In public sales, we know that the final bidder is the one who wins. But at what price? Game theory is a way to resolve this conflict. Two hours with Françoise Forges, economics professor at the Université Paris-Dauphine. Everything you need to know on strategic bidding behaviour.
Game theory: as a topic, it’s a little bit tough to grasp. Otherwise defined as the secret life of numbers, or how to formalise conflicting situations within communities of individuals when they interact, it can be used effectively, for example, at public sales. How can the strategic behaviour of bidders be analysed, anticipated, or even thwarted? In very basic terms, game theory deals with the formal resolution of conflicts. First of all, there’s one name you need to keep in mind: William Vickrey, who in 1961 matched game theory with auction mechanisms for the first time. A winner of the Nobel Prize in Economics, he was recognised for his research contribution to “the economic theory of incentives under asymmetric information”. This was the man who namely theorised on the interaction of strategies used by bidders. Let’s say that here, we flirt with the concept of the “Nash equilibrium” whereby a player cannot modify his or her strategy unilaterally without weakening his or her position. All clear? It may not be exactly straightforward, but understanding it can pay off. Thanks to game theory, it’s possible, for example, to identify the symmetries at work in auction rooms. Game theory also offers very practical applications for military defence, where the modelling of nuclear dissuasion can prove handy. In short, the field is vast, starting with economic sciences and the analysis of competing logics, and spreading to the political sciences, where game theory can apply to electoral jousts. In the social sciences, Lévi-Strauss, an enthusiast of complex systems, also incorporated it into his structural anthropology. But back to the public sale and its system of increasing bids, also known as the “English auction”, where the last bidder wins…
The earliest conceptualisation of auctions is fairly recent…
That’s right, it dates back to the start of the 1960s. It’s all thanks to an American economist, a professor at Columbia University, William Vickrey, who published an article in 1961 and was the first to see – literally see, like a visionary – that game theory is a way to understand the mechanisms of auctions, the strategies at work among bidders. It really reveals the necessity to be strategic. Vickrey turned his attention to auctions with two players. At its time of publication, the article went completely unnoticed. But it was from this point that game theory developed considerably. Launched at the end of the 1920s, research took off in around 1945, though by the 1960s it was still gaining little public attention – the importance of game theory being overlooked until Vickrey was crowned with a Nobel Prize in Economics in 1996, shared with James Mirrlees, in recognition of research carried out on the theory of incentives under imperfect information.
What is a bid in mathematical terms?
It should be understood that a bid is a game inasmuch as it is an interaction between individuals, which can be identified in strategic terms. But it’s a game which has the specificity of being subject to “incomplete information” because very often, players don’t know the value which another bidder will attribute to the object placed on auction. Incidentally, game theorists have had a great deal of trouble with modelling, abstractly and generally, this type of game. Modelling only became clear at the end of the 1960s, with the work of John Harsanyi, Hungarian in origin, who would also end up with a Nobel for Economics, in 1994, for his major contribution to game theory and “equilibrium choices”.
How was Vickrey’s discovery decisive?
What’s phenomenal about Vickrey’s work – and this is undoubtedly why he went unnoticed – was that he managed to make extremely insightful observations at a time when the concept of games “with incomplete information” wasn’t clear in the minds of other game theorists. What raised problems for these theorists, inevitably, was the following point: if the information held by other players isn’t known, the natural reaction, in this case, is to form beliefs. This is a reflex that comes to us as the legacy of statistics, where we apply probabilities on events which are uncertain. This is what we tend to do when facing climatic risks, for example. A winegrower will make preparations on the basis of the uncertainty he faces. It’s quite simple. The thing is, if we form beliefs on the nature of another individual, and therefore, on the strategies that are personal to him or her, we quickly realise that this person must also be doing the same thing. So very quickly, we develop beliefs on what the other person believes, and on what the other person believes that I believe. This leads to an infinite hierarchy of beliefs. And in the 1960s, many economists thought that we could never shake free from this problem…
So how do we do manage to do it?
