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

Abstract

The ViSE (Voting in Stochastic Environment) model studies voting strategies and social decision rules under a stochastic agenda of proposals with random gains. The pit-of-losses paradox established earlier shows that, in hostile ("hell") environments, majority voting can be worse than rejecting all proposals. We show a counterpart in favorable ("heaven") environments, where majority voting can be worse than accepting all proposals. A one-line identity underlies this symmetry: for an antisymmetric voting body, the expected capital gain of each agent under a gain generator of mean $μ>0$ exceeds that under the opposite generator by exactly $μ$. Complementary voting strategies together with a complementary social decision rule provide a broad sufficient, but not necessary, condition for antisymmetry. This includes symmetrized majority with individualistic, utilitarian-group, and majoritarian-group strategies. For symmetric location families, the result yields mirror symmetry of performance relative to the baseline that rejects all proposals in unfavorable environments and accepts all in favorable ones. We also strengthen the Gaussian pit-of-losses result: simple majority has a pit in every society of $n>2$ individualists, and every rule requiring at least $r$ of $n$ votes, $r<n$, has one as well, while unanimity avoids the pit. Finally, under complementary strategies, optimal voting thresholds inherit the mirror structure. For stochastic vote-count rules under the same condition, a complementary optimal rule can always be selected, and optimal total performance is even in $μ$ in symmetric location families. In the two-plus-trio paradox, the optimal vote-count rule outperforms all threshold rules even in a neutral symmetric environment.