Download Lecture 4 - ComLabGames

Lecture 4
Multiunit Auctions and Monopoly
The first part of this lecture put auctions in a more general
context, by highlighting the similarities and differences
between auctions and monopolies.
In this spirit we investigate the sale of multiple units by
auction, to see when the selling mechanism affects the
outcome, and how.
The latter parts of the lecture analyze other aspects of
monopolistic practices. We discuss mechanisms for setting
prices and quantities, the role of commitment, market
segmentation, and product bundling.
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?
We now begin to make the transition
between auctions and markets by noting the
similarities and differences.
Two main differences between
most auction and monopoly models
The two main differences distinguishing models of
monopoly from 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.
An agenda
for the first part of the lecture
We will focus on three issues:
1. How does a multiunit auction differ from a single
unit auction?
2. What can we learn about market behavior from
multiunit auctions?
3. How does a uniform pricing monopolist set price
and quantity?
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 will begin by considering situations where each
buyer wishes to purchase at most one unit of the good.
Decisions for the seller to make
in multiunit auctions
The seller must decide whether to sell the
objects separately in multiple auctions or jointly in
a single auction.
The seller must choose among different auction
formats.
Open auctions for selling identical
units
Descending Dutch auction:
Suppose the auctioneer has five units for
sale. As the price falls, the first five 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 five highest
bidders at the price the sixth bidder drop
out.
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.
Example: Descending price auction
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.
Example: Japanese auction
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,
either at the respective prices they posted, or at some
common price that all the buyers have indicated they
are willing to pay.
It could invite sealed bid offers from customers in a kth
price auction (where k could range from 1 to N.)
Example: Multiunit sealed bid
uniform price auction
Example: Multiunit sealed bid
discriminatory price auction
Repeated English auction
Prices follow a random walk
In the repeated Dutch and English auctions,
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.
Revenue equivalence revisited
Suppose that each bidder:
- knows her own valuation, or alternatively has an
independent signal about her valuation drawn
from the same probability distribution
- is risk neutral
Consider two auctions that have the same allocation
mechanism (the mapping from the valuations to the
winner(s) of the auction.
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).
Multiunit demanders
By a multiunit demander we mean that each bidder
might desire (and bid on) all Q units for himself. We now
drop the assumption that N > Q.
Relaxing the assumption that each bidder demands
one unit at most seriously compromises the applicability
of the Revenue Equivalence theorem.
Typically auctions will not yield the same resource
allocation even if the usual conditions are met (private
valuations, risk neutrality, lowest feasible expects no rent
from participation).
Example: Two unit demanders
in a third price sealed bid auction
Consider a third price sealed bid auction for two units
where there are two bidders, each of whom wants two
units. Thus N = Q = 2. Each bidder submits two prices.
We suppose the first bidder has a valuation of v11 for
his first unit and v12 for for his second, where v11 > v12
say. Similarly the valuations of the second bidder are v21
and v22 respectively, where v21 > v22.
Example continued
 The arguments given for single unit second price
sealed bid auctions apply to the highest price of each
bidder. One of his prices is highest valuation.
 There is some probability that each bidder will win
one unit, and in this case the price paid by one of the
bidders will be determined by his second highest bid.
Recognizing this in advance, he shades his valuation on
his second highest bid.
Vickery auctions defined
A Vickery auction is a sealed bid auction, and units are
assigned according to the highest bids (as usual).
To calculate how much each bidder pays for the unit(s) he
has won, we define the losing bids he displaced. The losing
bids he displaced would have been included within the
winning set of bids if the bidder had not participated in the
auction, and everybody else had submitted the same bids. In
a single unit auction this corresponds to the second highest
bidder.
The total price a bidder pays in a Vickery auction for all
the units he has won is the sum of the bids on the units he
displaced.
Vickery auctions are efficient
A Vickery auction is the multiunit analogue to a
second price auction, in that the unique solution (derived
from weak dominance) is for each bidder to truthfully
report his valuations.
This implies that a Vickery auction allocates units
efficiently, in contrast to many multiunit auction
mechanisms.
Choosing quantity
When analyzing monopoly, an important issue is the
quantity the monopolist chooses to supply and sell.
Regulators argue that compared to a competitively
organized industry where there are many firms
supplying the product, a monopolist restricts the
supply of the good and charges higher prices to high
valuation demanders in order to make rents out of his
position of sole source.
Is this true in practice?
