Collusion in Second-Price Auctions under Minimax Regret Criterion Greys Soˇsi´c Marshall School of Business, University of Southern California, Bridge Hall 401, Los Angeles, California 90089,
[email protected] June 2006; to appear in Production and Operations Management Abstract Collusion in auctions, with different assumptions on distributions of bidders’ private valuation, has been studied extensively over the years. With the recent development of on-line markets, auctions are becoming an increasingly popular procurement method. The emergence of Internet marketplaces makes auction participation much easier and more convenient, since no physical presence of bidders is required. In addition, bidders in on-line auctions can easily switch their identities. Thus, it may very well happen that the bidders in an auction have very little, if any, prior knowledge about the distributions of other bidders’ valuations. We are proposing an efficient distribution of collusive profit for second-price sealed bid auctions in such environment. Unlike some known mechanism, which balance the budget only in expectation, our approach (which we call Random k) balances the budget ex-post. While truth-telling is not a dominant strategy for Random k, it is a minimax regret equilibrium. Keywords: Auctions, collusion, bidding rings, minimax regret, budget balance
1.
Introduction
With the development of on-line markets, auctions are becoming an increasingly popular procurement method. The Institute for Supply Management (ISM) and Forrester Research Inc. state in their last “Report On Technology In Supply Management” (October 27, 2003) that 25% of organizations participated in on-line auctions. eBay, the biggest on-line marketplace, had 180 million registered users in 2005. During that year, 71.8 million active users bought or sold items on eBay auctions, resulting in $12 billion gross sales in the fourth quarter of 2005. The most common auction types are: first-price sealed bid auction, second-price sealed bid (Vickrey) auction, ascending open bid (English) auction, and descending open bid (Dutch) auction. In a first-price sealed bid auction, the bidders submit sealed bids and the item is awarded to the highest bidder, for the price he bid. The difference between the first-price and the second-price sealed bid auctions is that in the latter case, the highest bidder, who is awarded the item, has to pay the price of the second-highest bid. In the English auction, the price is successively raised until only one bidder remains. In the Dutch auction, the auctioneer calls an initial high price and then lowers the price until one bidder accepts the current price. For a detailed analysis of auctions, see, e.g., Vickrey (1961), McAfee and McMillan (1987), Milgrom (1989), Wilson (1993), or Klemperer (2003). In an effort to decrease competition and increase their profits, bidders in an auction may decide to form a bidding ring. The gain that a bidding ring realizes compared to noncooperative bidding is called the collusive surplus. Although collusion among bidders, in general, is not legal, formation of bidding rings is a common occurrence, and is being studied and documented in several papers. For instance, Pesendorfer (2000) analyzes the bidding for school milk contracts in Florida and Texas during 1980s, where firms were convicted of bidrigging, while Cramton and Schwartz (2000) analyze collusion in FCC spectrum auctions. For a good survey of collusion in auctions, see Hendricks and Porter (1989). The main questions that arise with the collusion of players are: • How can a bidding ring enforce the agreed upon mode of behavior? • How should the collusive surplus be allocated among the ring members? The literature on bidding rings assumes certain distribution of bidders’ private valuation, and analyzes allocation mechanisms that will induce bidders’ collusive behavior. In this 1
paper, however, we assume that each player knows only his own private valuation, and has no information about distributions of valuations of the other bidders. We provide answers to the above questions in this framework. In the next section, we give a brief literature review and discuss collusive behavior and some mechanisms for coordination of bidding rings, along with our motivation for this work. We then discuss our method for comparing alternative bidding strategies without knowing distributions of other bidders’ valuations in Section 3. We present our model in Section 4, and analyze core membership in Section 5. Finally, we give some concluding remarks at the end.
2.
