Quantifying coalition seat risk under D’Hondt from polling uncertainty

Abstract

We develop a modeling and reporting framework for quantifying how pre-electoral coalition formation may alter parliamentary representation under the Jefferson--D'Hondt apportionment rule. Polls are usually presented as point estimates of vote shares, yet the objects that matter for representation are integer seat vectors, generated by a discontinuous, piecewise-constant conversion from shares to seats. We treat the unknown true share vector as a simplex-valued random variable, anchored at the poll snapshot and calibrated by an effective sample size. For each draw we evaluate two deterministic regimes at the same realization: (i) parties run separately and the coalition total is obtained by summing member seats, and (ii) coalition members appear on a joint list and seats are allocated to the aggregated list. The difference defines an integer-valued coalition advantage and supports probability statements about strictly positive seat improvement. Because audiences naturally read a poll through its point-estimate seat outcome, we further classify positive-advantage draws into gain/protection/shortfall relative to the separate-lists coalition seat total at the poll point. Finally, we report an l 1 margin diagnostic that measures how far the snapshot lies from the nearest profile attaining a target advantage or a poll-benchmark uplift. A Croatian district poll case study demonstrates the workflow and how these summaries are read in practice.