Majority voting is not good for heaven or hell, with mirrored performance

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.