Novel uses of the Mallows model in coloring and matching

Abstract

A natural model of a highly ordered random ranking is the Mallows model. Disorder is measured by the number of inversions; these are pairs of elements whose order is reversed. The Mallows model assigns to each ranking of a finite set a weight proportional to a parameter, q, raised to the number of inversions. Rankings of a finite set may be regarded as permutations, and thus the Mallows model consists of a probability measure on permutations. Originally introduced by Colin L. Mallows in statistical ranking theory, this model has enjoyed a recent flurry of interest in contexts including mixing times, statistical physics, learning theory, and longest increasing subsequences. In this work, we apply the model in a novel manner, using its intrinsic mathematical properties to resolve open questions in probability theory. These questions involve neither permutations nor rankings. Instead, we will use the model to construct random proper colorings of the integers possessing certain properties whose joint satisfiability was previously unknown. The first of these properties is that the colorings are finitely dependent: restrictions of the colorings to subsets separated by a fixed, finite, distance are stochastically independent of one another. The second is that they are expressible as finitary factors of iid with finite mean coding radius: this technical condition means, roughly speaking, that the colorings can be efficiently constructed by a parallel distributed algorithm. In addition to coloring applications, we explore connections between the Mallows model and stable matching. The celebrated Gale-Shapley algorithm provided the first proof of existence of stable matchings for arbitrary preferences. This is a fundamental result in the economic theory of stable allocations, a theory which appeared in the citation of the 2012 Nobel Prize in Economics awarded to Lloyd Shapley and Alvin E. Roth. Variants of the Gale-Shapley algorithm play a crucial in practical applications of stable matching, including in the National Resident Matching Program, which matches medical school students with residency programs. We introduce a model of stable matching in which the preferences are Mallows-distributed. A novel feature of this model is that preferences are highly correlated. Perhaps surprisingly, this leads to a vastly greater number of stable matchings than in the uniformly random case: exponential in the number of individuals, rather than polynomial. Finally, we show that the Mallows permutation itself can be regarded as the solution of a stable matching problem. Here, one presupposes a global ranking of the individuals, with random and independent incompatibilities. It turns out that for a finite set of individuals, there is always a unique stable matching. We show that the conditional distribution of this matching, given that it is perfect, is the Mallows measure. We also consider infinite analogues of this result, uncovering new phenomena not present in the finite case.