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STOCHASTIC MONOTONICITY AND REALIZABLE MONOTONICITY 1

We explore and relate two notions of monotonicity, stochastic and re- alizable, for a system of probability measures on a common ﬁnite partially ordered set (poset) (cid:1) when the measures are indexed by another poset (cid:2) . We give counterexamples to show that the two notions are not always equivalent, but for various large classes of (cid:1) we also present conditions on the poset (cid:2) that are necessary and sufﬁcient for equivalence. When (cid:2) = (cid:1) , the condition that the cover graph of (cid:1) have no cycles is neces- sary and sufﬁcient for equivalence. This case arises in comparing applica-bility of the perfect sampling algorithms of Propp and Wilson and the ﬁrst author of the present paper.