Statistical models on full and partial rankings of n items are often of limited prac- tical use for large n due to computational consideration. We explore the use of non-parametric models for partially ranked data and derive ef(cid:2)cient procedures for their use for large n. The derivations are largely possible through combinatorial and algebraic manipulations based on the lattice of partial rankings. In particular, we demonstrate for the (cid:2)rst time a non-parametric coherent and consistent model capable of ef(cid:2)ciently aggregating partially ranked data of different types.

Predictive models learned from historical data are widely used to help companies and organizations make decisions. However, they may digitally unfairly treat unwanted groups, raising concerns about fairness and discrimination. In this paper, we study the fairness-aware ranking problem which aims to discover discrimination in ranked datasets and reconstruct the fair ranking. Existing methods in fairness-aware ranking are mainly based on statistical parity that cannot measure the true discriminatory effect since discrimination is causal. On the other hand, existing methods in causal-based anti-discrimination learning focus on classification problems and cannot be directly applied to handle the ranked data. To address these limitations, we propose to map the rank position to a continuous score variable that represents the qualification of the candidates. Then, we build a causal graph that consists of both the discrete profile attributes and the continuous score. The path-specific effect technique is extended to the mixed-variable causal graph to identify both direct and indirect discrimination. The relationship between the path-specific effects for the ranked data and those for the binary decision is theoretically analyzed. Finally, algorithms for discovering and removing discrimination from a ranked dataset are developed. Experiments using the real dataset show the effectiveness of our approaches.