Core deviation minimizing auctions

We study dominant strategy implementable direct mechanisms that minimize the expected surplus from core deviations. Using incentive compatibility conditions, we formulate the core deviation miminimization problem as a calculus of variations problem and then numerically solve it for some particular cases. JEL Classi…cations Codes: D44, C71 Keywords: Core, Auctions, Mechanism Design 1 Introduction In an economic environment with complementarities, Vickrey-Clark-Groves (VCG) mechanism may generate low revenues and therefore is vulnerable to collusion (of coalitions that necessarily include the seller). Motivated by this observation, core-selecting auctions (CSA) have been proposed as alternatives to VCG mechanism.1 In a complete information setting, di¤erent forms of CSA have been considered and shown to have superior properties in terms of revenue while ensuring noncollusive behavior.2 More recently, in a simple stylized environment with private information, Goeree and Lien (2009) and Ausubel and Baranov (2010) analyze how VCG and CSA perform in terms of revenue, e¢ ciency, and distance from core allocations. Their results demonstrate that We are grateful to Bilkent University for hosting us during the early stages of this research. We also thank Gokhan Apaydin and Mohamad Maleki for discussions about numerical optimization and Ersin Korpeoglu for excellent research assistantship. yTepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA E-mail: [email protected] zDepartment of Economics, Zirve University, K¬z¬lhisar Kampüsü, Gaziantep 27260, Turkey E-mail: [email protected]. 1See Day and Raghavan (2007) and Day and Milgrom (2008). 2There is a small but growing literature on CSA. See, for instance, Day and Crampton (2008), Crampton (2009), Baranov (2010), Erdil and Klemperer (2010), Lamy (2010) and Sano (2011). 1 VCG and CSA cannot be generally ranked in terms of above measures, and their relative performances depend on speci…cations of value distributions of the players. The stylized model for CSA considers two goods, two local bidders, and a global bidder. In this model, the global bidder views the two goods as perfect complements in the sense that he/she has positive valuation for the bundle, but zero valuation for a single good. Local bidder i; on the other hand, has positive value for object i and zero value for object j (fi; jg = f1; 2g). In the same stylized model, we ask a mechanism design problem: Among dominant strategy implementable direct mechanisms, which mechanism achieves an outcome that is closest to the core? When answering this problem, we measure the distance from the core as expected surplus from core deviations.3 Appealing to envelope theorem, we write the transfer function in terms of the allocation function. We then apply some calculus operations and write the mechanism design problem as a standard calculus of variations problem. We prove that the optimal mechanism should “favor the global bidder” in the sense that if global bidder’s value is greater than the sum of local bidders’values, then global bidder is always awarded both items. While an analytical closed-form solution to the calculus of variations problem turns out to be di¢ cult to obtain, we numerically solve it for two interesting cases. If bidders’value distributions are all uniform; interestingly, Vickrey auction turns out to be the optimal mechanism. On the other hand, for another case, we show that the optimal mechanism performs signi…cantly better as compared to the Vickrey auction. 2 Model There are two goods, goods 1 and 2, that are to be allocated among two local buyers, buyers 1 and 2, and one global buyer, buyer 3. Local buyer i earns a positive value only from good i and global buyer earns a positive value only if he obtains both goods. Local buyer i’s value vi is distributed over [0; 1] according to a distribution function F and global buyer’s value for the bundle v3 is distributed over [0; 2] according to a distribution function G: We assume that distributions F and G are atomless, continuous and di¤erentiable.4 Also de…ne v (v1; v2; v3) : In this environment, we consider the set of dominant strategy incentive compatible direct mechanisms that minimize the expected surplus from core deviations. Note that a deviating group has to include the seller, therefore it is composed either of the 3Goeree and Lien (2009) already established that there exists no (Bayesian) incentive compatible direct mechanism for which the value of the objective function is zero. We look for a second best, that is, given incentive constraints, how close can a mechanism get to the core according to this measure? 4Ausubel and Baranov (2010) also considered the case in which local bidders’valuations may be correlated. We do not analyze that speci…cation here.