Here is a game for two players. A dollar bill is up for auction. Bids go up in 5 cent steps, and the highest bidder wins the dollar. There is one twist: the second-highest bidder pays their bid too, and gets nothing. Economist Martin Shubik described it in 1971, and he reported that at parties it often ended with the two bidders paying a combined $3 to $5 for a single dollar.
You have a $5 budget. You have just opened with 5¢. Your rival has a secret budget of their own, and will keep topping you until it runs dry. Try to lose as little money as you can.
The white line marks the $1 prize. Your bid is the green bar.
Suppose the rival bids 95¢ and you hold 90¢. If you quit, you are out 90¢ for nothing. If you bid $1.00 and win, you get a dollar back, so you are out nothing at all. Raising looks clearly better than quitting. The rival then reasons the same way at $1.00 versus your $1.05, and so on.
Past a dollar, the logic does not change. Each new bid compares the cost of quitting (your whole bid, lost) with the cost of one more step (a small extra loss).
| You hold | Quit now | Raise 5¢ and win |
|---|---|---|
| $0.90 | -$0.90 | -$0.05 (win $1 for $0.95) |
| $1.50 | -$1.50 | -$0.55 (win $1 for $1.55) |
| $3.00 | -$3.00 | -$2.05 (win $1 for $3.05) |
At every row, raising loses less than quitting, if you are sure it wins. That "if" is the trap. The rival is doing the same sums, so nobody ever wins cleanly.
The money you have already bid is gone whether you stop or carry on. Economists call it a sunk cost, and the rational move is to ignore it. In this game you can't ignore it, because quitting is the only moment you have to admit the loss. Each player is trying to avoid being the one who "lost" most obviously.
Game theorists have worked out that the fully rational answer is not a bid at all but a mix: each player should drop out with a small probability at every step, so the escalation is cut off at random. Played that way, the average loss to each player is around 50¢. The practical advice is simpler: before you bid, decide the most you would pay for the dollar, and stop there.
The same structure turns up whenever the loser pays too: lawsuits where both sides pay legal costs, price wars, arms races and lobbying contests. In each, the prize is fixed but the spending is not. The cheapest strategy is often to not play, or to agree a limit with the other side before the bidding starts.