Simplex subtraction to estimate unanimity violations in sequential majority voting

Based on Saari’s geometry of voting (see Saari (1995)), we introduce a concise geometric “simplex subtraction” approach to measure the likelihood with which sequential majority voting overturns unanimous consent under the impartial anonymous culture assumption. This approach can be used as an alternative to Nehring et al. (2016) who establish worst-case bounds for discrete electorates. We treat vote shares as points in the reduced simplex after one candidate is unanimously rejected. By systematically removing the volumes of smaller corner sub-simplexes, we recover the probabilities determined by Nehring et al. (2016), which reaches its maximum for seven candidates. We then generalize our method to arbitrary supermajority rules and identify the exact threshold at which unanimity violations become impossible.