  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.