Very simply… John Harsanyi severed the Gordian Knot and found a way to synthesize information – including this whole hierarchy of beliefs – through the notion of player “type”. The idea is that different “types” exist, each of which hold all information. Harsanyi didn’t resolve the issue mathematically but rather conceptually. He said that a player’s information cannot be dispersed, so it bears a form of coherency. In this way, what characterises the person can be represented “compactly”. This was what Vickrey had understood very clearly. Rather than lingering on the difficulties of modelling the information, it’s necessary to model it in the simplest possible way. With the problem resolved this way, we could then forget about it.
And what is this “simple way”?
The simplest case is to consider a certain number of individuals, each of whom forms a valuation of the object. Every individual therefore knows the price that he is willing to pay. Vickrey imagined the case where this information is personal, independent of the valuation of third parties. In other words, valuations are completely subjective, and the individual is not influenced by the value attributed by others in this case. Vickrey thus invented the “second-price” auction procedure, which is not commonly implemented, imaginging that bids would be submitted in sealed envelopes (later he would make the link with oral bids) and everyone would make a written offer of the value that he was willing to pay for the object. The auctioneer would then open the envelopes and attribute the object to the highest bidder. Up to this point, nothing extraordinary. But what Vickrey added was that the price to pay wasn’t what the highest bidder offered, but the one just below it. In other words, the “second-price” one; this meant that the winner would pay the price corresponding to the second-best offer.
We are familiar with the principle of Dutch auctions in which prices are gradually lowered, namely used for the sale of tulips in the Netherlands, and then first-price and second-price sealed-bid auctions which you’ve just mentioned. Can you say a word about increasing-bid auctions practised by auctioneers?
The decreasing-price Dutch auction is more appropriate for objects of lesser value, such as the sale of fish, vegetables… English auctions have the characterstic – among others – of offering the seller, in many cases, higher income than other types of auctions. What Vickrey demonstrates here, by establishing the strategic equivalence between the increasing-bid auction, also called the “English” auction, and the second-price sealed-bid auction, is that we risk nothing by revealing one’s true valuation of an object. This is where Vickrey’s idea is absolutely ingenious: it is a way to break free from coordination. We can find a good strategy independently of what others are doing. It’s even optimal to reveal what you deem to be the true value, even if other bidders don’t do this. It’s a strategy that is described as “weakly dominant”. Which doesn’t mean that there aren’t any other strategies that are just as good, but with this one, we never lose anything by telling the truth, by revealing one’s true valuation, which of course remains subjective.
When several economic players interact, how does each determine his or her strategy?
Indeed, there are many games where optimal behaviour depends on what others are doing. For example, if it’s a matter of acquiring an oil well or a mobile phone licence… It goes without saying that an object’s value is not purely subjective. Here, what is used as a solution concept is the “Nash equilibrium”. Let’s just say that every bidder should bring the best response to the strategies of others. The whole problem, incidentally, lies in the “strategies of others”. Of course, in the case of artworks, where we come up against resale value, subjective aspects, or exercise of a certain type of social vanity, the question becomes even more complex.
In a context of strategic ineractions, is informational asymmetry always at the origin of conflict?
If bidders don’t agree between themselves, if they don’t form a cartel of dealers with a collusion strategy, united in a tacit agreement, then they are adversaries. They are competitors who have their own auction behaviour, who act in a context of strategic uncertainty. So yes, there’s conflict, whenever there’s only a single winner.
Could you go back on John Nash’s work on equilibrium?
Nash’s equilibrium is the most popular solution concept in game theory, imagined by mathematician John Nash at the start of the 1950s, at a time when ideas on the matter weren’t clear. The theory consists in identifying a strategy for each player, which the latter implements while taking into account the strategies of the other players. There is therefore a type of implicit coordination. We are talking about independent strategies, but they are related to one another by an equilibrium. In England, for example, it’s in my interests, when I drive, to keep to the left side of the road… In terms of auctions, it’s a little bit the same. If everyone thinks that each of the players is rational, we’ll play the equilibrium strategy – and each person can calculate on this basis, even if it proves quite complicated.
Still talking about auctions, but outside of the art market, what applications exist for game theory today?
