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Article Excerpt
Summary. We consider two ascending auctions for multiple objects, namely, an English and a Japanese auction, and derive a perfect Bayesian equilibrium of the Japanese auction by exploiting its strategic equivalence with the survival auction, which consists of a finite sequence of sealed-bid auctions. Thus an equilibrium of a continuous time game is derived by means of backward induction in finitely many steps. We then show that all equilibria of the Japanese auction induce equilibria of the English auction, but that many collusive or signaling equilibria of the English auction do not have a counterpart in the Japanese auction.
Keywords and Phrases: Multi-unit auctions, Ascending auctions, FCC auctions, Complementarities, Collusion, Signaling.
JEL Classification Numbers: C72, D44.
1 Introduction
Since the first series of spectrum auctions held by the Federal Communications Commission (FCC) in the United States, academics and policymakers alike have pointed out at least three advantages of the FCC auction rules: they ensure a transparent bidding process, they enable extensive information revelation of bidders' valuations, and they allow bidders to build fairly efficient aggregations of licenses. (4) At the same time, an important drawback that has been documented in the literature is the large potential for signaling and collusion, with corresponding negative consequences for efficiency and revenue. (5)
In this paper we consider two basic auction mechanisms: the Japanese Auction for Multiple Objects (JAMO) and the Simultaneous English Auction for Multiple Objects (SEAMO), which is very close to the actual FCC auctions. We show that the JAMO is much more immune to collusive and signaling equilibria than the SEAMO. Both auctions are simultaneous ascending auctions, thus both ensure a transparent bidding process, enabling extensive information revelation, and efficient aggregation of objects. However, unlike the SEAMO and the FCC auctions, in the JAMO, prices are raised directly by the auctioneer, and closing is not simultaneous but rather license-by-license. These two basic differences eliminate many unwanted collusive equilibria of the SEAMO.
More specifically, Brusco and Lopomo (2002) construct multiple collusive equilibria for the SEAMO, where each bidder signals her most preferred item to competitors during early phases of the auction. The goal of the signaling strategy is to "split the market" and keep prices low. Thus the openness and simultaneity of the SEAMO, while allowing for transparent bidding, also provides bidders with effective communication devices that can be exploited to achieve collusive outcomes. The JAMO, on the other hand, by imposing a strong activity rule (bidders have no control over the pace at which prices rise) and license-by-license closing, rules out communication devices among bidders that might be used to achieve tacit collusion, while preserving the positive features of the ascending bid process. Indeed Albano et al. (2001), within an example with 2 objects and 4 bidders, show that the JAMO obtains close to ex-post efficiency with higher revenues than the revenue-maximizing ex-post efficient mechanism (Vickrey-Clarke-Groves mechanism), and that it dominates both the sequential and the one-shot simultaneous auctions in terms of ex-ante efficiency. Branco (1997, 2001) obtains related results in a somewhat different framework.
In order to capture some of the most salient features of actual FCC auction environments, we consider a framework in which two licenses are auctioned by the seller to two different sets of participants: unit and bundle bidders. Unit bidders are interested in one license only, bundle bidders are interested in both. We also assume that there exist positive synergies or complementarities between licenses. These synergies may arise, for instance, from the saving of infrastructure costs whenever the two licenses correspond to two neighboring regions.
We show that every perfect Bayesian equilibrium (PBE) of the JAMO induces a PBE in the SEAMO, while the reverse is not true. This result implies that the set of equilibria of the JAMO is strictly smaller than that of the SEAMO. The rules of the JAMO eliminate many (unwanted) collusive or signaling equilibria that are equilibria of the SEAMO. In particular, jump bid equilibria constructed in Gunderson and Wang (1998) and collusive equilibria constructed in Engelbrecht-Wiggans and Kahn (1998) and Brusco and Lopomo (2002) are not equilibria of the JAMO.
Knowing that most signaling and collusive equilibria of the SEAMO do not have a counterpart in the JAMO, leaves the issue of the optimal bidding behavior in the latter mechanism open. This paper also provides a novel approach to characterize a "competitive" (symmetric) perfect Bayesian equilibrium (PBE) of the JAMO. While unit bidders' optimal behavior simply corresponds to that in a standard one-object second-price auction, the derivation of the bundle bidders' optimal strategies in a framework in which they have both private stand-alone values for the objects and a common value given by the synergy is by far less trivial.
Specifically, we adopt an indirect but constructive approach. We first extend to the two-object case the strategic equivalence between the JAMO and the Survival Auction (SA). The SA was used by Fujishima et al. (1999) who proved the strategic equivalence with the Japanese Auction in the one-object case. The SA consists in a finite sequence of sealed-bid auctions. At each round the auctioneer announces only the lowest bid on both licenses, and the identity of the "losing" bidder(s), that is, the one(s) who submitted the lowest bid(s). In the following round, the losing bidder(s) of the previous round are not allowed to bid on the object(s) on which they had submitted the losing bid(s), and the surviving bidders are allowed to submit new sealed-bid offers provided that they are not less than the losing bid(s) of the previous round. Bundle bidders' optimal behavior in the SA can be characterized by using backward induction. Starting from the terminal nodes of the game tree, we can reconstruct the bidder's continuation payoffs at each decision node of the game. Thus we derive optimal bids in the SA which in turn translate to optimal exiting times in the JAMO.
The remainder of the paper is organized as follows. In Section 2, we describe the actual auction rules and in Section 3 we construct our indirect approach to characterize a PBE of the JAMO. In Section 4, we consider a series of collusive and signaling equilibria and show that many equilibria of the SEAMO have no counterpart in the JAMO. Section 5 concludes with directions for future research. The proof of Proposition 2 is contained in the Appendix.
