Ballot Structure from the Preference-Dependency Graph

A ballot fixes which issues are decided together, which separately, and what each question may be contingent on---and almost every voting system picks one of two extremes. Direct democracy asks every issue alone: optimal when preferences are independent across issues, but able to elect a package a majority dislikes once they are coupled (the multiple-election paradox). Platform voting bundles everything into a few take-it-or-leave-it packages: faithful to any structure, but forced to offer a short menu of the $2^m$ combinations, leaving preferred combinations unexpressed. We treat the ballot as an object to be \emph{designed} from the electorate's preference structure. A ballot, in general, is a rooted forest of contingent questions---``if the tax passes, should the rebate follow?''---with the familiar designs as its contingency-free special cases. Any ballot forgoes welfare in exactly two ways---\emph{separability-distortion}, from deciding coupled issues in separate questions, and \emph{menu-distortion}, from offering fewer combinations than voters demand---and both are governed by the dependency graph of voter preferences. Losses come only from couplings the questions fail to cover: they vanish entirely under coverage and, on today's partition ballots, are bounded by the severed coupling mass. Contingent questions decide chain- and tree-structured coupling exactly at cost linear in the number of issues---the cost of a faithful ballot is exponential only in the graph's treewidth, not in the size of its coupled blocks. Because the graph is latent, we estimate it from Polis-style agree/disagree votes and read the ballot off the estimate, provably exactly when the structure is clean. We show who pays when a budget forces a cut, and how ballots are attacked (duplicate-flooding the agenda; menu riders) and repaired. The pipeline is validated on synthetic electorates and real Polis conversations---including vTaiwan's Uber deliberation, where the ballot recovered from votes alone matches, in outline, the agenda Taiwan ratified in 2016.