Satisfiability of threshold monotonicity in randomized apportionment

Determine whether there exists a randomized apportionment method that satisfies quota, ex-ante proportionality, and threshold monotonicity, where threshold monotonicity requires that for any fixed house size h, any two vote vectors v and v′ such that every party i in a coalition T has a weakly larger standard quota q_i′ = h·v_i′/(∑_j v_j′) than q_i = h·v_i/(∑_j v_j) and every party outside T has a weakly smaller standard quota, the probability that T is awarded at least s seats weakly increases for every threshold s ∈ ℕ.

Background

The paper introduces new monotonicity axioms for randomized apportionment, building on the rounding-rule framework and Grimmett’s quota and ex-ante proportionality requirements. After establishing that Sampford sampling satisfies selection monotonicity for rounding and showing several impossibility results (including the incompatibility of pairwise threshold monotonicity with full support and vote-count threshold formulations), the authors propose threshold monotonicity as the central extension of their axioms to apportionment.

Threshold monotonicity concerns the probability that a coalition exceeds any seat threshold when its vote shares increase and others’ decrease. Unlike stronger variants shown to be impossible, the authors do not resolve whether threshold monotonicity itself can be achieved, making its satisfiability a key open problem.