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

Liquid democracy is a mechanism for the division of labor in decision-making through the transitive delegation of influence. In essence, all individuals possess the autonomy to determine the issues with which they will engage directly, while for other matters, they may appoint a representative of their choosing. So far, the literature has studied the delegation structures emerging in liquid democracy as static. As a result, transitivity defined as the capacity to transfer acquired authority to another entity, has been identified as a concern as it would be conducive to unrestrained accumulation of power. Focusing on the implementation of liquid democracy supported by the LiquidFeedback software, we propose a novel approach to assessing the influence of voting nodes in a transitive delegation graph, taking into account the process nature of real-world liquid democracy in which delegation and voting are distinct and increasingly independent activities. By introducing a novel model of delegations in liquid democracy, we show how transitivity may in fact contribute to an effective regulation of deliberation influence and decision-making power. While maintaining the one-person, one-vote paradigm for all votes cast, the anticipated influence of an agent, to the extent it is stemming from transitivity, experiences a precipitous decline following an exponential trajectory. In general, it is our objective to move the first steps towards a rigorous analysis of liquid democracy as an adaptive democratic representation process. The adaptivity aspect of liquid democracy has not yet been explored within the existing academic literature despite it being, we believe, one of its most important features. We therefore also outline a research agenda focusing on this aspect of liquid democracy.

Many collective-action systems are neither fully decentralized nor fully centralized: agents voluntarily organize into institutions that share organizational costs and delegate decisions to representatives. Delegation can reduce individual costs, but it also transfers control to representatives whose private objectives need not align with those of their constituents. We study this tension in a repeated game with endogenous delegation to self-interested centers. A simple reduced-form model yields a scale--accountability tradeoff: larger constituencies offer greater cost-sharing benefits to members, yet the marginal value to a center of retaining any one constituent decreases as group size grows. Consequently, the prospect of individual exit can sustain non-exploitative delegation only up to an endogenous accountable constituency size. Additionally, we identify parameter regions in which fully delegated, accountable governance, when feasible, must be polycentric. We investigate these predictions through simulations with large language model (LLM) agents. Across five matched pairs of 30-agent simulations, polycentric structures repeatedly emerge but remain fluid, exhibiting cycles of concentration, exploitation, exit, and re-formation. We further find that models of different capacity exhibit similar willingness to assume leadership but differ substantially in their exercise of authority, with Sol initiating exploitation more frequently than Luna in matched decision states.

Majority voting is the default unsupervised aggregator for multi-sample LLM inference, but it discards two signals: within-group answer entropy and between-group reasoning geometry. We aggregate by delegation instead (Propagational Proxy Voting, PPV): each group of samples keeps weight on its own answer in proportion to its entropy-based confidence (When) and routes the rest to peers by reasoning-embedding similarity (Whom); the stationary distribution of the resulting delegation matrix picks the consensus answer. This requires neither gold labels nor training. On MMLU-Pro with 128 samples per question, delegation beats majority by +1.5 pp overall and +2.24 pp on non-trivial questions (McNemar p ~ 1.0e-14, n = 8,099), overturning wrong majorities whose answer cluster is geometrically incoherent while the correct minority is tight. We then characterize exactly when delegation overturns majority: a two-option model gives a closed-form flip condition on each option's confidence and the weight it routes to the other, with a do-no-harm corollary for near-unanimous questions. The condition calls the realized winner on 96.5% of non-trivial questions, and its predicted mass gap tracks the realized gap at r = 0.97. We did not find any other unsupervised ensemble methods that close the oracle gap.