The dollar auction is a non-zero sum sequential game explored by economist Martin Shubik to illustrate how a short-sighted approach to rational choice can lead to decisions that are, in the long-run, irrational.
A series of short-term rational bids will reach and ultimately surpass one dollar as the bidders seek to minimize their losses.
[1] The game is a type of bidding fee auction which is a discrete version of the war of attrition.
Like these games, the dollar auction has a symmetric mixed strategy equilibrium (there are also asymmetric pure equilibria).
[2] The maximum bid follows a binomial distribution with mean 0.5 (recall the bidders make zero profits each, on average).