How Likely Are Voting Rules Equitable?

In social choice, anonymity (treating all agents equally) and neutrality (treating all alternatives equally) are widely regarded as "minimal demands" and "uncontroversial" axioms of equity and fairness. However, due to the ANR impossibility theorem, no voting rule can simultaneously satisfy anonymity, neutrality, and resoluteness (always choosing a unique winner). While the worst-case ANR impossibility is well understood, its likelihood in probabilistic models remains underexplored. We address this question through comprehensive likelihood analysis under semi-random models, providing accurate bounds to quantify the advantage of the optimial tie-breaking mechanisms over other commonly-used tie-breaking mechanisms. Our characterizations reveal an $n^{\Theta(m!)}$ improvement in many cases, where $n$ is the number of agents and $m$ is the number of alternatives. We also prove a quantitative ANR impossibility theorem characterizing the tradeoff between equity (anonymity and neutrality) and a new efficiency criterion called $\delta$-anonymity. Our work advances the understanding of trade-offs between equity and efficiency in voting.