Majority voting is not good for heaven or hell, with mirrored performance
Pavel Chebotarev, Vadim Afonkin
TL;DR
This work investigates how majority voting behaves in stochastic environments using the ViSE model, revealing a mirrored pit of losses: in hostile settings, rejecting all proposals can outperform majority voting, while in highly favorable settings, accepting all proposals can outperform majority voting. The authors introduce antisymmetric voting bodies and establish a key lemma that the expected implemented gain under opposite generators differs by exactly the mean $\mu$ of the proposal generator, leading to a symmetric, even performance of certain social decision rules around $\mu=0$. They show that for location-scale families with symmetric baselines, the symmetrized majority rule has mirror-improved or worsened performance depending on $\mu$, and that the optimal threshold rule in societies of individualists exhibits reversal symmetry and avoids the pit of losses. These results illuminate fundamental limits of majority voting under stochastic environments and point to threshold-based rules as robust alternatives, with implications for designing decision mechanisms in uncertain, multi-agent settings. The findings are formalized via antisymmetric voting bodies, complementary strategies and rules, and detailed results for Gaussian-like generators, highlighting a concrete path to achieving balanced performance across Heaven and Hell-like environments.
Abstract
Within the ViSE (Voting in Stochastic Environment) model, we study the effectiveness of majority voting in various environments. As shown by the pit-of-losses paradox identified in previous work, majority decisions in apparently hostile environments tend to reduce the capital of society. In such cases, the simple social decision rule of ``rejecting all proposals without voting'' outperforms majority voting. In this paper, we identify another pit of losses appearing in favorable environments; here, the simple social decision rule of ``accepting all proposals without voting'' is superior to majority voting. We prove that, under a version of simple majority called symmetrized majority and under the antisymmetry of the voting body, this second pit of losses is a mirror image of the one arising in hostile environments, and we explain this phenomenon. Technically, we consider a voting society consisting of individualists who support all proposals that increase their personal capital and a group (or groups) whose members vote to increase their group's wealth. According to the key lemma, the expected capital gain of each agent under the social decision rule when the random gain generator is $X$ with mean $μ>0$ exceeds their expected gain under the reflected generator $-X$ by exactly $μ$. This extends to location-scale families of generators with distributions symmetric about their mean. This result reveals a mirror symmetry in the performance of the symmetrized majority rule relative to a baseline rule. The baseline rule accepts all proposals in favorable environments and rejects them in unfavorable (hostile) ones.