Collusion in Auctions
Ring membership requires both competitive and cooperative behavior. Each bidder individually decides, with the maximization of his individual profit in mind, whether to join a ring or not, what bid to submit at the main auction, and what individual valuation to report (if needed) to the ring or an independent party selected by ring members. On the other hand, decisions about the bids that should be placed in the main auction and about the distribution of the collusive surplus are done in a cooperative manner. There are two basic models of collusion – with and without side payments among the bidders. McAfee and McMillan (1987) refer to them as strong and weak cartels, respectively. Collusion models without any side payments are the most difficult to detect in practice. Members of a bidding ring with no side payments select a winner for each of the auctions in which they participate. The selection of the winner can be done in a number of ways – he or she may rotate among all of the ring members, or some external devices (e.g. phases of the moon) can be used. Since the communication among the ring members is minimal, detection of a weak cartel is quite difficult. However, the downside of this approach is that the auctioned item, in general, will not be awarded to the bidder who values it the most, thus implying inefficiency of the mechanism. When transfers among ring members are allowed, mechanisms can be implemented that lead to efficient outcomes. Graham and Marshall (1987) (see also Holmstrom and Myerson, 1983) provide a list of requirements that a mechanism for coordination of a ring’s behavior should satisfy and a classification of these mechanisms. Namely, a mechanism should induce voluntary participation of ring members, and it should induce reporting of bidders’ private valuations and submission of agreed upon 2
bids at the main auction. A mechanism is said to be durable if the ring members would never jointly deviate to another mechanism, even if they knew more than just their own valuations. A mechanism is said to balance the budget if for every realization of values, the net payments from agents sum to zero. Graham and Marshall (1987) suggest a mechanism called the second-price pre-auction knockout (PAKT), and show that it is an efficient and durable mechanism that guarantees voluntary participation, induces truthfulness in the pre-auction, and induces bidders to participate as agreed in the main auction. Truthfulness of bidders in the pre-auction results from the fact that shares of the surplus allocated to individual ring members do not depend on the values of their reported valuations. The PAKT mechanism balances the budget in expectation. That is, if the ring does not win the item, the ring center will make the payments to the ring members and receive nothing, and thereby make a loss. On the other hand, if the difference between the second-highest valuation within the ring and the highest valuation outside the ring is large, the ring center may keep some of the surplus. McAfee and McMillan (1992) develop a mechanism with a pre-auction first-price sealed bid knockout. For an all inclusive ring in an auction with a reservation price, they show that it is optimal for the winner to pay each ring member 1/(n − 1) fraction of the difference between his (winner’s) bid and the reservation price, which is the amount paid to the auctioneer. Their mechanism is efficient and budget balancing, guarantees voluntary participation, induces bidders to participate as agreed in the main auction, but is not truth-inducing. More specifically, the authors show that each bidder should bid the expected second-highest valuation conditional on their own valuation being the highest. While the above papers use pre-auctions to determine the share of the collusive surplus which is to be allocated to a ring member, Graham et al. (1990) analyze a post-auction mechanism, wherein nested knockouts are used to determine the share received by each bidder. They show that, with perfect information, the amount allocated to each bidder corresponds to the Shapley value and is contained in the core of a corresponding game. With imperfect information, bidders with lower valuations tend to overbid in the post-auction knockouts to increase their allocation. This is consistent with the observation that allocations depending on the bidders’ reported valuations, in general, do not induce truthfulness among the ring members, and that post-auction mechanisms can lead to inefficient results, since the amount that the ring paid for the item may exceed the highest valuation of its members. Unlike the afore mentioned papers, which assume homogeneous bidders, Mailath and 3
Zemsky (1991) analyze a case with heterogeneous bidders and give an efficient mechanism for allocation of collusive surplus among ring members. They also note that efficient collusion requires a bidder’s net payoff from collusion to be independent of his valuation. Their mechanism is budget balancing, but may require a losing bidder to make payments to other bidders. Our motivation for this work was twofold. First, note that all the mechanisms mentioned above assume prior knowledge of the distributions of bidders’ valuations. The emergence of Internet marketplaces makes auction participation much easier and more convenient, since no physical presence of bidders is required. As noted in Pinker et al. (2003), “...the participants in e-markets are less likely to be familiar with each other than participants in traditional markets...” In addition, as observed in Geoffrion and Krishnan (2003), online participation makes it easier for the participants to switch their identities in different auctions. Consequently, even in the repeated auctions, a bidder may change its identity in different rounds. Thus, in on-line auctions, it may very well happen that the bidders do not have much, or in fact any, prior information on distributions of other bidders’ private valuations of the items being auctioned. This, however, does not prevent them from attempting to collaborate and coordinate their bids. Note that this can also happen in traditional auctions, when bidders meet for the first time and do not have information about other auction participants, but as described above, it is much more likely to occur in an on-line environment. To analyze models with no prior knowledge of the distributions of bidders’ valuations, we develop a framework wherein each bidder knows only his own valuation of the item being auctioned, and neither he nor the auctioneer has any prior knowledge of the distributions of valuations of the other bidders. Our minimax regret criterion for comparing alternative bidders’ strategies is presented in the next section. Second, we wanted to find a budget balancing mechanism that guarantees voluntary participation and induces bidders to participate as agreed in the main auction, and for which truth-telling is an equilibrium. Krishna (2002) poses a question whether efficient collusion is possible without the need for outside financing. He proposes a budget balancing mechanism based on Vickrey-Clarke-Groves (VCG) mechanism (Vickrey 1961; Clarke 1971; Groves 1973), for which truth-telling is not a dominant strategy. Note that VCG mechanism corresponds to the second-price auction in an auction context. Krishna concludes that the ring center faces a trade-off between mechanisms that balance the budget in expected terms, for which truth-telling is a dominant strategy, and mechanisms that balance the budget, 4
but for which truth-telling is not a dominant strategy. Note that his model, like the models previously mentioned, assumes that all bidders know the distributions of valuations of other bidders. d’Aspermont and Gerard-Varet (1979) and Groves and Loeb (1975) conclude that it is, in general, not possible to find a budget balancing mechanism that guarantees voluntary participation and induces bidders to participate as agreed in the main auction, and for which truth-telling is a dominant strategy. However, we were able to find a budget balancing mechanism that satisfies the first two criteria and for which truth-telling is a minimax regret equilibrium. In addition, note that the above mentioned mechanisms may require that the ring center knows the distribution of the bidders’ valuation in order to determine their allocations; our mechanism does not impose such requirements.
