How to rationalise auction sales

How to rationalise
auction sales
Jean-Jacques Laffont
Thanks to the Internet in particular, auctions have become widespread.
Modelling these sales processes makes it possible to determine their
rules and the optimal strategies for using them.
uctions, as a means
of buying or selling, have
become widespread. This
is, in particular, the case
on the Internet, as testified by the striking success of sites such as eBay
where goods of all kinds
- from books to cars,
including art objects and
household appliances are bid on. As a method
for allocating scarce
resources, auctions are An auction at Christie's of works of artists of the 20th century. Each potential buyer behaves according
traditional in the live- to what he believes the others will do. Game theory analyses such situations and helps in finding optistock markets and for mal strategies (Photo Gamma Connection/Jonathan Elderfield)
agricultural produce
(fish, flowers, etc). They have recently been describes the marriage market in Babylon as
extended to more expensive goods, such as an auction which started with the most beauapartments, and to much more complex tiful young women who were sold to the
objects, such as licences for third generation highest bidder (i.e., the highest offer gets the
mobile telephony.
``object'' to be sold). In Asia, the oldest account
The use of the system of bidding is very old, of bidding relates to the sale of the effects of
and goes back to Antiquity. Thus, Herodotus
dead monks in the 7th century.
How to rationalise auction sales
The first ideas about auctions were
inadequate because they were too
If auctioning goes back almost to the dawn
of humanity, their conceptualisation is much
more recent. The first important academic
work devoted to the subject is a 1955 thesis,
whose author was the American L. Friedman.
It was one of the first theses on operations
research. It was devoted to developing auction strategies by companies on the occasion
of the sale of the petroleum drilling rights in
the Gulf of Mexico. These auctions were ``firstprice sealed-bid auctions'': in this procedure,
the offers are not made public
and it is the highest offer which
wins the auction.
The strategy adopted by
Friedman consisted simply in
maximising what is called profit
expectancy. If he wins the bid, the
bidder makes a profit equal to
the difference (v - b) between his
valuation v of the object put up
for sale and the price b which he
proposes to pay for it. The profit
expectancy is thus this difference
multiplied by the probability P(b)
of winning the bid at this price,
that is to say (v - b) P(b). The probability P(b) is a priori unknown;
but by making a statistical analysis of past biddings, one can discover ways of outbidding the
competitors; that makes it possible to determine an approximation to the function P(b) and
thus find the bid b* which maximises profit expectancy, i.e., such
that (v - b*)P(b*) is maximum
This method, which is widely used and has
been refined in many ways, is however extremely na{\"\i}ve. Indeed, it implicitly supposes
that the other bidders have not worked out
a strategy and that their future behaviour can
be easily deduced from their past behaviour.
In 1961, the Canadian William Vickrey (who
received the Nobel Prize for Economics in 1996,
two days before his death) posed the problem
differently, by using game theory.
The home page of the Internet auction site eBay-France.
Game theory and mathematical economics enter the scene for defining optimal strategies
Created by the famous mathematician of
Hungarian origin John von Neumann in the
years 1920-1940, in collaboration with the economist of Austrian origin Oskar Morgenstern,
game theory examines strategic interactions
of the players. It deals with any situation where
each player must make a decision which determines the outcome of the situation. Game
theory thus applies to many scenarios of the
economic, political, diplomatic or military
world. But let us return to our biddings. When
a bidder has to decide what to bid, he asks
himself what the behaviour of his competitors is going to be, and each bidder does this.
An equilibrium of this situation indicates for
the specialists a rather complex object: it is a
method of bidding - in other words a relation
between the valuation v of the bidder and his
bid b - which is the best way for the bidder to
bid taking into account what he anticipates
are the bidding strategies of the other actors
and his guess about their valuations. For
example, in a symmetrical situation where the
expectations of all the actors are the same,
the strategy of a bidder must maximise his
profit expectancy knowing that all the others
are using the same strategy as he is.
The concept, which we have just evoked,
is a generalisation of Nash equilibrium, adapted to the context of incomplete information
about the bids. What is it all about? The
American mathematician John Nash (Nobel
Prize for Economics in 1994) had proposed
around 1950 a very natural concept of equilibrium which generalises the one given in 1838
by the French mathematician and economist
Antoine Cournot. Given a set of actions from
L’explosion des mathématiques
which the players can choose, these actions
form a Nash equilibrium if the action each
player chooses is the best possible one for him,
knowing that the other players also are choosing the actions specified by the Nash equilibrium. In a Nash equilibrium no one finds it
beneficial to unilaterally change his action.
