Monotone Randomized Apportionment

Proportional apportionment is the fundamental task of translating vote or population shares into an integer allocation of seats. The work monotone randomized apportionment studies randomized apportionment rules that are ex ante proportional; each party receives, in expectation, its prescribed quota, whereas the realized allocation remains integral and respects the target house size. However, the correlations between the seats awarded to different parties may exhibit bizarre nonmonotonicities. When parties or voters care about joint events, such as whether a coalition of parties reaches a majority, these nonmonotonicities can cause paradoxes, including incentives for strategic voting. This work identifies Sampford’s method, a classical unequal-probability sampling scheme, as a principled randomized rounding procedure that rules out such nonmonotonicity paradoxes. In doing so, this work connects apportionment, dependent randomized rounding, and sampling theory, providing a rigorous probabilistic framework for fair seat allocation.