3.
Minimax Regret
As mentioned above, we assume that ring members do not have prior information on distributions of other bidder’s valuations. Since the expected utility of a strategy profile in such setting cannot be defined, we need to consider the decision analysis tools that do not use prior probabilities. Among the methods commonly used in such environment (see, e.g., Luce and Raiffa 1957), let us mention maximin criterion, maximax criterion, and minimax regret criterion. Maximin approach assumes that the decision maker selects decision that maximizes his worst-case payoff. By its definition, this approach is very conservative, as it focuses on the worst-case scenario. For instance, Perakis and Roels (2005) show that, in the newsvendor problem wherein the newsvendor knows only the mean of the demand distribution, the optimal decision under maximin approach is not to order at all. Maximax approach selects the decision that maximizes the best-case profit, and is consequently usually viewed as too optimistic. Thus, we propose the use of the minimax regret criterion, introduced by Savage (1951), which is defined as follows. The regret is defined as the loss realized when a particular state occurs and the payoff of the selected decision is smaller than the maximum payoff available under that state. Denote by D a set of possible decisions from which a decision maker has to make his choice, by N a set of possible states of Nature, and denote by Π(d, s) the payoff that decision maker receives when he picks decision d ∈ D and the realized state of Nature is s ∈ N . The maximum possible payoff that can be realized under the state of Nature s is given by max Π(δ, s) δ∈D
5
and the regret of d with respect to s is defined as R(d, s) = max Π(δ, s) − Π(d, s). δ∈D
The decision maker looks at all possible states of Nature, evaluates the maximum regret of each strategy d, max R(d, s), s∈N
and selects the one with the smallest value, d∗ = arg min{max R(d, s)}. d∈D
s∈N
Strategies selected in this manner do well when compared with other available strategies, regardless of the true state of Nature. Minimax regret equilibrium is then a strategy profile in which each bidder minimizes his maximum regret with respect to the strategies of all other bidders. Note that, although it represents an improvement over the maximin approach, minimax regret is still a conservative criterion. In the auction framework, we illustrate this by the following example. Example 1. Suppose that bidders in a first-price auction do not posses information about distribution of other bidders’ evaluations, and select their bidding strategies according to the minimax regret criterion. If bidder i submits bid of bi , his payoffs are vi − bi if bi > maxj6=i bj Πi = 0 if bi < maxj6=i bj . If there is a tie, we assume that the object goes to each winning bidder with equal probability. Let B = maxj6=i bj and let B ε = B + ε for some ε → 0+ . Note that the value of B and its relationship to vi in this case denote the state of Nature. Suppose first that bi ≤ vi . If bi > B, bidder i wins the object and receives a payoff vi − bi . The maximum payoff is realized if he bids bi = B ε , when he receives vi − B ε . Thus, his regret is R(bi , B < bi ≤ vi ) = bi − B ε , and it is maximized when B = 0, R(bi , 0) = bi .
6
Next, if bi ≤ B, i does not win the object and his payoff is 0 whenever bi < B. The maximum payoff in this case is vi − B, which is realized if i bids bi = B. Therefore, i’s regret when bi ≤ B is R(bi , B ≥ bi ) = vi − B, and it is maximized when B = bi , R(bi , bi ) = vi − bi . Now, the maximum regret for bidding bi ≤ vi can be found as max R(bi , B) = max{vi − bi , bi }. B
It is easy to evaluate that minimax regret requires bi = vi /2, that is, that i bids one half of his valuation. Next, assume that bi ≥ vi , that is, i bids more than his valuation. If bi ≥ B ≥ vi , i wins the object and receives a negative payoff, vi − bi . The maximum possible payoff in this case is realized when i bids below B, in which case his payoff is zero. Thus, the regret is R(bi , vi ≤ B ≤ bi ) = bi − vi . If bi ≥ vi ≥ B, then i’s payoff is again vi − bi , but the maximum payoff equals vi − B. The regret is, therefore, R(bi , B ≤ vi ≤ bi ) = bi − B, and it is maximized when B = 0, R(bi , 0) = bi . Finally, if vi ≤ bi < B, i does not win the object and his payoff is zero. This is also the maximum payoff that he can realize while bidding bi ≥ vi , thus R(bi , vi ≤ bi < B) = 0. Consequently, the maximum regret when bi ≥ vi is max R(bi , B) = max{bi − vi , bi , 0} = bi , B
which is minimized for bi = vi . Hence, i would never bid more than his valuation.
