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In recent years, it has become clear that rankings delivered in many areasneed not only be useful to the users but also respect fairness of exposure forthe item producers. We consider the problem of finding ranking policies thatachieve a Pareto-optimal tradeoff between these two aspects. Several methodswere proposed to solve it; for instance a popular one is to use linearprogramming with a Birkhoff-von Neumann decomposition. These methods, however,are based on a classical Position Based exposure Model (PBM), which assumesindependence between the items (hence the exposure only depends on the rank).In many applications, this assumption is unrealistic and the communityincreasingly moves towards considering other models that include dependences,such as the Dynamic Bayesian Network (DBN) exposure model. For such models,computing (exact) optimal fair ranking policies remains an open question.  We answer this question by leveraging a new geometrical method based on theso-called expohedron proposed recently for the PBM (Kletti et al., WSDM'22). Welay out the structure of a new geometrical object (the DBN-expohedron), andpropose for it a Carath\'eodory decomposition algorithm of complexity $O(n^3)$,where $n$ is the number of documents to rank. Such an algorithm enablesexpressing any feasible expected exposure vector as a distribution over at most$n$ rankings; furthermore we show that we can compute the whole set ofPareto-optimal expected exposure vectors with the same complexity $O(n^3)$. Ourwork constitutes the first exact algorithm able to efficiently find aPareto-optimal distribution of rankings. It is applicable to a broad range offairness notions, including classical notions of meritocratic and demographicfairness. We empirically evaluate our method on the TREC2020 and MSLR datasetsand compare it to several baselines in terms of Pareto-optimality and speed.

Pareto-Optimal Fairness-Utility Amortizations in Rankings with a DBN Exposure Model