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 secondhighest 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.
<http://www.math.hmc.edu/funfacts>.