It is interesting to note that minimax regret behaves much better in second-price auctions, as illustrated in the following example. 7
Example 2. Suppose that bidders in a second-price auction do not posses information about distribution of other bidders’ evaluations, and select their bidding strategies according to the minimax regret criterion. Let B = maxj6=i bj . If bidder i submits bid of bi , his payoffs are vi − B if bi > B Πi = 0 if bi < B. If there is a tie, we assume that the object goes to each winning bidder with equal probability. Thus, bidder i wins the object if he bids bi ≥ B, and his payoff does not depend on the actual value of his bid. Unlike the first-price auction, i can bid more than his valuation and receive a positive payoff, that is, we can have vi < bi . If the state of Nature is vi ≥ B, i has a positive regret of vi − B only when he bids bi ≤ B ≤ vi (when he does not win the item and receives a payoff zero), R(bi , bi ≤ B ≤ vi ) = vi − B, and it is maximized when B = bi , R(bi , bi ) = vi − bi . If the state of Nature is vi < B, i has a positive regret when he bids bi ≥ B ≥ vi (when he receives the item but generates a negative payoff), R(bi , bi ≥ B ≥ vi ) = B − vi , and it is maximized when B = bi , R(bi , bi ) = bi − vi . The maximum regret is, then, max R(bi , B) = max{vi − bi , bi − vi , 0}. B
Thus, minimax regret requires bi = vi , hence truth telling is a strategy that minimizes the maximum regret.
Minimax regret can help in comparing alternative strategies when decision maker does not know the probability of different outcomes. Minimax regret models have received increasing attention over the last years, and are applied in very different problem areas, see e.g. Drezner and Scott (1999), Averbakh (2000), Bossert and Peters (2001), Hyafil and Boutilier (2004), 8
or Perakis and Roels (2005). Observe that this approach becomes quite complex when the number of possible alternatives increases. In the next section, we present our model, and then evaluate bidders’ choices from the minimax regret perspective.
4.
The Model
An auctioneer is selling an item using a second-price sealed bid auction. Let vi denote bidder i’s valuation of the item, known only to him. Moreover, we assume that the auctioneer and the other bidders have no prior knowledge about distribution of vi . The auctioneer can select a reservation price, R, such that all bids must exceed R. In the case of multiple highest bids, the winner is determined randomly, so that each of the tied bidders receives the object with equal probability. Assume that a ring having n members is conducting a pre-auction to determine the winner, who will participate in the main auction. The pre-auction is sealed bid, with an independent bid taker, referred to as ring center, who collects all the bids and determines the winner. The winner, who is a member with the highest bid, participates in the main auction, while the remaining ring members either do not participate, or bid an agreed upon amount (e.g., R). Let us denote by bi a bid made by bidder i in the pre-auction. Thus, bi is i’s report of his private valuation, and we do not require this report to be truthful. We distinguish three values – the highest bid, bh , the second-highest bid, bs , and the lowest bid, bl . We allow for the possibility that several bidders make the same bid, in which case there is an equal chance that each of the bidders with the highest bid will participate in the main auction. Thus, it may happen that bh = bs and/or bs = bl . Let us denote by h the ring member with the highest bid who participates in the main auction. Suppose that h has won the item in the main auction and has paid v ∗ for it. If the ring is all-inclusive, v ∗ is usually equal to the reservation price, R (if one was announced in advance). We also allow for the possibility that the ring is not all-inclusive (that is, some bidders in the main auction do not belong to the ring), or that the auctioneer did not announce his reservation price in advance. The collusive surplus realized by the ring is, then, bs − v ∗ , provided this amount is positive. If bs − v ∗ > 0, the collusive surplus will be paid by h to the ring center, who will allocate it among the ring members. If bs − v ∗ ≤ 0, h makes no payments to the ring center, and each 9
ring member is allocated zero. Suppose bs − v ∗ > 0, and denote by ϕi an allocation of the collusive surplus, bs − v ∗ , to bidder i. As mentioned in Section 2, any allocation that depends on the bids made in the pre-auction is likely to induce non-truthful behavior among the ring members in the pre-auction. If a bidder’s allocation always increases with the amount he bids, he is likely to overbid. Overbidding is likely even if all bidders are allocated equal shares of the collusive surplus, since it may increase the amount being allocated among ring members. To avoid these incentives, we propose a mechanism which randomly determines whether bidders’ allocations are increasing or decreasing with their reported value, hereafter referred to as the Random k mechanism. The introduction of the random component makes it impossible for ring members to know in advance how the size of their bids influences their allocation (it can be either increasing or decreasing), and thus eliminates possible causes for consistent overbidding or underbidding. The model is described below. Suppose that α ∈ (0, 1) is selected before the pre-auction is conducted, and let S denote the sum of all bids in pre-auction, S=
n X
bj .
