Overbidding in Auction Rings
Pre-Auction Bidder Behavior under a Novel Cartel Mechanism
Master Thesis
Kas Buunk
(361766)
Economics of Management and Organisation
Erasmus University Rotterdam
Supervisor: Dr. J.J.A. Kamphorst
Co-reader: Dr. E. Maasland
July 2015
Abstract
This paper proposes a novel cartel mechanism. For cooperative bidding, ring members receive a compensation that depends on their pre-auction bid, which by cartel mechanism design depends on their valuation in equilibrium. Rather than receiving a lump sum transfer, as is proposed by most economic literature on collusive practices at auctions, ring members are compensated proportionately,to the extent to which they have contributed to a lower price at the main auction. While this creates incentives in favor of untruthful bidding, it is preferred by those bidders with high valuations, who are typically the initiators of cartel practices. The main results are that given the model’s specifications, (1) bidding one’s valuation in the pre-auction is suboptimal; and (2) an equilibrium is characterized in which it is optimal for ring members to overbid in the pre-auction.
Content
I – Introduction
II – Literature Review
III – Model
IV – Analysis
A: Main auction bidding strategies
B: Pre-auction bidding strategy
V – Concluding remarks
Appendix
Appendix 1
Appendix 2
Appendix 3
Appendix 4
References
I – Introduction
At auctions, the threat of bidder coalition has disturbed auctioneers and inspired theorists to design auction mechanisms to restrict the possibility to collude. Second-price auctions are particularly sensitive to a cartel, but remain the most dominant mechanism nonetheless. Both the Vickrey and English variants are equally sensitive, in the sense that a simple cartel mechanism is equally stable for both auctions. Bidder coalition, known as a ring, realizes a cut in price as soon as the ring contains the two bidders with the highest valuations of all bidders. The ring members agree that only one party bids, while the others forgo the main auction. Among the ring members, the good is auctioned at a pre- or post-auction to determine who is willing to pay most. Ring members are compensated for bidding cooperatively (Graham & Marshall, 1987).
At every auction ring, bidders face the problem of asymmetric information, either in terms of private valuations or private signals that convey information about the item’s common value. For the ring to coordinate on the determination of the final owner and the allocation of ring earnings, the ring members must resolve this information asymmetry through a cartel mechanism.
A vast body of literature has developed over the past few decades on auction theory, but collusive practices remain relatively unexplored. Graham & Marshall (1987)describe how ring members do a pre-auction, which proceeds as follows. First, each member submits a sealed bid to the ring center, which determines the highest and second-highest bid. The highest pre-auction bidder is selected as the ring leader and is allowed to bid at the main auction, while the other members aredictated to refrain from bidding. If the ring leader wins the main auction, he additionally pays the ring center the difference between the second-highest pre-auction bid and the resulting price at the main auction, provided this difference is positive. Note that this difference signals the ring earnings, i.e. impact of the coalition on the price. The ring center pays a lump sum to each ring member as compensation, which equals the expected ring earnings divided over the number of ring members.
The key result from their study is that the resulting equilibrium is incentive-compatible. This implies that all ring members in the pre-auction bid truthfully, i.e. bidding and thus revealing their valuation. However, their paper relies on the rule that all ring members are equally compensated. This encourages free-rider behavior, leading potentially to a ring within a ring, callednesting. This may in turn be detrimental to the stability of the ring.
Even though this distribution design has the advantage of being incentive-compatible, ring members with higher valuations of the good may feel the weight of free-riders to be too heavy. Graham et al.(1990) propose an alternative compensation formula that distributes the ring earnings more fairly. Although it is debatable how a fair distribution should be defined, it is deemed fair in the sense that those with greater contribution to a lower priceat the main auction should gain more than those with lower or no contribution. This fair distribution divides the ring earnings proportionally over the ring members, to the extent to which they have contributed to a lower price. This fairness should improve the ex-ante appeal of a ring for bidders with high valuations and discourage a ring within a ring.
Graham and Marshal (1987) use the pre-auction as a cartel mechanism, in which the distribution of ring earnings is equal across its members, while Graham et al. (1990) use a post-auction and proportionate compensation. The pre-auction is more attractive than the post-auction, while the proportionate compensation is preferred to the fixed one as is explained in section II.
Mailath and Zemsky (1991) show that efficient collusion by any subset of bidders in second-price private value auctions is possible. Collusion is efficient if the ring members do not have an incentive to deviate from the agreement regarding anyone’s participation in the main auction.For efficient collusion, however, a bidder’s payoff must be independent of her valuation. Thispaperinvestigates whether an incentive-compatible equilibrium exists for this compensation function. Indeed, given the model’s specification, results show that no equilibrium exists in which ring members bid and thus reveal their valuation in the pre-auction.
