Lecture 4 on Auctions
Multiunit Auctions
We begin this lecture by comparing
auctions with monopolies. We then discuss
different pricing schemes for selling
multiple units, the choice of how many
units to sell, and the joint determination of
price and quantity.
Are auctions just like monopolies?
Monopoly is defined by the phrase
“single seller”, but that would seem to
characterize an auctioneer too.
Is there a difference, or can we apply
everything we know about a monopolist
to an auctioneer, and vice versa?
How does a multiunit auction differ from
a single unit auction?
What can we learn about market
behavior from multiunit auctions?
Two main differences between
most auction and monopoly models
The two main differences distinguishing models of
monopoly from a auction models are related to the
quantity of the good sold:
1. Monopolists typically sell multiple units, but most
auction models analyze the sale of a single unit. In
practice, though, auctioneers often sell multiple
units of the same item.
2. Monopolists choose the quantity to supply, but most
models of auctions focus on the sale of a fixed
number of units. But in reality the use of reservation
prices in auctions endogenously determines the
number the units sold.
Other differences between
most auction and monopoly models
1. Monopolists price discriminate through market
segmentation, but auction rules do not make the winner’s
payment depend on his type. However holding auctions
with multiple rounds (for example restricting entry to
qualified bidders in certain auctions) segments the
market and thus enables price discrimination.
2. A firm with a monopoly in two or more markets can
sometimes increase its value by bundling goods together
rather than selling each one individually. While auction
models do not typically explore these effects, auctioneers
also bundle goods together into lots to be sold as
indivisible units.
Auctioning multiple units
to single unit demanders
Suppose there are exactly Q identical units of a good
up for auction, all of which must be sold.
As before we shall suppose there are N bidders or
potential demanders of the product and that N > Q.
Also following previous notation, denote their
valuations by v1 through vN.
We begin by considering situations where each buyer
wishes to purchase at most one unit of the good.
Open auctions for selling identical
units
Descending Dutch auction:
As the price falls, the first Q bidders to
submit market orders purchase a unit of
the good at the price the auctioneer
offered to them.
Ascending Japanese auction:
The auctioneer holds an ascending
auction and awards the objects to the Q
highest bidders at the price the N - Q
highest bidder drops out.
Multiunit sealed bid auctions
Sealed bid auctions for multiple units can be
conducted by inviting bidders to submit limit order
offers, and allocating the available units to the
highest bidders.
In discriminatory auctions the winning bidders pay
different prices. For example they might pay at the
respective prices they posted.
In a uniform price auction the winners pay the same
price, such as a kth price auction (where k could
range from 1 to N.)
Revenue equivalence revisited
Suppose each bidder:
- knows her own valuation
- only want one of the identical items up for auction
- is risk neutral
Consider two auctions which both award the auctioned
items to the highest valuation bidders in equilibrium.
Then the revenue equivalence theorem applies, implying
that the mechanism chosen for trading is immaterial
(unless the auctioneer is concerned about entry
deterrence or collusive behavior).
Prices follow a random walk
In repeated auctions that satisfy the revenue
equivalence theorem, we can show that the price
of successive units follows a random walk.
Intuitively, each bidder is estimating the bid he
must make to beat the demander with (Q+1)st
highest valuation, that is conditional on his own
valuation being one of the Q highest.
If the expected price from the qs+1 item exceeds
that of the qs item before either is auctioned, then
we would expect this to cause more (less)
aggressive bidding for qs item (qs+1 item) to get
a better deal, thus driving up (down) its price.
Multiunit Dutch auction
To conduct a Dutch auction the auctioneer
successively posts limit orders, reducing the limit
order price of the good until all the units have been
bought by bidders making market orders.
Note that in a descending auction, objects for sale
might not be identical. The bidder willing to pay the
highest price chooses the object he ranks most
highly, and the price continues to fall until all the
objects are sold.
Clusters of trades
As the price falls in a Dutch auction for Q units, no one
adjusts her reservation bid, until it reaches the highest bid.
At that point the chance of winning one of the remaining
units falls. Players left in the auction reduce the amount of
surplus they would obtain in the event of a win, and
increase their reservation bids.
Consequently the remaining successful bids are clustered
(and trading is brisk) relative to the empirical probability
distribution of the valuations themselves.
Hence the Nash equilibrium solution to this auction creates
the impression of a frenzied grab for the asset, as herd like
instincts prevail.
Why the Dutch auction
does not satisfy the conditions
for revenue equivalence
We found that the revenue equivalence theorem applies
to multiunit auctions if each bidder only wants one item,
providing the mechanism ensures the items are sold to
the bidders who have the highest valuations.
In contrast to a single unit auction, the multiunit Dutch
auction does not meet the conditions for revenue
equivalence, because of the possibility of “rational
herding”.
If there is herding we cannot guarantee the highest
valuation bidders will be auction winners.
Summary
We began this lecture by comparing auctions with
monopoly, and establishing some close connections.
We found the revenue equivalence theorem applies to
multiunit auctions if each bidder only wants one item.
Intermediaries exploit their monopolistic position, by
creating a wedge between their buy and sell prices.
Although fixed price monopolies create inefficiencies,
by restricting supply, perfect price discriminators
produce where the lowest value consumer only pays
the marginal production cost, an efficient outcome.