j=1
Player h, determined in the pre-auction, should bid his private valuation in the main auction, while all other ring members should either not participate, or bid the recommended value (e.g., R). After the main auction, if h wins the item and bs − v ∗ > 0, the collusive surplus is to be allocated among the ring members in the following way: a bidder, k, is randomly selected, so that each ring member is picked with equal likelihood, and a number, ∆, is defined as ∆ = max{bh − bk , bk − bl }. We note that our results would hold if any ∆ ≥ max{bh − bk , bk − bl } is selected. As will be shown later, this choice of ∆ guarantees voluntary participation. Each bidder is allocated ∗ k bs −v α + δ(1 − α) bi −b , i 6= k, n ∆ k ϕi = (1) bs −v∗ α + (1 − α) n − δ S−nbk , i = k, n ∆ where −1, if bk > Sn , 0, if bk = Sn , δ= 1, if bk < Sn . 10
Thus, the allocation received by bidder i depends on the relationship between the bid of bidder k and the average bid, and on the relationship between the bids of players i and k. Observe that the identity of bidder k is not known at the moment the bids are being made – each bidder has an equal probability to be selected as k. In addition, note that larger values of α lead to less variability in bidders’ allocations. α = 1 eliminates the random component entirely and allocates equal shares of the collusive surplus to all ring members. Whenever α > 1/2, the allocations to all ring members are strictly positive, provided that the collusive surplus is positive. However, note that Random k can result in negative allocations when α < 1/2, as illustrated in the following example. Example 3. Suppose that we have three bidders, with bids b1 = 100, b2 = 85, b3 = 60. Suppose that the object is purchased for v ∗ = 50, and that bidder 2 was selected as k, hence ∆ = 25. Figure 1 depicts how bidders allocations vary when α changes from 0 to 1. Note that bidder 1 receives negative allocations for small α.
Figure 1: Bidders allocations as functions of α with negative allocations for α < 1/2 To avoid the possibility of negative allocations and to simplify the analysis, we hereafter assume α = 1/2. This is the smallest value that guarantees positive allocations. At the same time, it maximizes the variability in bidders’ allocations, and thus leads to the highest regret values. Our results, though, carry over for any α ≥ 1/2. As will be shown later, this choice of α guarantees voluntary participation. When α = 1/2, it follows from (1) that bidder i is allocated ϕki =
bs −v ∗ 2n
bs −v ∗ 2n
·
∆+δ(bi −bk ) , ∆
i 6= k,
·
∆(n+1)−δ(S−nbk ) , ∆
i = k,
11
(2)
It is easy to check that ϕk is efficient, that is, that it allocates the entire collusive surplus among the ring members: Pn
k i=1 ϕi
=
bs −v ∗ 2n
(n−1)∆+δ [
P
i6=k bi −(n−1)bk
∆
]
+
∆(n+1)−δ(S−nbk ) ∆
= bs − v ∗ We now analyze some additional characteristics of the Random k mechanism. As stated in Section 2, we would like to show that under the Random k mechanism no bidder can improve his outcome by altering his decisions unilaterally if all ring members report their valuations truthfully, participate in the ring voluntarily, and submit the recommended bid in the main auction. Ideally, one would want to select a mechanism that is an ex-post Nash equilibrium or a dominant strategy. We analyze ex-post equilibria through the following example. Example 4. Suppose that we have three bidders, with valuations v1 = 100, v2 = 80, v3 = 60. Suppose that the object is purchased for v ∗ = 50. If all bidders reported truthfully, the collusive surplus is 30. Suppose that bidder 1 was selected as k, hence ∆ = 40. Then, the allocations would be ϕ11 = 12.5, ϕ12 = 7.5, and ϕ13 = 10. Now, suppose that bidder 2 decided to overbid, say b2 = 90, while the remaining bidders report truthfully. Then, collusive surplus is 40, and bidder 2 is allocated ϕ12 = 8.3 > 7.5. Thus, truthfulness is not an ex-post equilibrium. Similarly, suppose that bidder 2 decided to underbid, say b2 = 70, while the remaining bidders report truthfully. Then, collusive surplus is 20, and bidder 2 is allocated ϕ12 = 35/6 < 7.5. Thus, underbidding is not an ex-post equilibrium. Finally, suppose that bidders 2 and 3 report truthfully, while bidder 1 overbids, say b1 = 110. Then, collusive surplus is still 30, but ∆ = 50 and bidder 1 is allocated ϕ11 = 12 < 12.5. As a result, overbidding is not an ex-post equilibrium.
This leads to the following result. Proposition 1 Overbidding, underbidding, and truth-telling are not ex-post equilibrium strategies for Random k mechanism. In addition, truthfulness is not dominated by any other strategy. As we cannot select a dominant strategy or an ex-post equilibrium strategy, we evaluate bidders’ choices through the minimax regret criteria. In other words, we show that no bidder 12
can decrease his maximum regret by deviating from truthfulness, voluntary participation, and/or submitting the recommended bid in the main auction. As mentioned above, using minimax regret implies that one should evaluate the relationships between payoffs that a player may receive for a particular action under all possible outcomes. Note that the number of possible outcomes – combinations of bids submitted by individual bidders – is, in general, infinite. We first show that reporting one’s true valuation is, in fact, an optimal strategy under the Random k mechanism. Proposition 2 Truthfulness is a minimax regret equilibrium for the Random k mechanism. As the proof of this result is rather tedious, we present a sketch of the proof in the technical appendix. We next show that our mechanism induces voluntary participation of all ring members. Proposition 3 The Random k mechanism induces voluntary participation of all bidders. Proof: Let us first consider a bidder, i, with i 6= k, and assume bi ≥ bk . Then, bi ≤ bh implies 0 ≤ bi − bk ≤ bh − bk ≤ ∆, hence bi − bk ≤ 1. ∆ If bi ≤ bk , then bi ≥ bl implies 0 ≤ bk − bi ≤ bk − bl ≤ ∆, hence bi − bk ≥ −1. ∆ Thus, ϕki =
bs −v ∗ 2n
k 1 + δ bi −b ≥ ∆
13
bs −v ∗ 2n
(1 − 1) = 0.
