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Article Excerpt
Summary. Most of the literature on collusive behavior in auctions ignores two important issues that make collusion difficult to sustain at least in one-shot interactions: the detection of cheating and the verification of bids. Colluding bidders may deceive each other by using shill bidders. Also, if the identities of the bidders and their bids are not published then it would be difficult to verify the bid of a colluding bidder. This paper addresses these problems in one shot second price auctions where one bidder offers another bidder a side payment in exchange for not participating in the auction, while the number of other bidders is stochastic. In spite of the barriers to collusion mentioned above, a simple side payment mechanism which depends only on the auction price is introduced. It induces a successful collusion, eliminates the verification problem, provides no incentive for the use of shill bidders and guarantees that the proponent obtains ex-post non-negative payoff. The colluding bidders are ex-ante strictly better off compared with the competitive case, irrespective of their types.
Keywords and Phrases: Auctions, Collusion, Shill bidders, Signaling games.
JEL Classification Numbers: C72, D44, D82.
1 Introduction
The literature on collusion in auctions typically disregards two important issues that are serious barriers to collusion, especially in one-shot interactions. The first barrier is the verification problem which arises if the identities of the bidders and their bids are not published. In this case it is difficult to verify that colluding members behave in accordance to their agreement to manipulate the auction outcome. (3) The second issue is the possible use of shill bidders. If a bidder offers another bidder a side payment in return for not participating in the auction they may have incentive to use shill bidders, assuming that the set of other potential bidders is random or unknown to the other bidder (see McAfee and McMillan, 1987; Das Varma, 2002). If the proponent offers a side payment contingent on winning the auction, he himself has an incentive to use a shill bidder to avoid paying. If the proponent offers an unconditional side payment, the respondent has the incentive to use a shill bidder while collecting the side payment. The use of shill bidders may not be relevant in auctions for licenses or procurement auctions where verifying the "real" identity of the winner is straightforward, it is however relevant in auctions of high value items such as antiques, rare stamps and special pictures. This paper addresses these two problems in a simple form of collusion in a second price sealed bid auction for a single product. (4)
Most of the literature (with the exception of Eso and Schummer, 2004, hereafter ES, 2004) model collusion by assuming that an incentiveless third party designs and administers a revelation mechanism, making or receiving payments from bidders based on relevant information collected by the ring's members. (5) This paper examines a more-realistic scenario, especially in one-shot interactions, where collusion is initiated by one of the participants whose goal is to maximize his own surplus (and not the joint surplus).
There is another serious barrier for collusion of a different kind. This is the inability of the colluding bidders to write binding agreements, since their activity is illegal. For instance, the proponent can refrain from paying the side payment after the conclusion of the auction and the respondent has no means to retaliate. On the other hand if the proponent pays the respondent cash in advance, nothing prevents the respondent from bidding his true valuation. Thus, collusion in one-shot interactions requires the presence of a credible third party to enforce the deal. This will enable the proponent to deposit cash in the hand of the third party to guarantee that the respondent gets paid only after fulfilling his part of the agreement (assuming that this can be verified).
The verification problem and the use of shill bidders are discussed in the mechanism design approach in Marshall and Marx (2004) (hereafter MM, 2004). Following the framework of GM (1987), they propose two "knock-off auction" collusion mechanisms in which an incentiveless ring center makes payments to all ring members, then each ring member reports a type to the center. The center then imposes payments on the ring members based on these reports. If the bids of the ring members are not observable, they can use a bid coordination mechanism, by which the payment to the center is based only on the bidders' reports. If all bids are published, a bid submission mechanism can be designed in which the required payments are based on both the reports of the ring members and their bids at the auction. Taking into account that ring members may use shill bidders to bid on their behalf, MM (2004) show that for a second price auction there exists a profitable, ex-post efficient collusion mechanism that suppresses all ring competition and allows the ring to attain the entire collusive gain while for first price auctions collusive payoffs are weakly reduced.
In contrast to the mechanism design approach, this paper studies the impact of shill bidders in a one shot simple form of collusion. In our model two bidders (in addition to a random number of other bidders) attempt collusion to manipulate the outcome of a second price auction. One of them, the proponent, takes the initiative and offers a side payment to the other bidder, the respondent, in return for not participating in the auction. Every bidder privately evaluates the object and the side payment offer may convey information about the proponent's valuation. For simplicity we deal with the case where a bidder has at most three possible valuation levels (types).
The paper considers two scenarios. In the first scenario the identities of the bidders and their bids are published after the completion of the auction. The only relevant issue here is the use of shill bidders. For this scenario, one can use a straightforward side payment method. Namely, the proponent offers to pay the respondent a predetermined amount in return for not participating in the auction. This simple form of collusion has been already studied in ES (2004) for the continuum type case. In their paper however the use of shill bidders is excluded. They show that in every robust equilibrium a proponent having a type below a certain threshold reveals his type through his side payment offer, while all higher types "pool" and offer the same side payment. The respondent of a certain type accepts the offer of a proponent with a lower type, if the offer exceeds his expected payoff in the auction and the equilibrium results, with positive probability, with inefficient allocation. Every type of respondent accepts the offer of the proponent with the threshold type and hence the proponent types not below the threshold level "pool" and offer the same side payment.
