Pool Formation in Oceanic Games: Shapley Value and Proportional Sharing

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, “reward sharing schemes,” 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 that occur among the stakeholders of a blockchain. 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 in terms of Sybil attack susceptibility, while also not being far from optimal guarantees w.r.t. decentralization. Finally, we also 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 our results for the Shapley value scheme are maintained in comparison to these functions as well.