Next, consider bidder k. We first assume bk < Sn , which implies δ = 1. If ∆ = bk − bl ≥ bh − bk , then k = n − δ S−nb ∆
n(bk −bl )−S+nbk ∆
≥
n(bk −bl )−n(bh −bk ) ∆
≥ 0,
because S ≤ nbh . If bk − bl ≤ bh − bk = ∆, then k n − δ S−nb = ∆
n(bh −bk )−S+nbk ∆
=
nbh −S ∆
≥ 0.
Now, if bk ≥ Sn , then δ = −1. When ∆ = bk − bl ≥ bh − bk , then k n − δ S−nb = ∆
n(bk −bl )+S−nbk ∆
=
S−nbl ∆
≥ 0,
because S ≥ nbl . If bk − bl ≤ bh − bk = ∆, then k n − δ S−nb = ∆
n(bh −bk )+S−nbk ∆
≥
n(bh −bk )−n(bk −bl ) ∆
≥ 0.
Therefore, ϕkk ≥ 0, and the proof follows. Next, let us consider the bids that the ring members place in the main auction. For bidder h, bidding his true valuation is always a dominant strategy in the second-price sealed bid auction (Vickrey, 1961). For all other ring members, bidding higher than the suggested value may only decrease the collusive surplus and, consequently, decrease the allocations to the ring members. This leads to the following proposition, which we state without the proof. Proposition 4 The Random k mechanism induces all ring members to bid their recommended bids in the main auction. As mentioned earlier, one of the desirable properties for a mechanism that coordinates a ring’s behavior is budget balancing. We next show that the Random k possesses this property. Proposition 5 The Random k mechanism balances the budget. Proof: Recall that, when the collusive surplus is positive, it is entirely distributed among the ring members. When the collusive surplus is negative, h does not make any payments to the ring center, and each member receives zero allocation. Thus, the Random k mechanism balances the budget in an ex post sense. 14
Note that PAKT mechanism introduced in Graham and Marshall (1987) guarantees voluntary participation and induces the bidders to participate as agreed on in the main auction, but balances the budget only in expectation. In other words, it requires an outside party to finance its operation. However, it is weakly dominant under PAKT mechanism for each bidder to report his true valuation. Under Random k, truth-telling is not a dominant strategy, but it is a minimax equilibrium. The results from Propositions 2 – 5 are summarized in Theorem 1. Theorem 1 The Random k mechanism balances the budget and guarantees voluntary participation of ring members. Telling the true valuation in the pre-auction and participating as agreed on in the main auction is a minimax regret equilibrium. The papers mentioned earlier (e.g. Graham and Marshall 1987, Mailath and Zemsky 1989, McAfee and McMillan 1997) consider a possible seller’s response to the collusion. As they assume that the bidders and the seller know the distribution of the bidders’ valuations, the seller can set the reserve price so that it mitigates the effect of cartel formation, providing that he knows the size of the cartel. In calculating his optimal response, the seller uses the distribution functions of bidders’ valuations. As we assume that the seller does not posses this information, he cannot use this approach in our model. We also observe that the properties of the Random k mechanism do not change if the ring does not include all of the bidders. Since truth-telling is a minimax equilibrium, a bidder with highest valuation will represent the ring in the auction. If the object is awarded to a bidder outside the ring, the bidder who represented the ring makes no payments to the ring, and none of the ring members is worse-off than if he decided not to collude in the first place. However, the existence of non-ring members in the auction may decrease the collusive surplus allocated among ring members. Thus, it appears that it is in the ring-members’ interest to include all bidders. Note that we discuss in Section 5 when this statement may not be correct.
5.
Random k and the Core
Consider now a cooperative game induced by the Random k mechanism. We introduce characteristic function V : 2N → IR as follows: for any subset A ⊂ N , characteristic function assigns to coalition A the collusive surplus that can be generated if the ring contains only 15
members of A. Formally, denote by bA s the second-highest bid in the preauction held among the members of coalition A. In addition, let v ∗ (A) denote the value at which the item is sold in the main auction, where members of A placed their recommended bids. Then, ∗ V(A) = bA s − v (A).
