Monotone Randomized Apportionment

TL;DR

This paper explores randomized apportionment methods that achieve proportionality and monotonicity simultaneously, addressing paradoxes and strategic voting issues through new axioms and sampling schemes.

Contribution

It introduces monotonicity axioms to prevent paradoxes in randomized apportionment and demonstrates that Sampford's sampling scheme satisfies these axioms alongside Grimmett's.

Findings

Sampford's sampling scheme satisfies Grimmett's axioms.

The proposed axioms prevent non-monotonic joint seat allocations.

The approach addresses strategic voting and coalition paradoxes.

Abstract

Apportionment is the act of distributing the seats of a legislature among political parties (or states) in proportion to their vote shares (or populations). A famous impossibility by Balinski and Young (2001) shows that no apportionment method can be proportional up to one seat (quota) while also responding monotonically to changes in the votes (population monotonicity). Grimmett (2004) proposed to overcome this impossibility by randomizing the apportionment, which can achieve quota as well as perfect proportionality and monotonicity -- at least in terms of the expected number of seats awarded to each party. Still, the correlations between the seats awarded to different parties may exhibit bizarre non-monotonicities. When parties or voters care about joint events, such as whether a coalition of parties reaches a majority, these non-monotonicities can cause paradoxes, including incentives for strategic voting. In this paper, we propose monotonicity axioms ruling out these paradoxes, and study which of them can be satisfied jointly with Grimmett's axioms. Essentially, we require that, if a set of parties all receive more votes, the probability of those parties jointly receiving more seats should increase. Our work draws on a rich literature on unequal probability sampling in statistics (studied as dependent randomized rounding in computer science). Our main result shows that a sampling scheme due to Sampford (1967) satisfies Grimmett's axioms and a notion of higher-order correlation monotonicity.