Generalizing Liquid Democracy to multi-agent delegation: A Voting Power Measure and Equilibrium Analysis

Liquid democracy has gained popularity in recent years due to its ability to balance representation and delegation of power. In this work, we propose a generalization of the classic model that allows for fractional delegation of voting weight. Our approach enables agents to divide and delegate their votes to multiple agents, while retaining a portion of the voting power for themselves. We discuss the desirable properties of a reasonable generalization of the classic model and introduce a set of simpler voting measures that include a penalty factor on the length of delegation chains. We demonstrate that the proposed voting measure is a well-defined limit of these simpler measures when the penalty approaches zero, and inherits key features of the classic model. In the second part of the article, we investigate the existence of equilibrium states in a delegation game that employs the suggested measures. We show that this game has pure strategy Nash equilibria as long as a penalty on the length of delegation chains is enforced.