As will be shown below, the ring members’ payoff positively depends on their bid in the pre-auction, giving them an incentive to bid higher than their valuation. If a bidder has won the pre-auction, he is restricted at the auction to bid at least his pre-auction bid. The reason for this is simple. The purpose of the pre-auction is to act as a platform for ring members to communicate their valuations. For efficiency purposes, the bidder with the highest valuation should win the auction. The resulting price and compensations are ultimately trivial, since they only serve to distribute the ring earnings among its members. As the bidder with the highest valuation should win the auction, he should not win as a result of overbidding someone with a higher valuation. By restricting the bid of the ring leader to at least the pre-auction bid, the cartel mechanism discourages overbidding. A practical interpretation of the restriction could be that the ring leader would suffer immense reputation loss if he bids lower than his pre-auction bid, since this is abuse of a system that is meant to convey private information.
As a consequence, the first result is that there exists no equilibrium in which ring members always bid their valuation. Given that one bids his valuation in the pre-auction, the other has an incentive to bid higher than his valuation, and more so for lower valuations. The second main result characterizes the symmetric equilibrium and describes that, in equilibrium, all ring members overbid to some degree in the pre-auction, except for those whose valuation is the highest possible one.
The paperis organized as follows: Section II discusses the relevant economic literature on bidder collusion and how it relates to this paper. Section III describes the model. Section IV analyzes the model and describes the results. Section V offers some concluding remarks and discusses directions for further research.
II – Literature Review
For the past decades, auction theory has flourished into a vast body of literature. One assumption made in most analyses is that bidders act non-cooperatively. As cooperative bidder behavior is in fact widespread, Englebrecht-Wiggans (1980) and Milgrom and Weber (1982) have noted the need for a better understanding of the behavior and effects of bidder coalitions at auctions.
While Mead (1967) merely stated that oral auctions were more vulnerable to collusion than sealed (first-price) auctions, Robinson (1985) was the first to formally analyze the effect of different auction methods on the probability and stability of cartels among bidders. He shows that cartels are more stable in second-price auctions than first-price auctions. The crucial difference in terms of cartel stability is whether cartel members regret their strategies if deviation occurs. All of his results regarding the oral ascending auction apply, with minor changes, to the sealed-bid second-price auction. The oral descending and the sealed-bid first-price auction are also isomorphic.
In English auctions of,among others, antiquities, fine art, fish, timber and wool, a common cartel structure is for one arbitrary member to bid solely, without competition of his fellow members. Next, the good is auctioned in a post-auction to determine the final owner. The cartel shares the ring earnings among its members, the earnings being equal to the difference between the prices reached in the post-auction and in the main auction (Cassady 1967, Ch. 13; Cooper, 1977, pp. 35-38; Gruen, 1960). A drawback of this post-auction arises when the highest valuation outside the cartel is higher than the highest valuation within the cartel. Then, the ring will pay more for the item than the ring member with the highest valuation is willing to pay, resulting in negative ring earnings.
Graham & Marshall (1987) conducted interviews of auctioneers and of bidders who participated in rings. The valuable insights they provided led to some stylized facts: (i) rings exist and have a stable form of organization over time; (ii) rings adopt strategies that eliminate competition among its members at the main auction and yet ensure that no item will be sold to a non-ring bidder or be retained by the auctioneer at a price below the maximum of the individual ring members’ private valuations; (iii) ring earnings are shared among its members rather than accruing entirely to the ring leader; (iv) rings have open membership policies in the sense that bidders who are expected to be competitive at the main auction are invited to join; (v) the auctioneer responds strategically to the existence of a ring by means of a reserve price; (vi) rings attempt to conceal their existence from the auctioneer.
Pre-Auction Knockout
Graham & Marshall (1987) continue to describe the pre-auction knockout (PAKT), which is a lot like the pre-auction in this paper. In their PAKT, prior to the main auction, an independent agent outside the auction is appointed as the ring center and pays a non-contingent constant to each ring member. Heacts as both mediator and banker for the ring, selects the sole bidder at the main auction and distributes the ring earnings. The payment to each ring member is a lump sum transfer and hence cannot affect incentives.
While this makes the PAKT an incentive-compatible and durable mechanism, its distribution of earnings causes the ring to lack stability even so. As Graham & Marshall (1987) describe, bidders with higher valuations are encouraged to form a ring within the ring, called nesting, because bidders with lower valuations enjoy high benefits by free-riding; high in the sense that the ring member with the lowest valuation receives the same compensation as the one with the second-highest valuation. The former’s absence has had no or little impact, depending on the highest valuation outside the ring, while the latter’s absence in the main auction may single-handedly cause the price to drop.
