “one-dollar problem.”
This isn’t a cheap version of the famous seven Millennium Problems; the one-dollar problem originates from a simple “social” game. An auctioneer offers a dollar to the highest bidder. Bids must increase by at least 5 cents. However, unlike standard auctions, there’s an additional rule: the second-highest bidder must still pay their bid.
For example, if Eve offers her dollar, Bob bids 10 cents, and Alice bids 15 cents, Alice will win the dollar if everyone stops bidding. Alice will pay her 15 cents, while Bob must still pay his 10 cents. If a third participant, Charlie, enters and bids 20 cents, and no one raises the bid further, Charlie will win the dollar for 20 cents. Alice will still have to pay her 15 cents, while Bob, being the third-highest bidder, won’t pay anything.
This small additional rule, requiring the second-highest bidder to pay, seems harmless but generates true paradoxes where participants bid far more than one dollar to try to win it.
The social mechanism at play begins with a desire not to lose face: when two acquaintances compete for the prize, giving up might be seen as a sign of weakness. Additionally, a sense of having already invested too much to quit develops. For instance, if someone bids 70 cents and gets outbid, they might think, “I’ve already invested 70 cents; adding another 10 to bid 80 cents is a small cost compared to winning the dollar.” However, this short-sighted and opportunistic reasoning perpetuates itself in a cycle.
In its simplicity, this model reflects much more complex and frequently occurring real-life situations. It was extensively studied by mathematician and sociologist Martin Shubik, who named it “The Dollar Auction Game.” In various studies, it was observed that, in many cases, bids could even exceed $20!
This small problem is a lesson in decision-making: we should not base our choices solely on how much we’ve already invested. Instead, we must look ahead and consider whether continuing will truly improve the situation.
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