Within the ViSE (Voting in Stochastic Environment) model, we study the effectiveness of majority voting in various environments. By the pit of losses paradox, majority decisions in apparently hostile environments systematically reduce the capital of society. In such cases, the simple social decision rule of ``rejecting all proposals without voting'' outperforms majority. 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. We prove that under a version of simple majority called symmetrized majority and the antisymmetry of the voting body, the second pit of losses is a mirror image of the pit of losses in hostile environments and explain this phenomenon. Technically, we consider a voting society consisting of individualists whose strategy is supporting all proposals that increase their capital and a group (groups) whose members vote to increase the total group capital. According to the main result, the expected capital gain of each agent in the environment whose generator $X$ has mean $\mu>0$ exceeds by $\mu$ their expected capital gain under generator $-X$. This result extends to location families of generators with distributions symmetric about their mean. The mentioned result determines the symmetry of the difference between the expected capital gain under the symmetrized majority and that under the ``basic'' social decision rule that rejects/accepts all proposals in unfavorable/favorable environments.

翻译：在ViSE（随机环境中的投票）模型中，我们研究了多数投票在不同环境下的有效性。通过损失悖论，在明显敌对环境中的多数决策会系统性地减少社会资本。在此类情形下，"不经投票直接否决所有提案"这一简单社会决策规则的表现优于多数投票。本文识别了在有利环境中出现的另一种损失悖论。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投票。我们证明，在称为对称化多数的简化多数投票版本及投票机构反对称性的条件下，第二种损失悖论是敌对环境中损失悖论的镜像，并解释了这一现象。从技术层面，我们考察了一个由个人主义者（其策略是支持所有能增加自身资本的提案）与一个（或多个）集团（其成员通过投票增加集团总资本）组成的投票社会。根据主要结果，在生成元$X$的均值$\mu>0$的环境中，每个智能体的期望资本收益比在生成元$-X$下的期望资本收益高出$\mu$。该结果可推广至具有关于均值对称分布的位置族生成元。上述结果确定了在对称化多数投票下的期望资本收益与"基础"社会决策规则（在不利/有利环境中否决/接受所有提案）下的期望资本收益之差的对称性。