  There are several measures for fairness in ranking, based on different
underlying assumptions and perspectives. PL optimization with the REINFORCE
algorithm can be used for optimizing black-box objective functions over
permutations. In particular, it can be used for optimizing fairness measures.
However, though effective for queries with a moderate number of repeating
sessions, PL optimization has room for improvement for queries with a small
number of repeating sessions.
  In this paper, we present a novel way of representing permutation
distributions, based on the notion of permutation graphs. Similar to PL, our
distribution representation, called PPG, can be used for black-box optimization
of fairness. Different from PL, where pointwise logits are used as the
distribution parameters, in PPG pairwise inversion probabilities together with
a reference permutation construct the distribution. As such, the reference
permutation can be set to the best sampled permutation regarding the objective
function, making PPG suitable for both deterministic and stochastic rankings.
Our experiments show that PPG, while comparable to PL for larger session
repetitions (i.e., stochastic ranking), improves over PL for optimizing
fairness metrics for queries with one session (i.e., deterministic ranking).
Additionally, when accurate utility estimations are available, e.g., in tabular
models, the performance of PPG in fairness optimization is significantly
boosted compared to lower quality utility estimations from a learning to rank
model, leading to a large performance gap with PL. Finally, the pairwise
probabilities make it possible to impose pairwise constraints such as "item
$d_1$ should always be ranked higher than item $d_2$." Such constraints can be
used to simultaneously optimize the fairness metric and control another
objective such as ranking performance.
