Inequality in the Age of Pseudonymity

Inequality measures such as the Gini coefficient are used to inform and motivate policymaking, and are increasingly applied to digital platforms. We analyze how measures fare in pseudonymous settings that are common in the digital age. One key challenge of such environments is the ability of actors to create fake identities under fictitious false names, also known as ``Sybils.'' While some actors may do so to preserve their privacy, we show that this can inadvertently hamper inequality measurements. As we prove, it is impossible for measures satisfying the literature's canonical set of desired properties to assess the inequality of an economy that may harbor Sybils. We characterize the class of all Sybil-proof measures, and prove that they must satisfy relaxed version of the aforementioned properties. Furthermore, we show that the structure imposed restricts the ability to assess inequality at a fine-grained level. By applying our results, we prove that large classes of popular measures are not Sybil-proof, with the famous Gini coefficient being but one example out of many. Finally, we examine the dynamics leading to the creation of Sybils in digital and traditional settings.