An allocation ϕ is contained in the core of a cooperative game (Gillies 1959) if the following holds: X
ϕi ≥ V(A) ∀A ⊆ N,
i∈A n X
ϕi = V(N ).
i=1
Thus, if the allocation rule satisfies the core property, no subset of players has an incentive to secede from the coalition of all players (also known as the grand coalition) and form their own coalition, because they collectively receive at least as much as what they can obtain as a coalition. Thus, core membership is a desirable property of an allocation. However, we show below that the Random k mechanism does not belong to the core. Proposition 6 The Random k mechanism is not contained in the core of the corresponding game. Proof: Suppose that there is a player j ∈ N such that vj < v ∗ (N ), and that vj < bN s . Then, if j does not belong to the ring, it does not change the price at which the item is purchased in N \{j}
N the main auction, hence v ∗ (N ) = v ∗ (N \ {j}). At the same time, since vj < bN s , bs = bs
,
therefore V(N ) = V(N \ {j}). It follows from the definition of the core that, in order for ϕk to belong to the core, we must have X i∈N \{j} n X
ϕki ≥ v ∗ (N \ {j}), ϕki = v ∗ (N ).
i=1
Because v ∗ (N ) = v ∗ (N \ {j}), the inequality above has to be satisfied as equation, which implies ϕkj = 0. Recall that Random k mechanism allocates a non-negative allocation of the collusive surplus to each ring member. Thus, player j can receive a positive allocation. This implies that members of coalition N \ {j} are better off if they defect from the grand coalition and exclude bidder j from their ring. 16
We illustrate this result with the following example. Example 5. Suppose that we have three bidders, with valuations v1 = 100, v2 = 80, v3 = 60. Suppose that the object is purchased for v ∗ = 70. If all bidders reported truthfully, the collusive surplus is 10. Suppose that bidder 1 was selected as k. Then, ∆ = 40, and the allocations would be ϕ11 = 25/6, ϕ12 = 5/2, and ϕ13 = 10/3. Thus, ϕ11 + ϕ12 = 20/3. If players 1 and 2 form the ring alone, the collusive surplus is still 10, hence V({1, 2}) = 10 > 20/3.
∗ ∗ Clearly, in a bidding ring with bN s > v (N ) which contains some bidders with vj ≤ v (N ),
a group containing all bidders j with vj > v ∗ (N ) would, by deviating, realize the same surplus as the original ring and share it among fewer players. It would be beneficial for a ring to contain only bidders with vj > v ∗ (N ), or to devise and allocation rule that would assign zero to all bidders with vj ≤ v ∗ (N ). Unfortunately, without any prior knowledge on distributions of bidders’ valuations, it is impossible to include only bidders with vj > v ∗ (N ) in a ring. On the other hand, allocating zero to all bidders with vj ≤ v ∗ (N ) is likely to induce overbidding by bidders with low valuations in the pre-auction. Therefore, a modification of our mechanism, that would make it more stable by achieving core membership, is impossible with no prior information. Whenever additional information becomes available to the ring members, a subset of bidders is likely to deviate, as explained above.
6.
Conclusion
When bidders do not have prior information about distributions of other bidders’ private valuations of an item being auctioned, existing mechanisms for collusive behavior may not be appropriate. We have proposed a mechanism, Random k, with a random component for a second-price sealed bid auction, which eliminates the temptation of either underbidding or overbidding in the pre-auction. We have shown that it guarantees voluntary participation and induces the bidders to participate as agreed on in the main auction. While truth-telling is not a dominant strategy under Random k (as it is, for instance, under PAKT mechanism), it is a minimax equilibrium. In addition, Random k’s benefit over PAKT is that it balances the budget (which PAKT does only in expectation) and does not require the ring center to know distributions of bidders’ valuations in order to determine their allocations. However,
17
the assumption about the lack of prior information prevents Random k from being contained in the core of the corresponding game. We note that we do not address the issues of coalition formation and the optimal ring size in this paper. Our assumption about the lack of prior information on bidders’ valuations makes it impossible to evaluate criteria for ring participation. However, we do not assume that the ring is all-inclusive, and the results obtained in the paper hold both for the case when all bidders participate in the ring, and when some bidders are excluded from the ring. As mentioned earlier, minimax regret criterion is a rather conservative approach, and assigns equal probabilities to all states of Nature. We can assume that a bidder has some beliefs about what states or Nature are more or less likely to happen. A possible extension of this work would look into allocating different probabilities to different states of Nature when evaluating regrets from possible decisions. Further research may also include possible modifications of Random k that may lead to core membership.
Acknowledgments The author is grateful to Daniel Granot and two anonymous referees, who provided many useful suggestions that significantly improved the exposition and quality of this paper.
