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 basic action of ``rejecting all proposals without voting'' outperforms simple majority. We reveal another pit of losses appearing in favorable environments. Here, the simple action of ``accepting all proposals without voting'' is superior to simple majority, which thus causes a loss compared to total acceptance. We show that 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 individual agents whose strategy is supporting all proposals that increase their capital and a group whose members vote for the increase of the total group capital. According to the main result, the expected capital gain of each agent in the environment whose proposal generator $\xi$ has mean $\mu>0$ exceeds by $\mu$ their expected capital gain with generator $-\xi$. This result extends to the shift-based families of generators with symmetric distributions. The difference by $\mu$ causes symmetry relative to the basic action that rejects/accepts all proposals in unfavorable/favorable environments.