Nesting occurs when ring member types are considered to be heterogeneous. It is common then to observe a nested coalition structure in practice. This facilitates differential payments to the ring members, where a nested secondary pre- or post-auction is conducted at each level of nesting (Graham et al., 1990).
To further illustrate the frustration that ring members with high valuation may experience, consider the fact that McAfee & McMillan (1992) differentiate between weak and strong cartels. The strong cartel differs predominantly from the weak by its ability to exclude bidders who could never win in a non-cooperative auction, but seek to free-ride on the ring earnings.
A novel cartel mechanism
The bidders with the highest valuations are understandably the initiators of cooperative behavior and have little to gain by allowing bidders with low valuations to enjoy an equal share of the ring earnings. That forms the motivation for the analysis of ring members’ behavior undera compensation formula that would likely be preferred by those bidders whose participation in the ring is essential for any ring earnings.
In section III of this paper, the model prescribes that compensation of ring members depends on their pre-auction bid. The general model specifications presented here are in essence very similar to that of the model in Graham et al. (1990). Graham et al. describe side-payments that also depend proportionally on the ring members’ pre-auction bids. These side-payments are conceptually the same as the compensations in this paper. The same holds for the ring members’ payoff functions of their model and the ones in this paper. Their model and the one in this paper differ in some aspects. First and foremost, their model is a post-auction that depends on the crucial assumption that the highest valuation of the ring members is greater than the highest of the outsiders. This is due to the fact that a post-auction forces the ring to win the main auction before continuing to allocate the item and ring earnings. By rather doing a pre-auction, this assumption can be dropped, expanding the model beyond the case in which the ring necessarily wins the main auction.Second, though the main principles of the pre- or post-auction knockout are maintained, they are modeled in a more practical, realistic and direct way, while maintaining a simple analysis. This causes the payoff functions to differ in functional form, but not fundamentally.
One of the realistic model specifications that this paper proposes is the role of the ring leader. The highest bidder in the pre-auction becomes the ring leader, whose role encompasses the role of Graham & Marshall’s (1987) ring center. If he wins the main auction, he gains his valuation of the item sold and pays a price equal to the second-highest of all bids at the main auction. Those ring members whose reported PAKT bid is higher than the main auction price receive a compensation paid directly out of the ring leader’s pocket. The benefit of introducing the ring leader and him directly compensating other ring members is that it makes the model realistic and straightforward in transfers of payoff and players’ participation at the pre-auction and the main auction. The compensation is proportionate on the degree to which the respective ring member has contributed to a lower price and is expressed in a formula in section III-A, similar to Graham et al. (1990).
In addition to the compensation being equal among its members, Graham & Marshall (1987) model the compensation to be paid in advance, as a function of ex-ante expected ring earnings divided by the amount of members. Thus, if the ring does not win the main auction, every ring member still receives compensation. In this paper, compensation is conditional on the case in which the ring leader wins the main auction and pays a price lower than the second-highest PAKT bid. Again, this is in line with reality while maintaining a simple analysis.
Managingincentives
Recall that the purpose of the pre-auction is to serve as a platform to convey private information on ring members’ valuations. Given now that one’s compensation depends positively on his pre-auction bid, some mechanism should secure the usefulness of the pre-auction. If not, opportunistic behavior is encouraged; any arbitrary ring member could bid an outrageous amount in the pre-auction, after which to bid his valuation in the main auction, thus foregoing the objective of the pre-auction. If we extend the static model presented in section III to an infinitely repeated game, a dynamic equilibrium should exist in which the ring punishes a member if he gains ring leadership by overbidding in the pre-auction, after which he bids his valuation in the main auction. To ensure that members comply with the cartel mechanism, one could appeal to a grim trigger strategy in an infinitely repeated auction context. This is part of the Folk Theorem of repeated games (Aumann, 1981; Abreu, Pearce, and Stacchetti, 1986).
Even though the model lacks such dynamics, following the logic of McAfee & McMillan (1992), cooperation should be an equilibrium strategy in a repeated non-cooperative game. This cooperation could be interpreted as at least acting as if the pre-auction bid is one’s valuation. One way to model this in a static game is by simply restricting the ring leader at the main auction bid to bid at least his pre-auction bid.Deviation by the ring leader, i.e. bidding lower than one’s pre-auction bid, can be threatened with non-cooperative utility in all subsequent auctions, since this does not comply with purpose of the pre-auction. Such a threat will indeed discourage opportunistic behavior, provided that the discount factor is high enough. Although there may be multiple equilibria in this multi-stage setting, it obviously gives the cartel mechanism designer incentives to coordinate on this desired equilibrium.