Appendix Proof or Proposition 2: The regret of a bidder, i, is calculated with respect to the total payoff that i receives as a result of ring membership. Thus, if i does not participate in the main auction, his payoff corresponds to the allocation assigned by the Random k mechanism, ϕki . If i is the bidder with the highest bid selected to participate in the main auction, his payoff also includes the amount vi − v ∗ − (bs − v ∗ ) = vi − bs , and his total payoff is ϕki + vi − bs . To prove our result, we will analyze a bidder’s allocation under all possible states of Nature. To simplify further analysis, let us denote by Si the sum of all bids in the preauction with the assumption that i bids his true valuation, bi = vi . Thus, X Si = bj + vi . j6=i
18
Suppose first that i’s valuation exceeds all other bids, vi > bs . Then, whenever i reports truthfully or overbids, he is the bidder with the highest bid who participates in the main auction, while underbidding can change this situation. Let ε > 0 and η > 0 be defined as follows: whenever i bids bi = vi − ε, his bid remains the highest, vi − ε > bs . In other words, ε < vi −bs and the second-highest bid, bs , does not change. If, on the other hand, i underbids by η, bi = vi − η, then vi − η < bs . Thus, η > vi − bs and bs becomes the highest bid. We denote by bηs ≥ vi − η the new second-highest bid. Table 1 shows the allocations assigned to bidder i by the Random k mechanism when vi > bs and i 6= k. (Table 1 here.) Note that Table 1 contains only allocations assigned to player i because of ring membership; when i bids so that he remains the bidder with the highest bid (by overbidding, underbidding by at most ε, or reporting truthfully), he also receives vi − bs > 0 from the main auction, while he does not receive this amount if his bid becomes lower than the second-highest bid (when he underbids by η). As bidders do not know distributions of others’ evaluations, i cannot restrict his underbidding to eliminate the latter option (that is, i cannot decide to underbid by at most ε). This, together with the allocations in Table 1, can be used to evaluate that, among the three strategies, underbidding always leads to the bs −v ∗ . n
highest regret, vi − bs +
The regret from truthfulness and overbidding is
bs −v ∗ . n
Now, suppose that i is still a bidder with the highest bid, vi > bs , and that he is now selected as bidder k, i = k. Then, if vi − ε >
Si −ε n
(that is, ε
nvn−1 ), then the payoffs from overbidding and truthful reporting
and it can be as low as vi − ε
bs . We next evaluate the case vi ≤ bs . If i underbids or reports truthfully, he never participates in the main auction, and his payoff corresponds to the allocation assigned by the Random k mechanism. Note, however, that by overbidding i can become the player with the highest bid who participates in the main auction, which gives him an additional (negative) payoff vi − bs ≤ 0. He also has to pay the amount bs − v ∗ ≥ vi − v ∗ to the ring. Therefore, the total payoff that i receives as a result of overbidding can be negative, hence this strategy maximizes the regret value. Consequently, i should either report truthfully or underbid whenever vi ≤ bs . Suppose first that vi < bs . If we assume i 6= k, an analysis similar to that conducted for vi > bs shows that both underbidding and truthfulness lead to regret of
bs −v ∗ . n
If i = k and i Si , n Si +ε n
is the bidder with the lowest bid, then his valuation never exceeds the average bid, vi < and the same is true when i underbids, vi − ε < (that is, ε
Si −nvi ), n−1
(n−1)(bs −v ∗ ) . 2n
If,
overbidding results in allocation
bs − v ∗ Si − nvi − (n − 1)ε + ε) = n+1+ , 2n bh − vi − ε
which can lead to the highest payoff, and both truthfulness and underbidding lead to the same regret value,
bs −v ∗ . 2
A similar (but much longer) analysis shows that underbidding and
truthfulness lead to the same regret when i’s valuation exceeds the lowest bid and we can have both vi
Si . n
Therefore, when vi < bs , both underbidding and truthfulness
are equilibrium strategies. Finally, we consider the case in which vi corresponds to the second-highest bid, vi = bs . As mentioned earlier, when vi < bh , overbidding can make i the bidder with the highest bid 20
who participates in the main auction. In such a case, i’s payoff from the main auction is vi − bs = 0. In other words, he has to give to the ring the entire amount, bs − v ∗ = vi − v ∗ , and his total payoff corresponds to his Random k allocation. Thus, regardless of his strategy, i never receives any payoff other then his allocation from the ring membership. If vi < v ∗ , then both underbidding and truthfulness result in bs − v ∗ < 0. As a result, collusive surplus is zero, hence both strategies result in zero allocation. On the other hand, even when vi < v ∗ overbidding can lead to a strictly positive allocation, and its maximum (n+1)(bh −v ∗ ) . n h ∗ (n+1)(b −v ) . n
value is
Thus, both underbidding and truthfulness lead to the same regret value,
We can now evaluate the overall minimax equilibrium for the Random k mechanism. As shown above, there are several states of Nature in which all strategies (underbidding, overbidding, truthfulness) lead to the equal regret value. Note that regret from truthfulness does not strictly exceed regret from both remaining strategies under any state of Nature. In addition, we have shown that overbidding is the only strategy that can lead to a negative value of the total payoff received by bidder i. This can increase i’s regret from overbidding (with respect to regrets from other strategies) by bs −vi whenever bs > vi . On the other hand, when i’s evaluation exceeds other players’ bids, underbidding can lead to nonparticipation in the main auction. This increases i’s regret (with respect to regrets from other strategies) by vi − bs whenever bs < vi . As a result, truthfulness is a minimax equilibrium strategy.
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23
State \ Bid
vi + ε
vi
vi − ε
0
0
0
0
−v ∗
0
vi − η
i 6= k, bk − bl ≤ vi − bk − η bk > Sh n
Sh +ε n
< bk