Suppose you are hosting a silent auction to sell your antique car. The rules are: (1) prospective buyers bid for your car by placing their bids in sealed envelopes, (2) then after collecting all bids, you sell the car to the highest bidder for the price that he bid.
This may seem like a reasonable way to conduct a silent auction, but there is a risk involved for you, the seller. If your car is highly valuable, but all buyers think they are the only ones who recognize that it is, they may actually make bids that are LESS than they think the car is actually worth. As a result, your bids will be lower, and you suffer.
Are there rules for a silent auction that will induce people to bid what they think an object is truly worth?
The answer is yes, and is given by a Vickrey auction. The rules specify that each player make bids, and the car goes to the highest bidder, but at the second-highest bid!
Can you figure out why this induces people to bid truthfully?
Presentation Suggestions:
This may be fun for students to ponder, or try as a group experiment, before discussing the answer. Additionally, you could ask students to think about what behavior other kinds of bidding rules would induce.
The Math Behind the Fact:
Here is the reason it works— we’ll show that, when holding all other bids fixed (and unknown to a given bidder), that bidder’s optimal strategy is to bid what she thinks the car is worth.
Suppose the bidder’s name is Alice. Let V be the amount that Alice thinks the car is actually worth, and B the bid that she actually makes. Let M be the maximum of all other bids. If M is more than V, then Alice should set her bid B less than or equal to V, so that she does not get the car for more than she thinks it is worth. If M is less than V, then Alice should set B=V, because if she bids any less, she will not get the car any cheaper, and she may lose the car.
The study of mathematical models for decision making is called game theory.
How to Cite this Page:
Su, Francis E., et al. “Vickrey Auction.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
References:
Morton Davis, The Art of Decision Making, Springer-Verlag, 1986.
Fun Fact suggested by:
Francis Su