In this paper it is shown that when the use of shill bidders is possible a perfect Bayesian equilibrium satisfying the Intuitive Criterion (a "sensible" equilibrium) always exists. In contrast to ES (2004), no successful collusion occurs between a respondent of a certain type and a proponent of a lower type. A respondent who believes with positive probability that an offer is made by a lower type proponent is better off accepting the side payment offer while sending a shill bidder to earn in addition the competitive auction payoff. As a result, the allocation of the good is always ex-post efficient in every sensible equilibrium, whether or not collusion occurs.
Moreover, in the case where bidders have three possible values, a sensible separating equilibrium always exists but a full pooling sensible equilibrium does not exist. Opposite to ES (2004), the only possible semi-pooling sensible equilibrium is the one where the two low proponent types pool while the highest proponent type separates himself from the other two. The reason is straightforward. When the highest proponent type pools with a lower type and offers a positive side payment, the highest respondent type is best off accepting the offer while sending a shill bidder. To obtain a contradiction we use the Intuitive Criterion and show that if the highest proponent type increases his offer to a point where it becomes equilibrium dominated for the other two proponent types to mimic him then the respondent will accept the offer and will not use a shill bidder, irrespective of his type and the payoff of the high proponent type is increasing.
It is shown however, that if we further restrict our attention to perfect Bayesian equilibrium which satisfies the Forward Induction principle (rather than perfect Bayesian equilibrium which satisfies the Intuitive Criterion) only separating equilibrium exists.
The second scenario deals with the case where not all bids are published. It is shown that the proponent can make a cash offer contingent on the auction price, which induces a collusion, eliminates the need to verify the bids, provides no incentive to use shill bidders and guarantees that the proponent obtains ex-post a non-negative payoff (ignoring his bidding cost), irrespective of the strategy of the respondent.
The side payment method is defined as follows. The proponent announces a number (auxiliary price) and makes a commitment to pay the respondent the difference between this number and the auction price, provided that this difference is positive. Otherwise, he pays zero. The respondent is automatically eligible to collect the side payment irrespective of his bid and on whether or not the proponent wins the auction. Thus the verification problem is avoided. The auxiliary price can be properly chosen to provide the respondent with the right incentives to act in the best interest of the proponent.
This procedure affects the bidding behavior of the two colluding parties in the auction. For instance, it is no longer true that bidding truthfully is a dominant action in the auction or even the best reply action for the proponent. Nevertheless, it is shown that irrespective of the number of types in any strategy which is not weakly dominated, any type of proponent announces an auxiliary price not greater than his value, then bids truthfully and does not use a shill bidder.
For simplicity, the full characterization of the sensible equilibria is analyzed only for the two types case. It is shown that the game always has a sensible equilibrium. The parameter space is divided into three regions. In one of them the equilibrium must be separating, in the other region it must be pooling, and in the third region both types of equilibria exist. In a pooling equilibrium the two colluding parties always end up with a successful collusion, even though it may not be ex-post efficient. In a separating equilibrium, they do not collude if the proponent has the low type and the respondent has the high type. Otherwise, they collude successfully and the outcome is ex-post efficient.
This suggests that collusion is more likely to occur with the second method of side payment since in the region where only pooling equilibrium exists, successful collusion occurs irrespective of the types of the proponent or the respondent. The proponent may end up with an ex-post lose with the first method but he is guaranteed to obtain ex-post non-negative payoff (ignoring his bidding cost) with the second method, on or off the equilibrium path. Also, in the second scenario the proponent obtains a higher expected payoff and consequently, he is better off with the second method even when the bids and the identities of the bidders can be verified.
It is also shown that the colluding parties in both scenarios are ex-ante strictly better off compared with the competitive case, irrespective of their types.
Finally, methods of side payments that are contingent on whether or not the proponent wins the auction, induce the proponent to use a shill bidder to avoid paying the respondent, and consequently they in general induce no collusion.
The paper is organized as follows. The model is presented in the next section. The first scenario is studied in Section 3. The second scenario is studied in Section 4. Section 5 provides a short conclusion. Most of the proofs appear in the Appendix.
2 The basic model
A single object is put up for sale in a second-price sealed bid auction. Two bidders A and B consider a collusion in order to manipulate the outcome of the auction. It is assumed that other than A and B, there are m potential participants in the auction and m is a random variable with a certain commonly known probability distribution. Let M be the set of active bidders other than A and B.
Every one of the bidders assigns a private value to the object which can be one of the following n values : [v.sub.n], [v.sub.n-1],..., [v.sub.1] with probabilities [[alpha].sub.n], [[alpha].sub.n-1],.., [[alpha].sub.1] respectively, where [[alpha].sub.t] > 0, t = 1,... n, [[summation].sub.t=1.sup.n] [[alpha].sub.t] = 1 and [v.sub.n] < [v.sub.n-1...] < [v.sub.1]. Let T = {1,... n} be the set of types. For every t [member of] T, let [A.sub.t] and [B.sub.t] be the t-type proponent and respondent respectively. The types are selected independently across bidders. The seller's reserve price is denoted by [v.sub.0]. We assume that the seller is a non-strategic player who announces his true reserve price [v.sub.0], and [less than or equal to] [v.sub.0] < [v.sub.n].
Let [[beta].sub.0] be the probability that M = [null] (that is, the only potential bidders are A and B). For t [greater than or equal to] 1, denote by [[beta].sub.t], t [member of] T, the probability that M...
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