On the optimality of Shapley mechanism for funding public excludable goods under Sybil strategies
Bruno Mazorra
TL;DR
This work investigates how Sybil (false-name) strategies affect cost-sharing mechanisms for funding public excludable goods in permissionless settings. It establishes a fundamental impossibility result: any deterministic, anonymous, truthful, Sybil-proof, upper semicontinuous, and IR mechanism incurs a worst-case social cost of at least $\Omega(n)$, unless one sacrifices deficit-balance or other properties. To address this, it introduces Sybil Welfare Invariant mechanisms and proves that the Shapley value mechanism is SWI under symmetric submodular costs, achieving a worst-case social cost of $\mathcal{H}_n$ even when the number of participants is unknown or uncertain due to Sybil attacks. This indicates that anonymous funding of public excludable goods can be efficient in practice (e.g., for DAOs and DeFi platforms) by adopting SWI-compatible mechanisms, albeit with cryptographic or protocol-level measures to implement them in permissionless environments.
Abstract
In the realm of cost-sharing mechanisms, the vulnerability to Sybil strategies -- also known as false-name strategies, where agents create fake identities to manipulate outcomes -- has not yet been studied. In this paper, we delve into the details of different cost-sharing mechanisms proposed in the literature, highlighting their non-Sybil-resistant nature. Furthermore, we prove no deterministic, anonymous, truthful, Sybil-proof, upper semicontinuous, and individually rational cost-sharing mechanism for public excludable goods is better than $Ω(n)$-approximate. This finding reveals an exponential increase in the worst-case social cost in environments where agents are restricted from using Sybil strategies. To circumvent these negative results, we introduce the concept of \textit{Sybil Welfare Invariant} mechanisms, where a mechanism does not decrease its welfare under Sybil strategies when agents choose weak dominant strategies and have subjective prior beliefs over other players' actions. Finally, we prove that the Shapley value mechanism for symmetric and submodular cost functions holds this property, and so deduce that the worst-case social cost of this mechanism is the $n$th harmonic number $\mathcal H_n$ under equilibrium with Sybil strategies, matching the worst-case social cost bound for cost-sharing mechanisms. This finding suggests that any group of agents, each with private valuations, can fund public excludable goods both permissionless and anonymously, achieving efficiency comparable to that of non-anonymous domains, even when the total number of participants is unknown.ess and anonymously, achieving efficiency comparable to that of permissioned and non-anonymous domains, even when the total number of participants is unknown.