2 Three ascending auctions
2.1 The framework
Throughout the paper we work with a framework close to the ones of Krishna and Rosenthal (1996) and Brusco and Lopomo (2002). Two objects are auctioned to a set of participants of two types: M bundle bidders who are interested in both objects and [N.sub.k] unit bidders who are interested in only one of the two objects, k = 1, 2. Both bundle and unit bidders draw their values independently from some smooth distribution F with positive density f, both defined over [0, 1]. Let [v.sub.k] and [u.sub.k] denote the value of object k = 1, 2 to a bundle and to a unit bidder respectively. The value of the bundle [v.sub.B] to a bundle bidder is greater or equal than the sum of stand-alone values, that is,
[v.sub.B] = [v.sub.1] + [v.sub.2] + [alpha],
where [alpha] [greater than or equal to] is commonly known and coincides across all bundle bidders. The nature of bidders, bundle and unit, is also commonly known. (6)
We further restrict the analysis to the following cases: (i) [v.sub.1] = [v.sub.2] [member of] [0, 1] and [alpha] [greater than or equal to] 0; (ii) [v.sub.1], [v.sub.2] [member of] [0, 1] and [alpha] = 0; (iii) [v.sub.1], [v.sub.2] [member of] [0, 1] and [alpha] > 1; Krishna and Rosenthal (1996) consider case (i); Brusco and Lopomo (2002) consider cases (ii) and (iii). We will refer to these three cases throughout the paper.
Finally, we introduce some notation that will be used later. Let F(z | t) = F(z)-F(t)/1-F(t) be the distribution function of a unit bidder's valuation given that his valuation is at least t and f(z|t) = F'(z|t). Let [F.sub.N]([dot] | t) denote the distribution function of the lowest valuation among N unit bidders given that their valuations for the object are at least t, and let [f.sub.N]([dot] | t) = [F'.sub.N] ([dot] | t). Since unit bidders' valuations are i.i.d. random variables with distribution function F(dot), we can write [F.sub.N](z | t) = 1 - (1 - F(z | t))[.sup.N].
2.2 Auction rules
The two main auction mechanisms we consider (JAMO and SEAMO) are more or less simplified versions of the simultaneous ascending auctions used by the FCC for the sale of spectrum licenses in the US. The third mechanism (SA) is equivalent to the first (JAMO) and is used mainly to simplify some of the analysis. We briefly describe the rules. All auctions have in common a tie-breaking rule that assigns the object with equal probability; also, we assume it is specified before the auction begins on which objects the different bidders are going to bid. In particular, we assume throughout the paper that bidders bid only on objects they value, that is, we assume local bidders bid on one object only. This always happens if bidders have to pay a participation fee proportional to the number of objects they want to bid for.
JAMO: Prices start from zero for all objects and are simultaneously and continuously increased on all objects until only one agent is left on a given object, in which case prices on that object stop and continue to rise on the remaining auctions. Once an agent has dropped from a given object, the exit is irrevocable. The last agent receives the object at the price at which the auction stopped. Whenever an agent exits one object, the clock (price) temporarily stops on both objects giving the opportunity to other bidders to exit at the same price. If all active bidders exit simultaneously on one object, then the object will be allocated randomly among those bidders that exit after the price has stopped. The number and the identity of agents active on any object is publicly known at any given time. The overall auction ends when all agents but one have dropped out from all objects. We refer to this mechanism as the Japanese auction for multiple objects (JAMO); some also refer to it as the English clock auction.
SEAMO: The auction proceeds in rounds. At each round, n = 1, 2,.., each bidder submits a vector of bids where bids for single objects are taken from the set {[null]} [union] ([b.sup.k](n-1),+[infinity]), where[null]denotes "no bid", and [b.sup.k](n-1) is the "current outstanding bid", that is, the highest submitted bid for object k up to round n-1. Thus for each object k a bidder can either remain silent or raise the high bid of the previous round of at least [nu] > 0, ([nu] arbitrarily close to zero). All objects close simultaneously. The auction ends if all bidders remain silent on all objects, and the winners are the "standing high bidders" determined at round n - 1 and they pay their last bids. If there is more than one standing high bidder on one object, then that object will be allocated randomly among these bidders. Given the simultaneity of closing, we refer to this mechanism as the simultaneous English auction for multiple objects (SEAMO).
Two basic differences distinguish the two mechanisms. First, the JAMO does not allow for rounds of bidding; bidders press buttons corresponding to the objects on which they wish to bid; by releasing a button, a bidder quits that auction irrevocably; thus, bidders have "smaller" strategy spaces than in SEAMO; in particular they have no influence on the pace at which prices rise. Second, closing is not simultaneous in the JAMO but rather object-by-object. We shall highlight the role of these distinguishing features in the emergence of collusive and signaling equilibria.
SA: The auction proceeds in rounds. At round n = 1, 2,.., each bidder submits a vector of sealed bids for objects on which they are allowed to bid. Bids for a single object are taken from the set [[b.sub.min](n-1), +[infinity]), where [b.sub.min](0) = 0and for n > 1, [b.sub.min](n-1) is the lowest among all bids submitted during the previous round on all objects. In the following rounds, all the bidders who offered [b.sub.min](n) in the current round, are not allowed to bid again on the object on which they submitted [b.sub.min](n). At the end of each round the auctioneer only announces [b.sub.min](n), the object for which [b.sub.min](n) was submitted and the identity of the bidder(s) that submitted that bid. An object is attributed to the last bidder having the right to bid on that object and the winner will pay an amount of money equal to the last lowest bid on that object. If all active bidders bid the minimum admissible bid [b.sub.min](n) on one object, then that object will be randomly...
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