Pool Formation in Oceanic Games: Shapley Value and Proportional Sharing

2025-05-07

AI Summary

This paper addresses the reward allocation mechanism design for staking pools in proof-of-stake (PoS) blockchains, modeling it as a “oceanic game” featuring a few large players and numerous small ones. We introduce the Shapley value to this setting for the first time and systematically compare it with proportional allocation schemes in terms of decentralization and Sybil-resilience. Theoretically, we prove that the Shapley mechanism significantly mitigates stake-splitting attacks and reduces Sybil attack profitability. Its price of stability—quantifying the decentralization cost—is strictly superior to all proportional variants (including super- and sub-additive forms), while providing stronger incentives for pivotal participants and achieving near-optimal system-wide decentralization. This work establishes a novel paradigm and provides rigorous theoretical foundations for incentive mechanism design in PoS staking pools.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Blockchains and distributed ledgers
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Designing reward mechanisms for Proof of Stake blockchain pools
• Comparing Shapley value and proportional sharing in pool formation
• Analyzing decentralization and Sybil attack resistance in reward schemes

Innovation

• Uses Shapley value for blockchain reward distribution
• Compares Shapley and proportional sharing mechanisms
• Analyzes decentralization and Sybil attack resistance