Polynomial Voting Rules

Wenpin Tang () and David D. Yao ()
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Wenpin Tang: Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
David D. Yao: Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027

Mathematics of Operations Research, 2025, vol. 50, issue 1, 90-106

Abstract: We propose and study a new class of polynomial voting rules for a general decentralized decision/consensus system, and more specifically for the 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, whereas voter shares form a martingale process that converges to a Dirichlet distribution, their voting powers follow a supermartingale 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 aversion) in three categories, corresponding to the voter’s utility being a supermartingale, a submartingale, and a martingale. For each category, we identify the voter’s best strategy in terms of participation and trading.

Keywords: Primary: 60C05; 91B08; secondary: 60F05; 60G42; cryptocurrency; economic incentive; fluid limit; phase transition; polynomial voting rules; proof-of-stake protocol; stability; urn models (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:1:p:90-106

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