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.

翻译：在随机环境投票（ViSE）模型中，我们研究了多数表决在不同环境下的有效性。如先前研究所揭示的"损失陷阱悖论"所示，在明显敌对的环境中进行多数决策往往会减少社会总资本。在此类情形下，"不经投票直接拒绝所有提案"这一简单社会决策规则的表现优于多数表决。本文发现了在有利环境中出现的另一类损失陷阱；此时，"不经投票直接接受所有提案"的简单社会决策规则优于多数表决。我们证明，在称为对称化多数的简化多数表决版本下，结合投票机构的反对称性，这第二类损失陷阱与敌对环境中产生的损失陷阱呈现镜像对称关系，并对此现象进行了解释。从技术层面，我们考虑了一个由个人主义者（支持所有能增加个人资本的提案）与一个（或多个）团体（其成员投票以增加团体财富）组成的投票社会。根据关键引理，当随机收益生成器为均值$μ>0$的$X$时，每位智能体在社会决策规则下的期望资本收益，较其在反射生成器$-X$下的期望收益恰好高出$μ$。这一结论可推广至关于均值对称的位置-尺度分布族生成器。该结果揭示了对称化多数规则相对于基准规则的性能镜像对称性，其中基准规则在有利环境中接受所有提案，在不利（敌对）环境中拒绝所有提案。