Price discovery
Market models in financial mathematics often assume that buyers and sellers in the market
act as "price takers". This assumption is reasonable, for example, when
the number of buyers and sellers is big enough so that nobody believes
his actions will influence prices. This hypothesis is known as the
Competitive
market hypothesis. There exist however many
examples when this assumption does not hold and it becomes crucial to
explain where prices come from and the influence of players' actions on
them.
Auction theory provides one
explicit model of price making (ignoring bargaining
aspects of the process).
For instance, the New York Stock Exchange (NYSE) trades everyday in a
continuous time auction format. Specifically, buyers and sellers of a
determined stock submit an offer price and asking price, respectively
(see
double auction).
Whenever the ask price matches the offer price a transaction occurs and
the price of the transaction is the
market price
(quote). Most of the times the bid price is below the offer price. In
this case the difference between the asking price and offer price is
known as the
bidask spread.
This quantity measures the level of
liquidity of
the stock. For example, suppose that the market price of a share of
stock of ABC is $10 (the last price at which a transaction took place).
The maximum bid price is $9.9 and the minimum asking price is $10.1. In
this case the bidask spread is $0.2. Suppose that one second later the
market price increases to $10.05 (a transaction occurred at this
price). In this case the buyer with bid price $9.9 may be forced to
reconsider and offer a higher price, which tightens the bidask spread
(see figure). The transaction at $10.05 indicates that the stock is
more liquid and hence the bidask spread should be smaller.
Regarding the Competitive market hypothesis, it is important to observe
that most of the theory of
Mathematical
Finance has been developed by assuming this hypothesis to
hold. When this assumption is dropped the models become more complex.
One way to approach noncompetitive markets is by specifying in the
model, the price of the assets as a function not only of time but also
of the size of order. In other words for every asset in the market we
would have a family of price processes,
S(t,x) that
model, from the perspective of a large investor, the price of the asset
at time
t if
x shares of the
asset are bought (
x>0) or sold (
x
< 0). There are many assumptions that have to be made
in order to get nice results. The most important one of them is the
fact that as
x increases the price
S(t,x)
should either remain unchanged or increase as well (see figure).
This
accounts
for the fact that if a large number of shares have to be bought, since there is a limited ammount of cheap shares, the investor will have to
pay more on average. It excludes however, a phenomenon often seen in consumer products
when buyers get discounts for purchasing large quantities of a product.
Auction market
A more practical reason for studying auctions is that the
value and diversity of goods exchanged in
the auction market is huge. For instance in 2007 the total
revenue from goods and services sold at live auction was $270.7
billion. Among the many goods exchanged in auctions, excluding online
auctions, are: antiques, collectibles, wines, real estate property,
commodities, US Treasury bills, etc. Online auction companies have
grown in number during the past years. Among them we find eBay,
Amazon.com, Yahoo, etc. Therefore, the size of the auction market
highlights the importance of studying Auction Theory.
The Revelation Principle
From the gametheoretical point of view auctions are
games
with incomplete information or Bayesian games: the
player's payoff functions are not common knowledge. This characteristic
is due to the fact that in an auction, bidders are uncertain about
other players' valuation of the item or good, and sometimes are even
uncertain about their own valuation. A quite useful result in game
theory, states that any
Bayesian
Nash equilibrium in a Bayesian game (no matter the nature of
the game) can be represented as a Bayesian Nash equilibrium of a
particular type of auction in which:

The bidders simultaneously make (possibly dishonest) claims about
their valuations. Say, that there are N bidders
with valuations and claims v_{1},...,v_{N}
and c_{1},...,c_{N},
respectively.

Given the bidders' claims about their valuations each player i
pays x_{i}(c_{1},...,c_{N})
and receives the good with probability q_{i}(c_{1},...,c_{N}).
The sum q_{1}(c_{1},...,c_{N})+...+q_{N}(c_{1},...,c_{N})
must be less than or equal to 1.

Telling the truth is a Bayesian Nash equilibrium for each player.
An auction with the properties 1.3. is known as an
incentivecompatible
direct mechanism. This result is known as
The
Revelation Principle, an important tool for
designing games with private information and studying auction and
bilateral trading problems. This theorem simplifies the study of
auctions itself, but at the same time explains why understanding such
mechanisms may be of great importance in order
to
understand the behavior of other institutions.
Subsequent hypotheses
Before going ahead and studying particular examples of auctions, it is
important to state a precise definition of the concept and list the
assumptions we will make in the subsequent lessons. By relaxing or
completely changing some of these assumptions new results are obtained.
The study of more general models is out of the scope of these lessons,
however in
lesson 4 we
mention some results related to extensions of the model.
An
Auction, according to
McAfee and McMillan, 1987, is
"a
market institution with an explicit set of rules determining resource
allocation and prices on the basis of bids from the market participants".
In the auctions that we study in the following lessons there is a
unique
seller of the item or good, known as a
monopolist.
The organizer of the auction
commits himself in
advance to a set of policies, specifying how the
winner of the auction will be selected and the price the winner has to
pay for the item. Additionally we assume that:
 A1. The bidders are riskneutral.
For instance, obtaining a payoff equal to zero after taking part in the
auction is the same than obtaining a payoff of zero with out submitting
any bid.
 A2. Every bidder knows
precisely how much he/she values the item. Additionally, the valuations
among bidders are independent and the distributions of this valuations
are common knowledge. This is known as the independentprivatevalues
assumption. We will explain it in more detail in
the next lesson.
 A3. The bidders are symmetric
in the sense that the beliefs of the bidders and seller about the
others valuations are common.
 A4. Payment is a function of
the bids alone.
This model, represents the
benchmark model and
particular examples will be studied in the subsequent lesson.
Activity
Can you think of particular examples of auctions when some of the
hypotheses listed above are violated? How would you think the outcome
of the auction would change in these cases compared to the one observed
when A1A4 hold?