Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and optimality. We focus on AMD's paradigmatic problem: combinatorial auctions. We present a new generalization of the VC dimension to multivalued collections of functions, which encompasses the classical VC dimension, Natarajan dimension, and Steele dimension. We present a corresponding generalization of the Sauer-Shelah Lemma and harness this VC machinery to establish inapproximability results for deterministic truthful mechanisms. Our results essentially unify all inapproximability results for deterministic truthful mechanisms for combinatorial auctions to date and establish new separation gaps between truthful and non-truthful algorithms. * It is highly recommended to prefer the full version of this paper that can be found on the authors' webpages and on arXiv.
Preventing fake or duplicate e-identities (aka sybils) from joining an e-community may be crucial to its survival, especially if it utilizes a consensus protocol among its members or employs democratic governance, where sybils can undermine consensus, tilt decisions, or even take over. Here, we explore the use of a trust graph of identities, with trust edges representing trust among identity owners, to allow a community to grow without increasing its sybil penetration. Since identities are admitted to the e-community based on their trust by existing e-community members, corrupt identities, which may trust sybils, also pose a threat to the ecommunity. Sybils and their corrupt perpetrators are together referred to as Byzantines, and our overarching aim is to limit their penetration into an e-community. Our key tool in achieving this is graph conductance, and our key assumption is that honest people are averse to corrupt ones and tend to distrust them. Of particular interest is keeping the fraction of Byzantines below one third, as it would allow the use of Byzantine Agreement [13] for consensus as well as for sybil-resilient social choice [16,20] and sybil-resilient parameter update [19]. We consider sequences of incrementally growing trust graphs and show that, under our key assumption and additional requirements, including keeping the conductance of the community trust graph sufficiently high, a community may grow safely.
Sybil attacks, in which fake or duplicate identities (sybils) infiltrate an online community, pose a serious threat to such communities, as they might tilt community-wide decisions in their favor. While the extensive research on sybil identification may help keep the fraction of sybils in such communities low, it cannot however ensure their complete eradication. Thus, our goal is to enhance social choice theory with effective group decision mechanisms for communities with bounded sybil penetration. Inspired by Reality-Aware Social Choice [Shapiro and Talmon, 2018], we use the status quo as the anchor of sybil resilience, characterized by sybil safety -the inability of sybils to change the status quo against the will of the genuine agents, and sybil liveness -the ability of the genuine agents to change the status quo against the will of the sybils. We consider the social choice settings of deciding on a single proposal, on multiple proposals, and on updating a parameter. For each, we present social choice rules that are sybil-safe and, under certain conditions, satisfy sybil-liveness.2 Other variants, such as using any tournament solution, are also possible but not explored here.
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