  We introduce and study reward sharing schemes (RSS) that promote the fair
formation of {\em stake pools}\ in collaborative projects that involve a large
number of stakeholders such as the maintenance of a proof-of-stake (PoS)
blockchain. Our mechanisms are parameterized by a target value for the desired
number of pools. We show that by properly incentivizing participants, the
desired number of stake pools is a Nash equilibrium arising from rational play.
Our equilibria also exhibit an efficiency / security tradeoff via a parameter
that calibrates between including pools with the smallest cost and providing
protection against Sybil attacks, the setting where a single stakeholder
creates a large number of pools in the hopes to dominate the collaborative
project. We then describe how RSS can be deployed in the PoS setting,
mitigating a number of potential deployment attacks and protocol deviations
that include censoring transactions, performing Sybil attacks with the
objective to control the majority of stake, lying about the actual cost and
others. Finally, we experimentally demonstrate fast convergence to equilibria
in dynamic environments where players react to each other's strategic moves
over an indefinite period of interactive play. We also show how simple reward
sharing schemes that are seemingly more "fair", perhaps counterintuitively,
converge to centralized equilibria.
