On the optimality of Shapley mechanism for funding public excludable goods under Sybil strategies

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 that a Sybil-proof cost-sharing mechanism for public excludable goods under mild conditions is at least $(n+1)/2-$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 permissioned and non-anonymous domains, even when the total number of participants is unknown.