Optimal Partisan Gerrymandering Is Segregate-Pair at Every Level of Relative Uncertainty

Kolotilin and Wolitzky (2026a) show that the optimal districting plan is unique and segregate-pair when idiosyncratic uncertainty exceeds aggregate uncertainty, and conjecture that segregate-pair optimality holds more generally. We prove the conjecture under their assumptions on voter types and idiosyncratic shocks, at every level of relative uncertainty. The result holds for any log-concave density of the aggregate shock and, more generally, whenever the second derivative of the log density is bounded above, over the range of voter types, by a positive constant determined by the idiosyncratic shock and the range of voter types. This bound has the right order when idiosyncratic shocks are spread out. We also show that an S-shaped distribution of the aggregate shock is neither necessary nor sufficient for the conclusion.