Cost-Sharing mechanisms for DAOs: 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, where agents can create fake identities to manipulate outcomes, has not yet been studied. In this paper, we delve into the intricacies of different cost-sharing mechanisms proposed in the literature highlighting its non Sybil-resistance nature. Furthermore, we prove that under mild conditions, a Sybil-proof cost-sharing mechanism for public excludable goods is at least $(n/2+1)-$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 maintains its worst-case welfare under Sybil-strategies for every set of prior beliefs even when the mechanism is not Sybil-proof. Finally, we prove that the Shapley value mechanism for public excludable goods holds this property, and so deduce that the worst-case social cost of this mechanism is the $n$th harmonic number $\mathcal H_n$ even under equilibrium of the game with Sybil strategies, matching the worst-case social cost bound for cost-sharing mechanisms. This finding carries important implications for decentralized autonomous organizations (DAOs), indicating that they are capable of funding public excludable goods efficiently, even when the total number of agents is unknown.