Scarcity Is Not Enough: An Impossibility Result for Linear Sybil Cost Under Parallelizable Resources

2026-05-28

AI Summary

This study investigates whether scarcity alone is sufficient to defend against Sybil attacks, revealing that the structural properties of resources critically determine the cost of concentrating influence. By formally modeling adversarial cost as \(C(s,T)\), the work introduces the notion of "influence amortization" and demonstrates that when resources are divisible, exhibit additive influence, are temporally reusable, or permit identity transfer, an adversary can reduce the cost below linear—specifically, \(C(s,T) = o(sT)\). Conversely, resources that are throughput-constrained, non-transferable, and locally bounded in time achieve a lower bound of \(C(s,T) = \Omega(sT)\). This establishes a fundamental asymptotic separation between two classes of resources, providing theoretical limits and design principles for Sybil-resistant mechanisms.

Abstract

Permissionless systems resist Sybil attacks by binding influence to scarce resources. We show that scarcity alone is insufficient: the structural properties of the resource determine whether influence can be concentrated at sublinear cost through identity replication, delegation, or pooling. We model this through the adversarial cost C(s,T): the minimum expenditure required to achieve influence proportional to s independent participation units over T windows. We prove that any resource satisfying divisibility, additivity of influence, temporal reusability, and identity transferability admits influence amortization: C(s,T)=o(sT), regardless of protocol design. This is an impossibility result: no protocol rule can enforce linear cost of influence concentration over a structurally parallelizable resource. We further prove that throughput-bounded, non-transferable, window-local resources enforce C(s,T)=Omega(sT): each additional unit of sustained influence incurs marginal cost Delta(s,T)=Omega(T), growing with the time horizon. The two resource classes are asymptotically separated. As a direct design consequence, any mechanism targeting linear cost of influence concentration must ground participation in a resource that violates at least one parallelizability property.

Problem

• Sybil resistance
• resource scarcity
• influence concentration
• parallelizable resources
• impossibility result

Innovation

• Sybil resistance
• parallelizable resources
• influence amortization
• impossibility result
• adversarial cost