Auctions in Theory

‘Auction Theory is one of economics’ success stories. It is of both practical and theoretical importance: practical because many of the world’s most important markets are auction markets...; theoretical because lessons from auction the­ory have led to important insights elsewhere in economics’ (Klemperer 2004b: 1).

A new line of study started with Klemperer in the mid-1990s, coinciding with a renewal of an old partnership with Jeremy Bulow. However, it is impor­tant to note that Klemperer had already used an auction framework both in his 1987 paper with Peter Cramton and Robert Gibbons (see Cramton et al. 1987) and in Klemperer and Meyer (1989), discussed above. Significantly, for future research, the Cramton et al. paper was inspired by the Federal Communications Commission’s allocation of licences for cellular telephone franchises and also the Federal Aviation Administration’s allocation of landing slots. Cramton et al. showed that achieving the optimal allocation involved payments depending on all bids and not just the winning bidder paying—this is very different to standard auction mechanisms such as first- or second-price auctions. However, they also showed that standard mechanisms can achieve the efficient outcome if the initial partner shares are close enough to equal.

The title of Bulow and Klemperer (1994), “Rational Frenzies and Crashes”, does not mention auctions, but is about a Dutch auction where a seller has multiple units to sell and starts from a high price. Buyers have a valuation for buying one unit coming from a common distribution, which is known. The key point is that buyers can choose to bid or delay. If at least one buyer offers to buy at the current price, the other buyers are also asked if they wish to buy at the current price. This simple auction is solved using revenue equivalence. What Bulow and Klemperer showed is that once one person has bid and been allocated a unit, others may wish to also buy since removing the unit pur­chased will raise the price buyers can expect to pay.

In their 1996 paper “Auctions Versus Negotiations”, Bulow and Klemperer pose the question: when selling a firm, should you employ ‘an auction with no reserve price or an optimally-structured negotiation with one less bidder? We show under reasonable assumptions that the auction is always preferable’ (ibid.: 180). However, the paper is as important for the way it models the auc­tion process in terms of a monopoly, and looks at marginal revenues instead of prices. The price is the value of the bidder; the quantity is defined by the cumulative density of the bidder's value. Marginal revenue (MR) is then the derivative of price times quantity, with the optimal price for the monopolist occurring when MR is zero (assuming the seller has no cost). The analogy works because,

just as the expected revenue from a take-it-or-leave-it price can be calculated by multiplying that price by the probability of sale at that price, expected revenue can also be found by taking the area under the MR curve for all the values in excess of the take-it-or-leave-it price. The seller may be thought of as receiving, in expectation, the MR of the buyer when it is positive, and zero when the buyer’s MR is negative (ibid.: 183—184).

This approach was originally developed by Bulow and Roberts (1989) for bidders with independent private values, but the Bulow and Klemperer paper shows how to extend the approach to all auctions in which bidders’ signals are independent, whether values are private or common or something in between, and then—a key insight—to all ascending auctions, whether or not bidders have independent signals.

The Bulow and Klemperer paper also shows the usefulness of this approach. Using marginal revenue analysis, it is simple to show what was not otherwise obvious: that so long as all bidders are serious (in the sense that their valua­tions exceed the seller’s actual value), then adding an extra (serious) bidder to an ascending auction with no reserve price will, under mild conditions, always increase expected revenue more than adding the use of a reserve price.^[227]

Klemperer went on to promote his view that ‘connections between auction theory and standard economic theory run deeper than many people realise’ (Klemperer 2000: 2) and applied the auction-theoretic perspective to a wide variety of applications, in his aptly titled Econometric Society World Congress lecture, “Why Every Economist Should Learn Some Auction Theory”. Put simply, Klemperer believes that many markets can be understood as equivalent to particular type of auction markets. Alternatively, auctions are a way of setting up a market. A well-designed auction can yield a desirable out­come where for some reason no market exists.

In Klemperer (1998), “Auctions with Almost Common Values: The ‘Wallet Game' and Its Applications”, he considered the following classroom experi­ment: two students are picked, each checks how much is in his or her wallet; the combined contents of the wallets are then auctioned to the two students using an English auction. The problem for each student is that they only know the value of their own wallet, not the value of the other student's wallet. There are many equilibria to this, but only one symmetric equilibrium (both bidders offer up to twice what was in their own wallet). Klemperer went on to consider situations where bidders have almost common values: that is one bidder has a slightly higher value. He found that small differences in valua­tions would lead to far from symmetric outcomes.

In “The Generalized War of Attrition”, Bulow and Klemperer (1999) used the Revenue Equivalence Theorem from auction theory to solve the case where N+K firms were competing for N prizes. Such wars of attrition occur when a number of firms are competing for a fixed number of slots (for exam­ple, firms competing to supply wireless telephony in major US cities) or where there is a battle to control a new technology and set the standards. Previous analysis had focused on the two-firm case. Bulow and Klemperer instead looked at the more general natural oligopoly case: ‘The natural oligopoly case yields a striking result: there is “instant sorting”, so K-1 firms will exit imme­diately, leaving only N+1, or one too many firms to battle for the N prizes' (ibid.: 177). This instant sorting result is a general feature of these games.

Bulow and Klemperer (2002), “Prices and the Winner's Curse”, developed themes from Klemperer (1998). In particular, they showed how in ascending- price auctions increasing supply might in fact lead to higher prices because it encourages weaker bidders to participate, since the greater supply alleviates the winner's curse they face. More active bidders can result in a higher price. Furthermore, when there are even small deviations from common value, the auction can behave very differently from the common value auction. This insight was to have important implications when it came to designing auc­tions in practice because in real life, pure common values are almost never found. It is a case of economic theorists spending time looking at the case that is easy to solve. If auction theorists want to produce relevant models, they need to stop looking at the simple models and look at something more realis­tic. Even small deviations from the simple case can lead to very different outcomes.

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