There's a sport, supposedly popular in Afghanistan: two players bang their head together until one gives up. The other is declared the winner.
A dollar auction is the financial equivalent of that sport. You bid on an item and the bid costs you a dollar; the other guy bids, costs him a dollar; and it goes back and forth until one of you gives up. The price of the item doesn't change but it's attractive enough to make you start playing.
I tried, without any luck, to analyze the auction.
First, there's not much reason for the auction to ever end. Each round of the game is identical, except that each player is a dollar poorer and each player has learned that the other is determined (or foolish) enough to play this far.
Then I thought: well, maybe that's enough. Dollar auctions (this has been tried in real life) do end. Maybe one player just ran out of money or decided that the other player was crazier than he was.
Given that, is there any reason to start playing -- especially given that most dollar-auctions end with both players paying much more than the item is worth? Well, if you decide that no, it isn't worth playing, your opponent has a powerful reason to play: with no opposition, he can snap up the item for a single dollar!
But hey, you could be the one reason that your opponent has decided to forfeit -- or you can convince him to forfeit by boldly bidding the first time. Of course, he might treat that first bid as a bluff and bid back, but if you bid twice...
And it's on. The basic problem is that whatever strategy one player adopts, the other player has incentive to adopt the opposite strategy, and I haven't figured out any way to get off the merry-go-round. I'm open to suggestions.