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. a 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 simplistic 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 57 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. 58 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 intuitively... In the auction field mathematics makes it possible to model the behaviour of bidders, 59 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 refutable.. 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 60 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).

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