## Game Theory applied to “Dollar Auctions”

Link:

http://www.math.toronto.edu/mpugh/Teaching/Sci199_03/dollar_auction_1.pdf

I found this academic paper to be very relevant to the concepts relating to game theory and auctions discussed in lecture. The paper describes the process of multiple people betting on a one dollar bill in 5 cent increments. It operates like a simple ascending bid auction, but there is a catch; not only does the top bidder pay his or her bid if they win, but the second highest bidder also pays his or her respective bid. Bidders start out bidding small amounts of change. However, it becomes interesting when the bidding approaches the one dollar mark; when one player bets one dollar for the dollar bill, the second player may choose to continue betting, and the total bid continues to climb.

If analyzed as a simple ascending bid auction, this behavior makes no sense. The value of a dollar bill should be one dollar for all bidders, and the dominant strategy in these types of auctions is to never bet more than your value for the item in question. However, the price tends to climb to over a dollar, which seems irrational. It is the requirement that the second bidder pay his or her bid that creates this paradox. In game theory, it is the goal of all active players to attempt to maximize the payoffs they receive from playing the game. In this case, however, once the dollar limit is reached, it becomes a game where the top two bidders attempt to minimize their losses. For example, if bidder A bets 95 cents, and then bidder B decides to raise the bid to a full dollar, bidder A now can make two choices. He or she can choose let bidder B win the auction. Bidder B would then have a net payoff of 0, but bidder A, due to the rules of the game, would have a net payoff of negative 95 cents. However, if bidder A chose to raise the bid, his or her new payoff would be negative 5 cents, while bidder B would have a new net payoff of negative one dollar. This thought process would continue for each player as they made successive bids, and the price begins to escalate for seemingly unreasonable values for a one dollar bill. The author of this paper notes that it’s common to see prices escalate to between three and five dollar for a one dollar bill. Eventually, one of the bidders would back off and accept losing their bid.

If this price escalation occurs, the auctioneer is the only one who benefits from this game. In fact, he or she can only lose if the bids stay below 50 cents for each player. Perhaps, if the players can realize what will happen, two players can agree to keep bids low and split the dollar among them, but if players compete to maximize their own profit at the beginning, this probably will not happen.  So if you’re a bit short on cash, consider auctioning a dollar.