## Tuesday, November 15, 2011

### Martin Shubik's Dollar Auction Game

Martin Shubik's endowed chair at Yale University is the Seymour Knox Professor Emeritus of Mathematical Institutional Economics. Most of the time, "mathematical" and "institutional" economists are separate people. But Shubik's career has combined both deep mathematical insights about strategic behavior and also applications to financial, corporate, defense, and other institutions. The opening pages of the just-arrived October 2011 issue of the American Economic Review offer a short tribute to Shubik, who was named a Distinguished Fellow of the American Economic Association in 2010. A list of past Distinguished Fellows is here; the short description of Shubik's work from the AER is here.

One of my favorites of Shubik's papers, in part because it is so accessible that it can readily be used with introductory students and in part because it gives a sense of how his mind works, is the Dollar Auction Game. The Dollar Auction Game is in some ways similar to the better-known prisoner's dilemma, because it illustrates how two parties each pursuing their own self-interest can end up with an outcome that makes both of them worse off. The first published discussion of the game is in the Journal of Conflict Resolution, March 1971, pp. 109-111, which is not freely available on-line but can be found on JSTOR.

The rules of the Dollar Auction are deceptively simple: "The auctioneer auctions off a dollar bill to the highest bidder, with the understanding that both the highest bidder and the second highest bidder will pay. For example, if A has bid 10 cents and B has bid 25 cents, pay a dollar to B, and A will be out 10 cents."

Now consider how the game unfolds. Imagine that two players are willing to bid small amounts, enticed by the prospect of the reward. Then the logic of the game takes hold. Say that player A has bet 20 cents and player B has bet 25 cents. Player A reasons: "If I quit now, I lose 20 cents. But if I bid 30 cents, I have a chance to the \$1 and thus gain 70 cents." So Player A bids more. But the same logic applies for Player B: lose what was already bid, or bid more.

But if both players continue to follow this logic, they find their bids steadily climbing past 50 cents apiece: in other words, the sum of their bids exceeds the dollar for which they are bidding. They approach bidding \$1 each--but even reaching this level doesn't halt the logic of the game. Say that A has bid 95 cents, and B has bid \$1. Player A reasons: "If I quit now, I lose 95 cents. But if I bid \$1.05, and win the dollar, I lose only 5 cents." So Player A bids more than a dollar, Player B, driven by the same logic, bids higher as well.

Apparently, Shubik and his colleagues liked to play these games at parties. As he writes in the 1971 article: "In playing this game, a large crowd is desirable. Furthermore, experience has indicated that the best time is during a party when spirits are high and the propensity to calculate does not settle in until after at least two bids have been made. ... Once two bids have been obtained from the crowd, the paradox of escalation is real. Experience with the game has shown that it is possible to "sell" a dollar bill for considerably more than a dollar. A total of payments between three and five dollars is not uncommon. ... This simple game is a paradigm for escalation. Once the contest has been joined, the odds are that the end will be a disaster to both. When this is played as a parlor game, this usually happens."

With any game of this sort, two sorts of questions arise: Under what conditions can the players sidestep the escalation? And does this simple game address real-world phenomena?

Of course, the two players can avoid the escalation if they communicate with each other, agree not to increase their bids, and perhaps also agree to split the gains. It may be necessary in this situation to enforce this agreement with a threat: for example, I will stop bidding and you will also stop bidding, but if you bid again, I will immediately jump my bid to \$1, and force us both to take losses. Or unless you stop bidding, I vow to bid forever, no matter the losses. Of course, whether these threats are credible and believable would be an issue.

Another exit strategy from the game is for one player to see where the game is headed, and to stop bidding. Notice that by bailing out of the game, the player who stops is a "loser" in a relative sense: that is, the other player gets the dollar. But by bailing out sooner, the player who stops actually prevents both players from further escalation and ending up as even bigger losers. A more extreme version of this strategy is that a player may refuse to follow the rules, perhaps declaring the game to be "unfair," and refuse to be bound by paying that player's previous bid. To avoid this possibility, perhaps the bids would need to be handed to the auctioneer as the bidding proceeds.

Again, the bottom line of the Dollar Auction game is to illustrate a simple setting in which self-interested behavior leads to losses for both players--in this case, to escalating losses until one of the players decides that enough is enough.

The Dollar Auction Game is simple enough that it doesn't fit perfectly with real-world situation. However,
John Kay argued in a Financial Times op-ed in July that many of our decisions in the last few years about bailing out financial institutions and now countries like Greece have a "dollar auction" aspect to them, in the sense that governments keep thinking that if they just make one more bid, they will have gains--or at least they will reduce the size of their losses. What the governments don't seem to realize is that the other parties in the economy will keep making another bid as well, forcing the government to make yet another bid.

Perhaps the deeper wisdom here is that when entering into a competitive situation, it's useful to look ahead and have a clear vision of what the end-game would look like. If you find yourself in a situation of escalation, by all means try instead to negotiate for a way in which both sides can combine a strategy of lower bids and bid-curdling threats--and then end up sharing the prize. But if negotiation seems impossible, and appealing to the rationality of the other player doesn't5 work it is better to bail out from the bidding rather than continue escalation with an irrational player. Better to take the immediate loss, and to let the less-rational player win the dollar, than to build up to larger losses. As John Kay writes: "In the dollar bill auction, one party eventually scores a pyrrhic victory and takes possession of the dollar bill. Both parties lose, but the smaller loser is the person who sticks out longest. That is not usually the rational player."