Majority voting is not good for heaven or hell, with mirrored performance
Majority voting can be worse than rejecting all proposals in an unfavorable stochastic environment and worse than accepting all proposals in a favorable one. We connect these effects in the ViSE (Voting in Stochastic Environment) model, where proposals specify random capital gains for the voters. If the voters' strategies and the social decision rule jointly interchange acceptance and rejection probabilities when gains and the environment are reversed, expected implemented gains satisfy a shift identity: each agent's expected gain under a gain generator with mean $μ>0$ exceeds that under the opposite generator by exactly $μ$. For families of gain distributions obtained by shifting a symmetric distribution, this implies mirrored performance relative to rejecting or accepting all proposals. Using uniformly distributed gains, we also prove a converse for a society of individualists voting for or against proposals according to their own gains: if a fixed vote-count rule is used, mirrored performance implies symmetry between acceptance and rejection. We further study gain-maximizing rules. A group-voting result shows exactly when an optimum must accept a lower vote total while rejecting a higher one. In this class, optimal decisions have a weighted-majority form: the group's common vote is weighted by its expected absolute internal vote margin rather than its size. For strategies that interchange support and opposition when all gains are reversed, optimal thresholds in opposite environments are mirror images, and, on symmetric families of gain distributions, an optimal vote-count rule can be chosen with mirrored performance as well.