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Collusion in Multi-Unit Auctions

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Collusion in Multi-Unit Auctions Honor Among Thieves — Collusion in Multi-Unit Auctions Yoram Bachrach Microsoft Research Cambridge UK [email protected] ABSTRACT tacit collusion. Certain types of collusion are illegal in many We consider collusion in multi-unit auctions where the allo- countries, due to competition law. However, many types of cation and payments are determined using the VCG mecha- tacit collusion are hard to detect. This paper studies one nism. We show how collusion can increase the utility of the form of collusion called bid rigging, where participants in colluders, characterize the optimal collusion and show it can an auction change their bids in an attempt to lower prices. Specifically, we consider the domain of multi-unit auctions easily be computed in polynomial time. We then analyze 1 the colluders’ coalition from a cooperative game theoretic under the Vickrey-Clarke-Groves (VCG) mechanism . perspective. We show that the collusion game is a convex In a multi-unit domain there are multiple identical items game, so it always has a non-empty core, which contains to be allocated. Each agent thus only cares about the num- the Shapley value. We show how to find core imputations ber of items she receives. We assume free disposal, so agents and compute the Shapley value, and thus show that in this always value obtaining more items at least as much as ob- setting the colluders can always share the gain from their taining less items. Agents can thus express their preferences manipulation in a stable and fair way. This shows that this as a valuation function mapping the number of items they domain is extremely vulnerable to collusion. obtain to their valuation for this quantity of items. Given the above valuation functions, a central mechanism can easily allocate the items in a way that maximizes social Categories and Subject Descriptors welfare. The center can ask agents to specify the maximal F.2 [Theory of Computation]: Analysis of Algorithms amount they would be willing to pay to obtain various quan- and Problem Complexity; tities of items. These specifications are the agents’ bids. I.2.11 [Artificial Intelligence]: Distributed Artificial In- One major problem that plagues such domains is the fact telligence—Multiagent Systems; that the valuation functions are private information of the J.4 [Computer Applications]: Social and Behavioral Sci- agents. Agents may bid strategically, by misreporting their ences—Economics valuation functions in order to achieve a better outcome for themselves. The VCG payment scheme is the canonical General Terms method for incentivizing the agents to bid truthfully (i.e. reveal their true valuation function). Algorithms, Theory, Economics Although VCG has many desirable properties, it is known to be susceptible to collusion. Although any single agent Keywords is incentivised to truthfully reveal her valuation function, Collusion, Cooperative Game Theory, Bid Rigging, Shapley several agents may agree to misreport their valuations in a Value, Core, Weber Set coordinated way, and split the gains from this manipulation. In this paper we show how agents can collude in VCG multi-unit auctions. We provide a simple polynomial algo- 1. INTRODUCTION rithm for finding the optimal way to collude, given a specific Collusion is an agreement between two or more agents to coalition of colluders. We then analyze how the colluders limit competition by manipulating or defrauding in order to can split the gains from such a manipulation using coopera- obtain an unfair advantage [26]. Such manipulations include tive game theory, by modeling the situation as a coalitional agreements to divide the market, set prices or limit produc- game we call the collusion game. We show that this game tion or bids. For example, in oligopoly where there are few is a convex game, so it has a non-empty core that contains firms producing a certain good, the decision of a few firms the Shapley value. This means the coalition of colluders can to limit production can significantly affect the market as a form stable agreements, in a way that guarantees no bick- whole. Cartels are a special case of explicit collusion, where ering among the colluders. In fact, the colluders can even firms coordinate prices. Collusion which is not overt is called split the gains in a fair manner, based on the contribution Cite as: Honor Among Thieves — Collusion in Multi-Unit Auc- of each colluder to the coalition. We also provide algorithms tions, Yoram Bachrach, Proc. of 9th Int. Conf. on Autonomous to compute these colluder utility distributions. These dis- Agents and Multiagent Systems (AAMAS 2010), van der