A party game
Martin Shubik advised starting the auction when spirits are high, and wrote that total payments of three to five dollars for one dollar are not uncommon.
Bids rise in steps of 5 cents. The highest bid gets the dollar, but the second-highest also pays its amount and gets nothing. Nobody has to bid, and anyone may stop at any moment. How much will the seller collect in total for one bill? Lock in your answer, then check it with a game against bots, a simulation and the explanations.
Each single step really does look worth it: quitting means a sure loss of your bid, while 5 more cents buys a chance that the rival gives up. So the bids pass the dollar, and the seller pockets both. Max Bazerman and Don Moore sold a 20-dollar bill this way to students and managers for years, and the bidding typically ended between $20 and $70. Check it with the game, the simulation and the explanations.
The economist Martin Shubik described this game in 1971. He suggested playing it at a lively party with plenty of people.
The host sells an ordinary one-dollar bill. To everyone it is worth exactly one dollar, no more and no less.
Anyone may call a bid. Each new bid is at least 5 cents above the last. The auction ends when nobody raises any more. Nobody has to take part, and anyone may stop at any time.
The highest bid gets the dollar and pays its amount. The second-highest bid also pays its amount to the seller but gets nothing. Everyone else pays nothing.
How much will the seller collect in total for one dollar bill when ordinary people bid?
"Nobody pays more than a dollar for a dollar"
So it seems before the first bid. But a bidder at 95 cents against a dollar no longer chooses "buy a dollar for more or not" but "lose 95 cents for sure, or risk 10 more". The second looks better, and so it goes at every step, above a dollar too.
Shubik recommended a large crowd: people only start calculating once two bids have been made. From then on the top two decide everything, so here we start with two.
Such auctions have been run in classrooms and at parties with real people for decades. How much do the two top bidders usually pay in total for a one-dollar bill?
Pick an answer. Data from real experiments and a simulation of 10,000 auctions between our bots will appear here.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
Suppose the bids have reached 95 cents for you and a dollar for your rival. What do you lose if you give up now? And if you raise to $1.05 and the rival gives up?
Now look through the rival's eyes one step later. Is there anything in this reasoning that stops the bidding at exactly one dollar?
The seller gets both final bids. How high must the last bid be for the two together to pass a dollar? And how often does the bidding stop below that?
You bid for a dollar against five bots in turn. The bot opens with 5 cents, then you take turns raising by 5 cents. At each step decide: raise or give up. The bot's budget is secret; you only see its temper. At the end of the series you get your score and every bot's budget.
| # | Bot | Budget | Your result |
|---|
Two random bots bid against each other. The seller gets both final bids.
Set a rule in advance: "I keep raising while my bid stays within the limit". A limit of 0 means "I never enter". The simulation runs thousands of auctions against the chosen bots and averages what the rule gains. The line is the exact calculation for every limit, the dots the simulation.
The first layer has no formulas, the second has a chart, a table and formulas, the third covers the rational solution and the game's history.
“In plain words” is hidden on all puzzles.
You have bid 90 cents, the rival 95. If you give up, you lose 90 cents for sure. If you raise to a dollar and the rival gives up, you break even. Giving up is worse, so you raise.
Now the rival sees the same: giving up means losing 95 cents, raising to $1.05 means losing only 5. At every step, above a dollar too, one more bid looks better than a sure loss. This card explains why the bidding passes a dollar, but not where it stops: that depends on how much each is willing to spend. The next cards cover that.
The seller does not need anyone to pay more than a dollar for the dollar. It is enough that the two final bids together pass a dollar, which happens as soon as the last bid is above 50 cents. Here the winner even made 35 cents, and the seller is still 25 cents up.
Your earlier bid is already gone: it is your loss whatever you do next. A sound decision looks only ahead: how much more you must pay and what the chance is that the rival gives up. So deep into the auction carrying on can be genuinely right, and the rival reasons the same way. The mistake came earlier: when both entered without knowing who would last longer.
Against a cautious bot that never goes above $1.20, it pays to raise all the way: you always outlast it and make per auction on average. Against a stubborn one, with a budget of up to $3, it pays to stay out. And against a bot of unknown temper the worst is to stop halfway, around $1.40: you have already bid a lot, yet you rarely outlast it.
“Rigorous” is hidden on all puzzles.
Your bid trails by 5 cents. Raising costs 10 more cents: your bid goes from b − 5 to b + 5. If the rival gives up right away with probability q, you get the dollar. Compare "raise, then give up" with "give up now": the difference is q · $1 − $0.10. Raising pays when q ≥ 10%, and your earlier bid does not enter this difference at all.
| Your bid limit | Cautious | Gamblers | Stubborn | Any |
|---|
The best row in each column is underlined. Against an unknown bot, every limit from 10 cents to $3 loses money on average; you lose least by staying out. But this is no law of nature: against a rival you are sure to outlast, the same auction makes money.
One raise: 10 extra cents against a chance q of getting the dollar. The sunk bid b − 5 is in both options and cancels out.
The chance the bot gives up after your bid y, given it has just bid y − 5: either its budget ran out or it quit with its own probability p. The budget B is uniform from L to H.
