Adrian Fry L&S Math & Physical Sciences
General Nash Equilibrium for the Multi-Player Dollar Auction
The dollar auction was developed by economist Martin Shubik as an example of how immediate decisions which seem reasonable can lead to irrational choices in the long run. The form of the auction is as follows: A dollar bill is auctioned to n players who may bid in ascending five cent increments. In a typical round, any player except for the current highest bidder can place a bid which is five cents higher than the current highest bid. If multiple players bid at once, the bid is assigned to one bidder uniformly at random. The dollar bill is given to the highest bidder once there is a round in which no players bid. Unlike traditional auctions, however, every other player must also pay their highest bid!
In practice, bidders will often bid over a dollar in order to minimize their losses- for example if the current highest bidder is at a dollar, the bidder who is at 95¢ may want to bid $1.05 in order to lose only a nickel (rather than 90¢ more).
This project will try to derive the Nash equilibrium strategy, meaning it will be a function which takes in a game state and a player’s current ranking among the other bidders, and returns a probability that they should bid.
Message To Sponsor
Thank you for your support. Funding will allow me to spend my summer making meaningful progress on this problem. Auctions like the one I am looking at have probabilistic strategies to accommodate the conditions on the problem, and I am excited to find one and prove that it is a subgame perfect equilibrium strategy, i.e., it is rational at every step. I am grateful for the opportunity to dedicate a significant portion of my summer towards applying the math I've learned to answer this problem.