Polynomial Voting Rules.

The article introduces and analyzes a new class of polynomial voting rules, denoted **Poly(α)**, designed for decentralized consensus systems, with a focus on the proof-of-stake (PoS) blockchain protocol. Unlike traditional voting where voting power equals voter share, Poly(α) decouples these by scaling voting power with a parameter α, allowing voting power to diminish over time, which enhances security by preventing any single voter from dominating the process. The study rigorously characterizes the stochastic behavior of voter shares and voting powers, showing that shares form a martingale converging to a Dirichlet distribution, while voting powers form a supermartingale that decays to zero, with explicit convergence rates. It also identifies a phase transition in the stability of voter shares based on initial stake size and extends the model to include stake trading among bidders, analyzing optimal strategies under varying risk sensitivities and demonstrating conditions under which bidders prefer nonparticipation or buyout strategies. The work further discusses practical implications for controlling block validation times by dynamically adjusting α and outlines potential extensions involving market impact models.