  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 $\Omega(n)$-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.