More or less sophisticated auction procedures are very commonly used in the public domain, namely to attribute ownership or user rights, where the approach would once have been to draw lots. To ensure allocative efficiency for offshore drilling or mobile phone licences in the United States, the Grand Committee gets advice from game theorists, and apparently companies as well, which have backed an equilibrium. This is a case in which game theory can be very useful, when we don’t know what price to apply, as in the allocation of such licences. The US State felt, at the time, that it could not decide administratively. Who would dispose of what and at what price? To determine the “right price”, game theorists were asked to conceptualise the auction. Would all the licences be sold at the same time, would California be treated like New Jersey, would the operation be carried out once, globally, or periodically?
It’s said that there’s a “winner’s curse”…
This is a phenomenon that has been genuinely observed, namely with respect to the sale of oil resources, for drilling rights for wells or mining platforms, for example in the Gulf of Mexico, sold to private companies in the United States. It’s often been observed – in hindsight, obviously – that a good’s real value is far below the price paid by the last bidder. We’ve also come across this case recently in France, when mines related to the nuclear industry were bought at outlandish prices. So we often see over-payment. The fact of winning typically reveals an over-estimation of the good’s objective value, which it should have been necessary to scale downwards, by anticipating that assessment might be too high… It’s true that we can never be absolutely certain of winning except by paying too much. But to avoid the “winner’s curse”, a person should concede a small probability of losing, in order to pay the “right price” when they do win!
What exactly are the “economics of uncertainty”?
This is the aspect of microeconomics that takes into account the fact that agents don’t have a perfect knowledge of the environment in which they must take decisions. These may be climatic risks as mentioned above, but also social or political situations… The field is a priori far vaster than that of game theory, as it covers all market phenomena related to insurance, finance, far beyond the identifiable strategic interactions of a few individuals… In a certain way, game theory on incomplete information is part of the economics of uncertainty, but it also has other applications, in political science for example.
More personally, what is your relationship to auctions?
Above all, I see them as a research topic for game theory… but I also sometimes buy at auctions! I collect a little. Paintings from the Barbizon School…
A flashback on a certain Nash
If you believe that happiness lies in harmony, then the “Nash equilibrium” is tailor-made for you! What is it exactly? It’s about choosing an “optimal” strategy when facing individuals confronting the same problem. This type of interaction, which crops up in card games such as bridge, can also be experienced in more serious contexts. For example, when bidding in an auction room, every potential buyer will be wondering about the behaviour of others in the room. This is quite a natural approach to anticipating how to bid oneself. According to American mathematician John Nash, winner of the Nobel Prize for Economics in 1994, the equilibrium is reached when each bidder no longer has any interest in deviating from his or her position, given the positions that he or she anticipates, correctly, from fellow bidders. The problem, in the context of auctions, is that all “players” are unaware of the expertise and/or tastes which motivate the intentions of their rivals. Information is therefore said to be “incomplete”. It was William Vickrey, the other Nobel Prize winner, who understood this problem in 1961. Vickrey’s approach to equilibrium takes into account the anticipations and hidden valuations of every bidder. We thus speak of a “Bayesian Nash equilibrium” – a notion (from Thomas Bayes, an English mathematician from the 18th century) that allows conjectures to be made on the information held by rational decision-makers and also on the way that they behave. As in the world of statistics from which this notion derives, the Bayesian variant enables players to update the beliefs that they have formed from partial information. In short, game theory offers to shed light on optimal strategies. According to economist Jean-Jacques Laffont, a great specialist on incentive theory, “in the field of auctions, mathematics have enabled the modelling of player behaviour, which leads to predictions on how they bid”. Good to bear this in mind!
You can find out more by reading Ivar Ekeland’s La Théorie des jeux et ses applications à l’économie mathématique, published by Presses Universitaires de France (PUF), or Gabrielle Demange and Jean-Pierre Ponssard’s Théorie des jeux et analyse économique, also published by PUF, with a chapter on auctions in the 1994 edition. Finally, we can also mention Jean-Jacques Laffont’s article “Game theory and empirical economics: the case of auction data”, published in the European Economic Review (volume 41).