Dollar auctions and giving-up time
The economist Martin Shubnik (1971) is responsible for a fiendish yet rather entertaining game called the dollar auction. I recommend trying the game next time you’re out with friends, so here are some brief instructions and a couple of things to look out for:
- Put a pound up for auction and invite some friends to bid for it (two works best)
- They may bid any amount and could technically buy the pound for just one penny but, unlike in traditional auctions, the agreement is that both winner and loser must pay
- Because the eventual loser must pay money in return for nothing, there is a large incentive for all parties to keep bidding once they have made their first bid
- The moment someone bids fifty pence you are guaranteed to earn your pound back on the next bid; any further bids represent pure profit
- If there is a bid of one pound, any further bid will commit the bidders to pay more than one hundred pence in order to win the pound! Whilst this is clearly irrational, on a bid-by-bid basis it is a more attractive option than paying a similar amount on a losing bid.
I was first introduced to this game when reading William Poundstone’s excellent book, Prisoner’s Dilemma (1993). Poundstone points out that we encounter similar scenarios all the time, such as when a company puts our phone call on hold or we wait in a queue for a ride at a theme park. In fact, examples are so exasperatingly common that I have found the term ‘dollar auction’ to be a depressingly useful bit of shorthand for explaining some of life’s most irritating encounters. However, it occurred to me recently that dollar auctions aren’t just a frustrating aspect of modern life, they are a frequent problem for foraging animals as they address the question: when to give up?
Giving-up time refers to the period between the last successful capture of a prey item and the time at which an animal decides to leave a foraging patch (McNair, 1982). Optimising giving-up time is an interesting alternative strategy to optimising within-patch time (also known as residence time). In both cases the animal must develop some idea of the distribution of resources within a patch and act accordingly, but in the case of giving-up time the animal need only keep track of the time since finding its last prey item when deciding whether to continue foraging in the current patch. This decision bears a striking resemblance to decisions in dollar auctions. For example, if a bird has been successfully foraging for ripe fruit in a tree but found nothing over the past couple of minutes, it has two options. It could leave the tree, accept that it has wasted time foraging for no return and incur the cost of travelling to the next tree in the hope of finding fruit more easily. Alternatively it could remain in the current tree in the hope that the recent difficulties don’t represent a severe overall decline in resources.
McNair, J. N. (1982). Optimal giving-up times and the marginal value theorem. The American Naturalist, 119, 511-529.
Poundstone, W. (1993). Prisoner’s dilemma: John von Neumann, game theory, and the puzzle of the bomb. Oxford University Press.
Shubik, M. (1971). The dollar auction game: a paradox in noncooperative behavior and escalation. The Journal of Conflict Resolution, 15, 109-111.