Repeated auctions with impatient bidders and externalities.

• Author: Rhee, Ki-Eun

1. Introduction

   Collusive bidding behavior has been conventionally studied in a static setup, often relying on some unspecified repeated-interaction story for the enforcement of a cartel. (1) It is not clear, however, whether the results from static auctions immediately carry over to a repeated setup when bidders are impatient and the enforcement or the participation constraint is binding. Moreover, many empirical analyses for detecting collusion rely on an explicitly repeated structure, rendering a theoretical analysis of collusion in a repeated setup necessary. (2) Several recent theoretical developments in repeated auctions have been made by Aoyagi (2003, 2007), Skrzypacz and Hopenhayn (2004), and Blume and Heidhues (2006). Taking standard auctions as stage games, these articles present various collusive bidding schemes with asymmetric punishments that outperform the bid-rotation scheme, which is known to be optimal in static auctions (McAfee and McMillan 1992).

   This article studies collusive bidding behavior in repeated auctions, among both patient and impatient bidders, when the outside option of the participation constraint is endogenized because of externalities. A stage game involves negative externalities among the bidders when any losing bidder is worse off if a rival wins the good than if no one wins it. Auctions with externalities have been introduced most notably by Jehiel and Moldovanu (1996) and Jehiel, Moldovanu, and Stacchetti (1996). These auctions describe environments in which the outcomes of auctions affect the nature of ensuing market interactions among potential bidders. A wide variety of situations in which bidders are a set of oligopoly firms competing in the same market apply to this setting. For example, procurement auctions for highway construction contracts, studied empirically by Porter and Zona (1993), may qualify as repeated independent private value auctions with negative externalities except that the lowest bidder wins. (3) Consecutive highway lettings were distributed by the Department of Transportation (DOT) to a set of oligopoly firms competing in the same industry. Negative externalities may arise because a firm that wins the auction may experience learning by doing, which provides the winning firm with a comparative cost advantage, or because the winning firm may gain a reputation that gives the firm an advantage in future private sector competition. As Jehiel and Moldovanu (1996) and Jehiel, Moldovanu, and Stacchetti (1996) discuss, the issue of participation is nontrivial in auctions with externalities because the ex ante payoff a bidder expects by not participating is no longer zero as in standard auctions but is rather endogenously determined by the actions of the other bidders. Thus, an analysis of collusive bidding behavior under both nonbinding and binding participation constraints seems especially worth studying for these auctions.

   The main findings of this article are that (i) there are no bidding wars along the equilibrium path, and bidders expect to receive the maximal continuation values when they are both patient and impatient; when bidders are impatient, the optimal collusive bidding scheme involves (ii) a lower threshold type above whom bidding starts when externalities are small or (iii) more frequent jumps (i.e., more sorting of bidders' types) when externalities are large. In both cases, seller's ex ante profit is higher when bidders are impatient.

   If we consider patient bidders to be the ones that are interacting in a stable market, (4) results carry an empirical implication: either less frequent sales or fewer bid levels in auctions that are let out in stable markets will be observed. The fact that the cartel manipulates both the amounts of bids and the probabilities of sales is worth noting. While one might wonder how a cartel can manipulate the probability of sales and how a sale can actually not occur in reality, such an incidence has been reported to happen. Porter and Zona (1993) note a spell of periods of nonbidding in their study of auctions for highway construction contracts. In the case of a contract let out in Long Island in February 1983, the DOT did not award the contract because eight bids that were submitted were unusually high relative to its own estimate. The same contract was let out again in May 1983. This time, the lowest bid was 20% higher and was submitted by the same lowest bidder of the February auction. The contract was once again not awarded. Porter and Zona (1993) note that such unusual bidding patterns caused the contract not to be awarded until 1987.

   This article is closely related to Rhee (2007) and Athey, Bagwell, and Sanchirico (2004). Rhee (2007) studies tacit collusion in auctions with externalities in a static setup. We extend the model to a repeated structure and analyze the case in which the enforcement of the cartel becomes an issue as bidders become impatient. Analytically, we draw on Athey, Bagwell, and Sanchirico (2004), who study infinitely repeated Bertrand games in which firms receive a privately observed i.i.d, cost shock in each period. For the case in which demand is inelastic, their analysis parallels that of a repeated first-price auction without externalities. We borrow the technical tools of Athey, Bagwell, and Sanchirico (2004) to transform the dynamic problem into a static mechanism design problem. The result that the cartel manipulates probabilities of sales can be also related to the optimal bidding scheme provided in Skrzypacz and Hopenhayn (2004), in which bidders collude by temporarily excluding a portion of bidders from the bidding process.

