Non-Rarity Metrics for Non-Fungible Tokens

Abstract

Within the new asset class of non-fungible tokens, personal profile picture (PFP) collections are avatars-with-benefits. Like Pok'emon cards, the value of a token should depend on its rarity, indeed some very rare tokens have sold for several millions of dollars. However, each Pok'emon card carries symbols from which its rarity can be determined at-a-glance, whereas the rarity of a PFP token must be measured from the characteristics defined in the metadata of the collection when it is minted to a blockchain. Unfortunately, there is nothing yet in the public domain that measures rarity correctly, except for a few special collections that have been designed to have independent traits. Although numerous metrics are now used by the PFP industry, different metrics can give vastly different results. This obscures any fundamental relationship between rarity and price and leads to great inefficiency in the PFP market. Our invariance results clarify the state-of-the-art, allowing classification of the numerous supposedly-different metrics into just four distinct cases, each a special case of weighted power mean. Importantly, neither the NFTGo Jaccard distance nor the OpenRarity Shannon entropy differ from the most commonly used metrics, i.e. the Pythagorean means, so they are not new, as claimed. We derive tests for trait independence, showing that most of the ~200 PFP collections analysed have dependent traits, so none of these metrics is mathematically correct. For traders in PFPs we present two novel visualization tools, with code, which identify how discordant different rarity rankings can be, depending on the marketplace used. (This paper supersedes its previous version ``Rarity Metrics for Non-Fungible Tokens'')

Keywords: Personal profile picture, Pythagorean mean, Weighted power mean, Trait normalization

JEL Classification: C60, G10, O30, O35, O36