  Proof-of-stake (PoS) is a promising approach for designing efficient
blockchains, where block proposers are randomly chosen with probability
proportional to their stake. A primary concern with PoS systems is the "rich
getting richer" phenomenon, whereby wealthier nodes are more likely to get
elected, and hence reap the block reward, making them even wealthier. In this
paper, we introduce the notion of equitability, which quantifies how much a
proposer can amplify her stake compared to her initial investment. Even with
everyone following protocol (i.e., honest behavior), we show that existing
methods of allocating block rewards lead to poor equitability, as does
initializing systems with small stake pools and/or large rewards relative to
the stake pool. We identify a \emph{geometric} reward function, which we prove
is maximally equitable over all choices of reward functions under honest
behavior and bound the deviation for strategic actions; the proofs involve the
study of optimization problems and stochastic dominances of Polya urn
processes, and are of independent mathematical interest. These results allow us
to provide a systematic framework to choose the parameters of a practical
incentive system for PoS cryptocurrencies.
