  We propose and study a new class of polynomial voting rules for a general
decentralized decision/consensus system, and more specifically for the PoS
(Proof of Stake) protocol. The main idea, inspired by the Penrose square-root
law and the more recent quadratic voting rule, is to differentiate a voter's
voting power and the voter's share (fraction of the total in the system). We
show that while voter shares form a martingale process that converge to a
Dirichlet distribution, their voting powers follow a super-martingale process
that decays to zero over time. This prevents any voter from controlling the
voting process, and thus enhances security. For both limiting results, we also
provide explicit rates of convergence. When the initial total volume of votes
(or stakes) is large, we show a phase transition in share stability (or the
lack thereof), corresponding to the voter's initial share relative to the
total. We also study the scenario in which trading (of votes/stakes) among the
voters is allowed, and quantify the level of risk sensitivity (or risk averse)
in three categories, corresponding to the voter's utility being a
super-martingale, a sub-martingale, and a martingale. For each category, we
identify the voter's best strategy in terms of participation and trading.