The particular difficulty in auctions is that
each bidder is the only one who knows his
own valuation of the goods to be sold and
that he does not know the valuations of other
The American mathematician John Forbes Nash, born in 1928,
received the Nobel Prize for Economics in 1994, in particular for
his work on game theory. Around the age of thirty, Nash started suffering from serious mental disorders from which he made a spectacular recovery in the middle of the 1980’s. His life was the subject
of a biography ``A beautiful mind'', which inspired a film of the
same title. (Photo University of Princeton)
How to rationalise auction sales
potential buyers. It is thus necessary to generalise the concept of the Nash equilibrium to
this situation where information is incomplete.
This is what was carried out intuitively by
Vickrey in 1961; the American of Hungarian
origin John Harsanyi did it more precisely
around 1967-1968, which won him as well the
Nobel Prize in 1994. One thus arrived at the
notion of a Bayesian Nash equilibrium, a
concept of equilibrium which allows one to
put forward a conjecture about the way in
which rational bidders must bid in an auction.
In the context of auctions, a strategy from
the mathematical point of view is a function
S which associates to a bidder's valuation his
corresponding bid. In other words, for any
particular valuation v, this function must specify the bid b* = S(v) which maximises his profit expectancy as calculated from the rules of
the auction, supposing that the other players
use the same strategy. That means that in a
symmetric Bayesian Nash equilibrium, if the
others bid in the same way, employing the
same strategy, this way of bidding is optimal.
Why have we used the adjective Bayesian?
Because the player calculates his profit expectancy starting from his beliefs about the valuations of the other bidders (in probability and
statistics, the Bayesian point of view - named
after Thomas Bayes,a British mathematician
of the 18th century - consists in evaluating probabilities on the basis of the available partial
information and on a priori beliefs).
When theory confirms and extends
the utility of sales methods arrived at
In the auction field mathematics makes it
possible to model the behaviour of bidders,
which leads to a prediction about their way
of bidding. That has led to progress in two
directions. On the level of positive knowledge,
it has become possible to compare the data,
i.e., the bids of the players in different types
of auctions, with those predicted by the theory.
The theory thus acquires a scientific status:
one could reject it if one finds data which
contradict the predictions; the theory is thus
At the level of establishing standards, the
consequences have been even more important. Within the framework of the assumptions of the theory of auctions thus constructed, one could prove a rather fascinating
theorem: the revenue equivalence theorem.
Without going into the details, this theorem
shows that the first-price or second-price (the
winning bidder pays only the second-highest
price, not the highest price) sealed-bid auctions, ascending (English) or descending
(Dutch) oral auctions are equivalent for the
seller and they are, moreover, often optimal.
Thus, sales methods which were used pragmatically in particular cases have turned out
to be, in the light of theory, the optimal ways
to allocate scarce resources. Hence the new
enthusiasm for extending these methods to
all kinds of economic activities. Finally, in more
complex circumstances than the sale of a
simple object, theory makes it possible to
conceive generalisations of simple auctions in
order to optimise even more the seller's
income, or social well-being if the organiser
of the auction is a State concerned with this
aspect of things. Thanks to mathematics, it
has been possible to understand the meaning
and the importance of an ancestral practice
and thereafter to transform human intuition
into a true tool for economic development.
With the emergence of the Internet and
L’explosion des mathématiques
the new communication technologies, auctions have found an immense field for experimentation. The Internet offers new possibilities for this system of selling, which theory
will help to evaluate and to exploit. For
example, in an auction an anonymous seller
should a priori suffer from the asymmetry of
information - he is the only one who knows
the quality of the goods he is selling - and
would therefore manage to obtain only a very
low selling price; but by repeated sales of quality objects to a priori unknown potential
buyers, he can build a reputation little by little,
thanks to the satisfied comments of his previous buyers. The quality of the transactions
can thus be improved by creating a place
where one can build a reputation for quality
and honesty, something to which an Internet
site lends itself easily.
Jean-Jacques Laffont
Institut d’économie industrielle,
Université des sciences sociales,
Manufacture des tabacs, Toulouse
Some references:
• I. Ekeland, La théorie des jeux et ses applications
à l’économie mathématique (P.U.F., 1974)
• A. Cournot (1838), Recherche sur les principes
mathématiques de la théorie des richesses
(Calmann-Lévy, Paris, rééd. 1974).
• J. Crémer et J.-J. Laffont, « Téléphonie
mobile », Commentaire, 93, 81-92 (2001).
• L. Friedman, « A Competitive bidding strategy »,
Operations Research, 4, 104-112 (1956).
• J. Harsanyi, « Games with incomplete information
played by bayesian players », Management Science,
14, 159-182, 320-134, 486-502 (1967-1968).
• J.-J. Laffont, « Game theory and empirical economics : the case of auction data », European
Economic Review, 41, 1-35 (1997).