Learning in sequential auctions.

• Author: Jeitschko, Thomas D.

1. Introduction

   A fundamental question of economics concerns the impact of market structure and institutions on the allocation of resources. A class of trading mechanisms that has received much attention in this respect over the past several years is the transfer of resources through auctions. While tremendous advances have been made in understanding single-unit auctions (see the surveys by Milgrom 1985; McAfee and McMillan 1987; Wilson 1992; and Wolfstetter 1996), only recently have auction theorists begun to embed this analysis in broader and more realistic institutional settings.(1)

   Auctions are often part of an institutionalized market in which trades are recurring events, for example, markets for government treasury securities, used cars, fresh fish, and many wholesale agricultural commodities (flowers, cheese, tobacco, and so on). The focus of this study is the potential impact of information transmission and learning when goods in these markets are auctioned sequentially.

   The transmission of information and the resolution of uncertainty during a sequential auction can have important implications for bidding behavior and auction outcomes. First, each participant learns about the valuations of other bidders before it is "too late" to act on the information. Specifically, a trader is able to use the behavior of rivals in earlier rounds to update beliefs about rivals' valuations for the item currently on the trading block. This leads to the second informational effect. When deciding on bids in earlier auctions, bidders must take into account how these bids affect the flow of information. Specifically, bidders must account for how their own bids affect the information they obtain about their rivals in later auctions. Hence, sequential auctions exhibit two learning effects not present in static auctions: a direct effect in which information gleaned from earlier rounds is used to formulate current bids and an anticipation effect in which bidders incorporate the impact of their earlier bids on the direct effect in future rounds.

   Despite the potential significance of information transmission in sequential auctions, most of the existing literature in this area has focused on other aspects of these markets. Gale and Hausch (1994), for example, present a model in which two "stochastically equivalent" objects are auctioned in sequence to two bidders with unit demands. However, because there is only one bidder left in the second auction, the learning effects described above are not present in this setting. In similar models, Engelbrecht-Wiggans (1994) and Bernhardt and Scoones (1994) consider scenarios in which values are assumed to be independent across both bidders and auctions and the bidders do not know their value of a particular object until it is up for sale. Together, these assumptions eliminate any learning during the sequence of sales. Finally, several papers on sequential auctions (e.g., Black and de Meza 1992) examine auction formats in which bidders have dominant strategies that do not depend on information.

   Most research studying informational issues in sequential auctions has focused on the common value paradigm (see, e.g., Engelbrecht-Wiggans and Weber 1983; Hausch 1986; and Bikhchandani 1988). The focus of this study, however, is how bidding behavior is affected when bidders have independent private values, so that learning is not about the value of the objects for auction but about opponents' types. Since many sequential auctions are wholesale markets in which the goods purchased are retailed in downstream markets, the independent private values assumption is relevant if demand conditions in the downstream markets are independent of each other. Moreover, even if there is a common value component to bidders' values, the private values paradigm is relevant if all bidders are equally (not necessarily fully) informed about the common value elements of the good, so that the opponents' strategies reveal no additional information about one's own value of the good.

   The papers most closely related to this investigation are Milgrom and Weber (1982), Weber (1983), and McAfee and Vincent (1993). In these studies, identical goods are sold sequentially through first price sealed bid auctions to bidders who have independent private values and unit demands. Rather than information transmission, however, the focus of these papers is the equilibrium price path. In fact, information transmission is of little importance in these models. In equilibrium, bidders assess their option value of continuing in further auctions if they lose the current auction. This option value depends solely on the probability of winning or losing the first auction, not, however, on estimations of how high the winning bid actually turns out to be. Given this option value, bidders bid their estimation of the highest valuation of their opponents, conditioning on their valuation being the highest. Because of this conditioning, bids are independent of the prior history of the game. In sum, bidders' strategies depend only on which auction they are currently in but not on any past price realizations or on estimations of future price realizations. Hence, bidders do not update their beliefs about their opponents in the course of the auction and therefore need not anticipate learning either.

   The reason for the absence of learning and the anticipation of information is that valuations are modeled to come from a continuous distribution, so there is zero probability that any two bidders have the same valuations. If, however, a bidder thinks there is a positive probability that another bidder has the same valuation for an object for auction, this bidder can no longer condition his bids on him having the highest valuation. In this instance the bidder uses a mixed strategy to avoid tying bidders of the same type. This strategy depends critically on the probability of another bidder having the same valuation. Bidders' beliefs about this probability are updated in the course of the auction, and thus bidders' expected payoffs are affected by their learning about rivals' types. This illustrates the direct effect of information transmission mentioned above. To appreciate the indirect effect, note that in the first auction bidders must trade off the benefits associated with winning the auction and retiring against the benefits of losing the first round and learning more about their rivals' valuation in the second auction. In sum, learning and the anticipation of information being generated affect bidders' optimal strategies and hence expected payoffs throughout the sequence of sales.

   The remainder of this paper contains three sections. In section 2 the model is formalized and the equilibrium discussed. In section 3 properties of the equilibrium are studied. It is shown that bidders who learn from the outcome of the first auction have higher expected payoffs. Moreover, the anticipation of information leads to greater payoffs when compared to "myopic" bidding in the first auction because myopic bidders are unaware of the trade-off between winning the first auction and being better informed in the second auction.

   In equilibrium, prices may fluctuate. The probability that prices increase or decrease depends on the information generated in the first auction. However, regardless of this information, and thus independent of the probability of increasing or decreasing prices, the price in the second auction is, on average, the same as the price in the first auction. This result is also obtained in the case for a continuum of bidders' types (see, e.g., Weber 1983). Thus, although bidders' strategies are affected differently when types are distributed discretely as opposed to continuously, the equilibrium price path forms a martingale in either case. Another analogy is found when considering the institution chosen for the sale of the objects, namely, contrasting the sequential auction with a simultaneous (static) auction. Despite the fact that learning has no role, the static auction yields the same expected final allocation of goods and bids, as does the sequential auction. Thus, expected revenue equivalence studied in Weber (1983) and Engelbrecht-Wiggans (1988) carries over to the case of a discrete distribution of types.

   Some brief concluding remarks appear in section 4, and the appendix contains the proofs to the propositions and theorems.

2. The Model and the Equilibrium

   Two identical objects are auctioned in sequence to three bidders. Each bidder can be one of two possible types: have either a high valuation for a unit or a low valuation for a unit. Without loss of generality, valuations are normalized to 1 and 0. Bidders have an ex ante probability of [Rho] [Epsilon] (0,1) of having a high valuation for a unit of the goods, and 1 - [Rho] that they have a low valuation. They know their own valuation for a unit of the good and have beliefs [Rho] that any given opponent has a high valuation for a unit of the good. Bidders have unit demands; that is, a bidder's value for a second unit is nonpositive. Bidders have a reservation utility of zero, which they obtain if they do not win one of the goods; otherwise, their utility is given by the difference between their valuation of a unit and the price they pay. In both auctions the strategy space is the real line; that is, negative bids are permitted. Ties are broken by the roll of a die. The auction format is a first price sealed bid auction in which only the winning bid is announced.(2)

   Notice that the auction format does not maximize expected revenue of the sellers. However, in most markets in which sequential auctions are used, the institutional framework is agreed on by both sellers and buyers, as these markets are often arranged by trade associations or the like. Thus, one would expect these markets to be efficient yet not necessarily producer surplus maximizing.(3) The model under consideration, which...