Reservation prices for auctions
One reason for an auctioneer to set a reservation
price is because of the value of the auctioned item to
him if it is not sold. This value represents the
opportunity cost of auctioning the item. For example
he might sell it at another auction at some later time,
and maybe use the item in the meantime.
Should the auctioneer set a reservation above its
opportunity cost?
Auction Revenue
What is the optimal reservation price in a private
value, second price sealed bid auction, where bidders
are risk neutral and their valuations are drawn from
the same probability distribution function?
Let r denote the reservation price, let v0 denote the
opportunity cost, let F(v) denote the distribution of
private values and N the number of bidders. Then the
revenue from the auction is:
N?1
K
F
r v 0 +NF
r 
1 ?F
r+N Xr v
N ?1
F
vN?2 F v
v
dv
N
Solving for the optimal reservation price
Differentiating with respect to r, we obtain the first
order condition for optimality below, where r0 denotes the
optimal reservation price.
Note that the optimal reservation price does not
depend on N.
Intuitively the marginal cost of the top valuation falling
below r, so that the auction only nets v0 instead of r0,
equals the marginal benefit from extracting a little more
from the top bidder when he is the only one bidder to
beat the reservation price.

ro ?v 0 
F v
ro 1 ?F
ro 
The uniform distribution
When the valuations are distributed uniformly with:
F
v 
v ?v 
/v ?v 
then:
o
r  v
+
v0 
/2
Designing a monopoly game
with a quantity choice
In the game below, the valuations of buyers are
uniformly distributed between $10 and $20.
Each buyer is endowed with $20.
The monopolist’s production capacity is 100 units of
the good. The marginal cost of producing each unit up
to capacity is constant at $10.
What is the equilibrium quantity bought and sold?
The game
A static approach
The traditional argument can be framed as follows.
Let:
c denote the cost per unit produced,
consumers demand quantity q(p) when the
price is p and the function
Assume q(p) is differentiable and declining in p, and
write p(q) as its inverse function. That is q(p(q)) =
q.
The monopolist chooses q to maximize
(p(q) – c) q
Marginal revenue equals marginal cost
Let qm denote the profit maximizing quantity supplied by
the monopolist. Then qm satisfies the first order condition
for the optimization problem, which is:
p(qm) + p’(qm) qm = c
The two terms on the left side of the equation comprise
the marginal revenue from increasing the quantity sold.
When an additional unit is sold it fetches p(q) if we ignore
any downward pressure on prices.
The traditional argument is that the monopolist will only
produce sell an extra unit if the marginal revenue from
doing so exceeds the marginal cost.
Demand schedule
In this example the marginal cost is $10.
Price
20
19
18
17
16
15
14
13
12
11
10
Quantity Revenue Marginal Total costs Profit
revenue
1
2
3
4
5
6
7
8
9
10
11
20
38
54
68
80
90
98
104
108
110
110
18
16
14
12
10
8
6
4
2
0
10
20
30
40
50
60
70
80
90
100
110
10
18
24
28
30
30
28
24
18
10
0
Eleven buyers and one seller
20
19
18
17
16
15
14
13
12
11
10
-
MC=10
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
|
10 11
q
Solution to game
There are two outputs that yield the
maximum profit, which is $30.
If the monopolist offers 6 units for sale,
the market will clear at a price of $15.
If the monopolist offers 5 units for sale,
the market will clear at a price of $16.
The solution illustrated
Price in dollars
20
inverse demand curve
Uniform price
solution
unit cost
10
marginal revenue curve
quantity
0
Uniform quantity
solution
Intermediaries with market power
We typically think of monopolies owning the property
rights to a unique resource. Yet the institutional
arrangements for trade may also be the source of the
monopoly power. Consider the NYSE, in which a dealer for
each stock sets the spread, and traders only have the
opportunity to make market buy and sell orders.
In auction terminology, each dealer is playing a multiunit
Dutch auction on both sides of the market, simultaneously
determining the units to be traded.
How should a dealer set the spread on the stock he
manages? A small spread encourages greater trading
volume, but a larger spread nets him a higher profit per
transaction.
Trading on a Dealer Market
A Summary
This first part of this lecture has emphasized the links
between auctions and monopoly, and thus established
the close connections between auctions and markets.
We discussed pricing multiunit sales, setting
quantities, segmenting the market to achieve
discrimination, the role of commitment in multiunit
settings, and product bundling which induce
consumers to self select product packages.
As we noted in our application to the dealer markets,
the lessons form auction theory carry over to more
complicated settings than single unit auctions, but
nevertheless they are harder to apply.