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There are several measures for fairness in ranking, based on differentunderlying assumptions and perspectives. PL optimization with the REINFORCEalgorithm can be used for optimizing black-box objective functions overpermutations. In particular, it can be used for optimizing fairness measures.However, though effective for queries with a moderate number of repeatingsessions, PL optimization has room for improvement for queries with a smallnumber of repeating sessions.  In this paper, we present a novel way of representing permutationdistributions, based on the notion of permutation graphs. Similar to PL, ourdistribution representation, called PPG, can be used for black-box optimizationof fairness. Different from PL, where pointwise logits are used as thedistribution parameters, in PPG pairwise inversion probabilities together witha reference permutation construct the distribution. As such, the referencepermutation can be set to the best sampled permutation regarding the objectivefunction, making PPG suitable for both deterministic and stochastic rankings.Our experiments show that PPG, while comparable to PL for larger sessionrepetitions (i.e., stochastic ranking), improves over PL for optimizingfairness metrics for queries with one session (i.e., deterministic ranking).Additionally, when accurate utility estimations are available, e.g., in tabularmodels, the performance of PPG in fairness optimization is significantlyboosted compared to lower quality utility estimations from a learning to rankmodel, leading to a large performance gap with PL. Finally, the pairwiseprobabilities make it possible to impose pairwise constraints such as "item$d_1$ should always be ranked higher than item $d_2$." Such constraints can beused to simultaneously optimize the fairness metric and control anotherobjective such as ranking performance.

Probabilistic Permutation Graph Search: Black-Box Optimization for Fairness in Ranking