   Despite the vast literature on collusive behavior in auctions, studies of collusion in a repeated auction setup are quite limited in number. It may be more appropriate to study the enforcement issue of the collusive effort in a repeated setup. Aoyagi (2003), Skrzypacz and Hopenhayn (2004), and Blume and Heidhues (2006) have shown the existence of dynamic bidding schemes that yield higher payoffs than the static bid-rotation schemes. These articles share a common theme, as in Athey and Bagwell (2001), that asymmetric continuation values may act as implicit transfers among bidders. The analysis here differs in that this article focuses on symmetric cases in which transfers among bidders are not allowed. Thus, in the terms of McAfee and McMillan (1992), we focus on "weak" cartels rather than "strong" cartels. This implies that any exchange of information that may happen in the preauction stage carries no value to bidders because there is no means with which the bidders can distribute the gains from informational advantage. Fabra (2003) is another article that studies repeated auctions and focuses on the enforcement problem faced by the cartel. Fabra (2003) compares collusive efforts under uniform versus discriminatory auctions and shows that bidders' incentives to deviate are weaker in uniform auctions.

   This article is organized as follows. Section 2 introduces the model and formalizes the repeated game. A couple of static results for auctions with externalities are also reported in this section. The cartel's maximization problem for patient bidders is solved in section 3. In section 4, the critical discount factor below which bidders are considered impatient is defined, and the maximization problem for impatient bidders is solved. Section 5 concludes.

2. The Model and Preliminaries

   The Stage Game

   The stage game is a standard first-price independent private value auction with a seller's reservation price, except for the existence of negative externalities so that all losing bidders incur a cost whenever there is a winner. There are n risk-neutral potential bidders indexed by i = 1, ..., n. Each bidder's private valuation of the good is denoted by [v.sub.i]. (5) Bidder i's valuation is assumed to be independently and identically distributed on [V.sub.i] [equivalent to] [[x.bar], [v.bar]], with a distribution function F(x) and a density f(x). It is assumed that F(x) is twice continuously differentiable and f(x) is strictly positive on [V.sub.i]. The seller's reservation price is denoted by R [member of] ([b.bar], [bar.v]), and a > 0 denotes the symmetric negative externalities suffered by all losing bidders whenever there is a winner. No explicit side payments are available among bidders.

   Because of externalities, there are three possible outcomes of the auction for any potential bidder i: (i) bidder i wins the auction and gets [v.sub.i] - [b.sub.i], where [b.sub.i] is the submitted bid; (ii) no one wins, and bidder i gets 0 utility; and (iii) some other bidder j [not equal to] i wins, and bidder i's payoff is -[alpha]. The third possible outcome creates an incentive for any bidder i to prevent his competitors from winning, which is a consequence of assuming externalities.

   Let [b.sub.i] [not member of] [R.sub.+] denote the bid submitted by bidder i and let [q.sub.i]: b [right arrow] [summation] denote the equilibrium probability of winning, where b = ([b.sub.1], ..., [b.sub.n]) is the vector of bids and [summation] = {q [member of] [R.sup.n.sub.+]|[summation][q.sub.1] [less than or equal to] 1} is the set of probability vectors. A bidding strategy for bidder i is a mapping b(x) from the set of valuations or types to bids. A bidder's interim utility, which is the expected utility for bidder i when he realizes a type [v.sub.i] and bids [b.sub.i] while expecting others to bid according to b(x), is denoted by [bar.u]([b.sub.i], [v.sub.i]; b(x)), where the expectation is taken over other bidders' types. Because bidders suffer negative externalities when there is a winner, a bidder is no longer guaranteed a zero utility even when he submits a bid below the seller's reservation price or stays out of the auction. That is, any bidder should expect two different forms of interim utilities that are dependent on whether he submits a bid above the seller's...