Pool Formation in Oceanic Games: Shapley Value and Proportional Sharing

TL;DR

This paper analyzes pool formation in Proof of Stake blockchains using game theory, comparing the Shapley value and proportional sharing schemes for reward distribution, focusing on decentralization and resistance to strategic attacks.

Contribution

It introduces a non-cooperative game-theoretic analysis of the Shapley value in blockchain pools, highlighting its advantages over proportional schemes in decentralization and attack resistance.

Findings

Shapley value can improve decentralization compared to proportional sharing.

Proportional scheme is more susceptible to Sybil attacks.

Shapley value remains competitive even with variations in reward sharing functions.

Abstract

We study a game-theoretic model for pool formation in Proof of Stake blockchain protocols. In such systems, stakeholders can form pools as a means of obtaining regular rewards from participation in ledger maintenance, with the power of each pool being dependent on its collective stake. The question we are interested in is the design of mechanisms that suitably split rewards among pool members and achieve favorable properties in the resulting pool configuration. With this in mind, we initiate a non-cooperative game-theoretic analysis of the well known Shapley value scheme from cooperative game theory into the context of blockchains. In particular, we focus on the oceanic model of games, proposed by Milnor and Shapley (1978), which is suitable for populations where a small set of large players coexists with a big mass of rather small, negligible players. This provides an appropriate level of abstraction for pool formation processes among the stakeholders. We provide comparisons between the Shapley mechanism and the more standard proportional scheme, in terms of attained decentralization, via a Price of Stability analysis and in terms of susceptibility to Sybil attacks, i.e., the strategic splitting of a players' stake with the intention of participating in multiple pools for increased profit. Interestingly, while the widely deployed proportional scheme appears to have certain advantages, the Shapley value scheme, which rewards higher the most pivotal players, emerges as a competitive alternative, by being able to bypass some of the downsides of proportional sharing, while also not being far from optimal guarantees w.r.t. decentralization. Finally, we complement our study with some variations of proportional sharing, where the profit is split in proportion to a superadditive or a subadditive function of the stake, showing that the Shapley value scheme still maintains the same advantages.