Strategy-proof mechanisms for the k-winner selection problem

The 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.