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Majority voting is not good for heaven or hell, with mirrored performance
arXiv:2401.00592v5 Announce Type: replace-cross
Abstract: 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 $\mu>0$ exceeds their expected gain under the reflected generator $-X$ by exactly $\mu$. 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.
Abstract: 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 $\mu>0$ exceeds their expected gain under the reflected generator $-X$ by exactly $\mu$. 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.
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