Strategies in Playing the Dollar Auction
The dollar auction is a famous paradox in which you are willing to pay more than the actual value of the product. A 2-player game works as such: the auctioneer sells a dollar bill. Player who pays the most takes away the dollar, and the other player also loses his bid to the auctioneer and gets nothing. A comic by Ryan North found on qwantz.com illustrates the problem here:
As we can see, bidder 1 starts off with a small amount of money, where $0.00<b1<$1.00, hoping to get a profit of $1.00 – b1, but bidder 2 would offer b1<b2<$1.00 to get $1.00 – b2 profit. Now that bidder 2 would profit and bidder 1 would lose his bid, bidder 1 raises his bid, and so does bidder 2. Their bids get closer and closer to $1.00, and will eventually go over $1.00 because bidding $1.01 and losing 1 cent is better than losing all the $0.99. Trying to lose less, the bid will keep increasing, and the only winner is the auctioneer who takes away both bidders’ money by spending only a dollar.
Then what to do? Does playing the game always result in a loss? Here are some possible strategies for not losing in the dollar auction:
- The trivial solution – do not enter the game. However, this is not at equilibrium because now that your profit is $0.00. You can change your decision, bid slightly more than the other bidder, and gain more than $0.00 without the other bidder’s changing his decision. Therefore, not entering the game would not likely happen, but it is a way not to lose any money.
- Bid $0.99 before anyone else bids. This way, you can still earn a cent and it reaches an equilibrium that the other bidder cannot gain by bidding any amount. However, keep in mind that it is possible that the other bidder really hates you and wants you to lose the $0.99 by bidding $1.00 for the dollar. This will break the equilibrium and result in the loss we mentioned before.
- Bid $1.00 before anyone else bids. You will not lose any, and the other bidder will not bid. This reaches an equilibrium that any change in strategy of the other bidder will result in a loss for both of you. This strategy is very similar to the dominant strategy for the English Auction we mentioned in class, that “you should bid the value you think the product worths”. Unlike products we discussed in class, which everyone values differently, here, obviously, a dollar worths a dollar to everyone, so bidding $1.00 is safe and logical.
- Cooperate with the other bidder by promising him to share the profit with him. This violates the assumption we had for auctions, because the other bidder will know what value you are willing to pay before you bid. Plus, it is possible that the other bidder betrays you, hoping to get more than what you promised him. However, this can, if the other bidder trusts you, result in a maximum profit of $0.99.
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