Strategy-Proof Mechanisms for the k-Winner Selection Problem

Abstract

AbstractThe goal of this paper is to develop a strategy-proof (SP) mechanism for the k-winner selection problem, which finds a set of (at most) k winners among participants. Here, we assume the winners can have positive/negative externalities with each other; the gross utility of a winner not only depends on whether she wins, but also on the other winners. If the types of agents, i.e., the gross utilities of agents, are known, we can obtain a Pareto efficient (PE) allocation that maximizes the sum of the gross utilities of winners in polynomial time, assuming k is a constant. On the other hand, when the types of agents are private information, we need a mechanism that can elicit the true types of agents; it must satisfy SP. We first show that there exists no SP mechanism that is PE, individual rational (IR), and non-deficit (ND) in a general setting where we put no restrictions on possible agent types. Thus, we need to give up at least one of these desirable properties.Next, we examine how a family of Vickrey-Clarke-Groves (VCG) based mechanisms works for this problem. We consider two alternative VCG-based mechanisms in this setting, both of which are SP and PE. We show that one alternative, called VCG-ND, is ND but not IR, and the other alternative, called VCG-IR, is IR but not ND. Also, we show special cases where VCG-ND satisfies IR, or VCG-IR satisfies ND. Moreover, we propose mechanisms called VCG-ND+ and VCG-IR+, which can be applied to a general setting, where a mechanism designer has partial knowledge about the possible interactions among agents. Both VCG-ND+ and VCG-IR+ are SP, IR, and ND, but they are not PE. Finally, we present a concise graphical representation scheme of agant types.KeywordsMechanism DesignerIndividually RationalCombinatorial AuctionSocial SurplusTrue TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.