Polynomial Voting Rules

23 Pages Posted: 30 Jun 2022 Last revised: 21 Aug 2023

See all articles by Wenpin Tang

Wenpin Tang

Columbia University - Department of Industrial Engineering and Operations Research

David Yao

Columbia University

Date Written: June 21, 2022


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.

Keywords: Cryptocurrency, economic incentive, fluid limit, phase transition, polynomial voting rules, Proof of Stake protocol, stability, urn models

Suggested Citation

Tang, Wenpin and Yao, David, Polynomial Voting Rules (June 21, 2022). Available at SSRN: https://ssrn.com/abstract=4141789 or http://dx.doi.org/10.2139/ssrn.4141789

Wenpin Tang (Contact Author)

Columbia University - Department of Industrial Engineering and Operations Research ( email )

500 W. 120th Street #315
New York, NY 10027
United States

David Yao

Columbia University ( email )

New York, NY
United States

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