The gain of the rule "limit X": a sum over all budgets and all moments someone gives up. A win at your bid y gives 100 − y; a loss takes your last bid.
The seller's takings: both final bids. They differ by 5 cents, so the takings exceed a dollar as soon as the last bid is above 50 cents.
“Deeper” is hidden on all puzzles.
In 1986 the political scientist Barry O'Neill solved the game for two bidders with equal known budgets, bids in multiples of 5 cents (jumps allowed), and the rule "when it makes no difference, I don't bid". The answer: the first bidder opens with a precisely chosen bid, and the second never enters. With $2.50 budgets it is 60 cents, as the paper itself says. Here backward induction finds the same answer by working through every possible auction from the end.
Why it works. Call a bid a "fortress" if the rival gains nothing by topping it: the budget minus 95 cents, minus $1.90 and so on. Whoever stands on a fortress cannot be profitably overtaken, because taking the dollar would force the rival all the way up to your budget. The opening bid is the lowest fortress. The formula: the opening bid in 5-cent steps is ((b − 1) mod 19) + 1, where b is the budget in steps and 19 is a dollar minus one step.
O'Neill's solution rests on both knowing each other's budgets and both calculating perfectly. O'Neill himself concludes that what makes the dollar auction a trap is not the logic of the game but some irrationality in the bidders. Once budgets are secret, as with our bots, the opening bid no longer guarantees anything.
In 1989 Wolfgang Leininger showed that if bidders may randomise their moves, the game also has equilibria with escalation: the bidding drags on without either player making any mistake.
Martin Shubik described the game on three pages in the Journal of Conflict Resolution in 1971. He wrote that experience shows a dollar can be sold for considerably more than a dollar, and that total payments of three to five dollars are not uncommon.
The psychologist Allan Teger devoted a book to such auctions, "Too Much Invested to Quit" (1980): bidders enter because they don't expect the bids to pass a dollar, then hold on to justify what they have already put in.
First the maths (the all-pay auction and the war of attrition), then the logic (escalation of commitment), then practice (online penny auctions).
n bidders write bids in sealed envelopes for a $1 prize at once. Everyone pays; the highest bid gets the prize. They play the equilibrium random strategy.
The dollar auction belongs to a big family: all-pay auctions and wars of attrition. In the latter, described by the biologist John Maynard Smith in 1974, two animals contest a resource by display: whoever lasts longer gets it, and both spend the time.
In the symmetric equilibrium of the all-pay auction each bids a random amount with distribution F(x) = x1/(n−1), and on average each breaks even. So together they pay exactly the prize on average: $1. Competition moves the bidders' whole gain into the seller's pocket, even when everyone plays perfectly.
A project has already cost X. Finishing needs 4 million more, and once finished it will bring in 3. Change what has been spent.
However much has been spent, finishing brings 1 million less than stopping. But the more has been spent, the harder that is to admit. In 1976 the psychologist Barry Staw gave business school students a role-play of allocating research and development money: those who had personally chosen a direction that then failed put even more into it. He called this escalation of commitment.
The same year Richard Dawkins and Tamsin Carlisle named this error the "Concorde fallacy", after the supersonic airliner that Britain and France, as the story goes, kept funding because they had already put in too much. The dollar auction is the same trap, only with a rival who drives you into it with every bid.
Each bid costs the bidder $0.75, non-refundable, and raises the price by $0.15. The winner pays the final price. The lot: a $100 gift card.
Each bid here costs 5 times what it adds to the price, so the seller gets six times the final price. It is already ahead when the $100 card "sells" for just $16.80. The economist Ned Augenblick studied 166,000 such auctions on the Swoopo site over four years: the seller's average margin exceeded 50%, and in auctions for cash the takings averaged 204% of face value.
And just as in the dollar auction: the more bids a player has already made, the less likely they are to leave. In Augenblick's data a player with only a few bids had more than a 10% chance of leaving at the next one, while one with hundreds of bids had less than 1%. The same logic keeps long projects and patent wars going: "we've invested too much to stop".
Martin Shubik advised starting the auction when spirits are high, and wrote that total payments of three to five dollars for one dollar are not uncommon.
Max Bazerman and Don Moore sold a $20 bill to students and managers. The bidding typically ended between $20 and $70 and now and then reached $100. Over twenty years this came to more than $30,000, which went on food and drinks for the class or to charity.
Barry O'Neill worked out that with known budgets of $2.50 each, the first bidder should bid exactly 60 cents and the second should keep quiet. He compared the dollar auction with the escalation of international conflicts.
On the penny auction site Swoopo, more than 2,000 auctions for cash prizes brought in 204% of face value on average. Even so, Swoopo went bankrupt in 2011: rivals copied the model easily.
The name "Concorde fallacy" was coined by the biologists Richard Dawkins and Tamsin Carlisle in a Nature paper on parental investment in animals: an animal that has already invested a lot in its young need not invest more because of it.
In 13 million bids on Swoopo, Ned Augenblick saw that a player with a few bids left the auction with probability above 10%, while one with hundreds of bids left with less than 1%. Sunk costs kept people in the game.
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