Hoek, turbing results indicate that VCG multi-unit domains are Kaminka, Lespérance, Luck and Sen (eds.), May, 10–14, 2010, Toronto, Canada, pp. XXX-XXX. 1 Copyright °c 2010, International Foundation for Autonomous Agents and See Section 5 and [21] for more information regarding the Multiagent Systems (www.ifaamas.org). All rights reserved. origins and properties of the VCG mechanism. extremely vulnerable to collusion. have v(C0) · v(C), and is super-additive when for all dis- The paper proceeds as follows. Section 2 contains some joint coalitions A; B ½ N we have v(A) + v(B) · v(A [ B). preliminary definitions and notation. Section 3 discusses In super-additive games, it is always worthwhile for two sub- possible ways of collusion in multi-unit VCG auctions. In coalitions to merge, so eventually the grand coalition con- Section 4 we analyze the colluders coalition using tools from taining all the agents will form. cooperative game theory. Section 5 contains references for The characteristic function only defines the gains a coali- important related work. We conclude in Section 6. tion can achieve, but does not define how these gains are distributed among the agents who formed the coalition. An imputation (p1; : : : ; pn) is a division of the gains of the grand 2. PRELIMINARIES Pn coalition among the agents, where pi 2 R, such that pi = We consider collusion in multi-unit auction under the VCG i=1 v(N). We call pi the payoff of agent i, and denote the payoff mechanism. We begin with a brief review of VCG. P of a coalition C as p(C) = i2C pi. Obviously, a key issue We have a set N of n agents, f1; 2;:::; g. The mechanism is choosing the appropriate imputation for the game. Coop- needs to choose one of a set of possible alternatives K. Each erative Game theory offers several answers to this question. agent reports a type θi 2 Θi, representing the agent’s pref- A basic requirement for a good imputation is individual erences over the different alternatives in K. Each agent has rationality, which states that for any agent i 2 N, we have a different valuation of the mechanism’s chosen alternative that pi ¸ v(fig)—otherwise, some agent is incentivized to k 2 K, vi(k; θi). The mechanism chooses the outcome ac- work alone. Similarly, we say a coalition B blocks the payoff cording to a choice rule k :Θ1 £:::£Θn ! K. Each agent is vector (p1; : : : ; pn) if p(B) < v(B), since the members of B also required to make a payment pi to the mechanism. The can split from the original coalition, derive the gains of v(B) mechanism chooses the payment of each agent according to in the game, give each member i 2 B its previous gains pi— a payment rule ti :Θ1 £ ::: £ Θn ! R. If the agents have and still some utility remains, so each member can get more quasi-linear utility functions, then the agents have utility utility. If a blocked payoff vector is chosen, the coalition is ui(k; pi; θi) = vi(k; θi) ¡ pi. An agent might not report her somewhat unstable. The most prominent solution concept true type, but has to always choose a type to report to the 0 focusing on such stability is that of the core [13]. mechanism. Thus, agent i reports a type θi = si(θi), accord- ing to its own strategy si. In Groves mechanisms, the mech- 0 0 0 Definition 1. The core of a coalitional game is the set anism’s choice rule given the reported types θ = (θ1; :::; θn) maximizes the sum of the agents’ utilities, according to their of all imputations (p1; : : : ; pn) that are not blocked by any reported types, as seen in Equation 1. coalition, so that for any coalition C, we have the following equation: p(C) ¸ v(C). Equation 1 Groves Allocation Another solution concept is the Shapley value [23] which X ¤ 0 0 defines a single value division. The Shapley value focuses k (θ ) = arg max vi(k; θi) (1) k2K on fairness, rather than stability. The Shapley value fulfills i several important fairness axioms [23, 29] and has been used to fairly share gains or costs. The Shapley value of an agent The payment rule in Groves mechanisms is given in Equa- depends on its marginal contribution to possible coalition permutations. We denote by ¼ a permutation (ordering) of tion 2. where hi :Θ¡i ! R may be any function that only the agents, and by Π the set of all possible such permuta- tions. Given permutation ¼ 2 Π = (i1; : : : ; in), the marginal Equation 2 Groves Payments ¼ n ¼ worth vector m [v] 2 R is defined as mi1 = v(fi1g) and for X ¼ 0 0 ¤ 0 k > 1 as mik [v] = v(fi1; i2; : : : ; ikg) ¡ v(fi1; i2; : : : ; ik¡1g). ti(θ ) = hi(θ¡i) ¡ vj (k ; θj ) (2) The convex hull of all the marginal vectors is called the We- j6=i ber Set. Weber showed [28] that the Weber set of any game contains its core. The Shapley value is the centroid of the marginal vectors.
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