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We consider the problem of computing a sequence of rankings that maximizesconsumer-side utility while minimizing producer-side individual unfairness ofexposure. While prior work has addressed this problem using linear or quadraticprograms on bistochastic matrices, such approaches, relying on Birkhoff-vonNeumann (BvN) decompositions, are too slow to be implemented at large scale.  In this paper we introduce a geometrical object, a polytope that we callexpohedron, whose points represent all achievable exposures of items for aPosition Based Model (PBM). We exhibit some of its properties and lay out aCarath\'eodory decomposition algorithm with complexity $O(n^2\log(n))$ able toexpress any point inside the expohedron as a convex sum of at most $n$vertices, where $n$ is the number of items to rank. Such a decomposition makesit possible to express any feasible target exposure as a distribution over atmost $n$ rankings. Furthermore we show that we can use this polytope to recoverthe whole Pareto frontier of the multi-objective fairness-utility optimizationproblem, using a simple geometrical procedure with complexity $O(n^2\log(n))$.Our approach compares favorably to linear or quadratic programming baselines interms of algorithmic complexity and empirical runtime and is applicable to anymerit that is a non-decreasing function of item relevance. Furthermore oursolution can be expressed as a distribution over only $n$ permutations, insteadof the $(n-1)^2 + 1$ achieved with BvN decompositions. We perform experimentson synthetic and real-world datasets, confirming our theoretical results.

Introducing the Expohedron for Efficient Pareto-optimal Fairness-Utility Amortizations in Repeated Rankings