Tradeoffs in Hierarchical Voting Systems

Condorcet's jury theorem states that the correct outcome is reached in direct majority voting systems with sufficiently large electorates as long as each voter's independent probability of voting for that outcome is greater than 0.5. Yet, in situations where direct voting systems are infeasible, such as due to high implementation and infrastructure costs, hierarchical voting systems provide a reasonable alternative. We study differences in outcome precision between hierarchical and direct voting systems for varying group sizes, abstention rates, and voter competencies. Using asymptotic expansions of the derivative of the reliability function (or Banzhaf number), we first prove that indirect systems differ most from their direct counterparts when group size and number are equal to each other, and therefore to $\sqrt{N_{\rm d}}$, where $N_{\rm d}$ is the total number of voters in the direct system. In multitier systems, we prove that this difference is maximized when group size equals $\sqrt[n]{N_{\rm d}}$, where $n$ is the number of hierarchical levels. Second, we show that while direct majority rule always outperforms hierarchical voting for homogeneous electorates that vote with certainty, as group numbers and size increase, hierarchical majority voting gains in its ability to represent all eligible voters. Furthermore, when voter abstention and competency are correlated within groups, hierarchical systems often outperform direct voting, which we show by using a generating function approach that is able to analytically characterize heterogeneous voting systems.