Impossibility of Full Decentralization in Permissionless Blockchains

Bitcoin uses the proof-of-work (PoW) mechanism where nodes earn rewards in return for the use of their computing resources. Although this incentive system has attracted many participants, power has, at the same time, been significantly biased towards a few nodes, called mining pools. In addition, poor decentralization appears not only in PoW-based coins but also in coins that adopt proof-of-stake (PoS) and delegated proof-of-stake (DPoS) mechanisms. In this paper, we address the issue of centralization in the consensus protocol. To this end, we first define (m, ε, δ)-decentralization as a state satisfying that 1) there are at least m participants running a node, and 2) the ratio between the total resource power of nodes run by the richest and the δ-th percentile participants is less than or equal to 1 + ε. Therefore, when m is sufficiently large, and ε and δ are 0, (m, ε, δ)-decentralization represents full decentralization, which is an ideal state. To ascertain if it is possible to achieve good decentralization, we introduce conditions for an incentive system that will allow a blockchain to achieve (m, ε, δ)-decentralization. When satisfying the conditions, a blockchain system can reach full decentralization with probability 1, regardless of its consensus protocol. However, to achieve this, the blockchain system should be able to assign a positive Sybil cost, where the Sybil cost is defined as the difference between the cost for one participant running multiple nodes and the total cost for multiple participants each running one node. Conversely, we prove that if there is no Sybil cost, the probability of achieving (m, ε, δ)-decentralization is bounded above by a function of fδ, where fδ is the ratio between the resource power of the δ-th percentile and the richest participants. Furthermore, the value of the upper bound is close to 0 for small values of fδ. Considering the current gap between the rich and poor, this result implies that it is almost impossible for a system without Sybil costs to achieve good decentralization. In addition, because it is yet unknown how to assign a Sybil cost without relying on a TTP in blockchains, it also represents that currently, a contradiction between achieving good decentralization in the consensus protocol and not relying on a TTP exists.