Compact Representations of Impact-Based Fair-Ranking Policies

On two-sided platforms such as e-commerce marketplaces, recommender systems deliver personalized item rankings to consumers and serve as a primary channel through which transactions occur between item providers and consumers. At the same time, promoting the activities of item providers requires the platform to balance consumer satisfaction with fair exposure across items. Recent work formulates this objective as impact-based fair ranking, which maximizes a Nash social welfare criterion over stochastic ranking policies. However, the resulting policies must be served as user-specific mixtures of deterministic rankings, whose support size can grow combinatorially. This imposes substantial storage and serving overhead, driving up operational cost in production deployment. Moreover, unseen users must fall back to a generic ranking rule. We therefore develop compact representations for serving impact-based fair-ranking policies. We first prove a tight bound that every optimum admits an impact-preserving mixture with support size at most the number of items. Using a dual reformulation, we then replace stored user-specific ranking mixtures with item-indexed dual vectors and compress the resulting sequence of dual vectors by weighted random subsampling. The resulting deployment representation is independent of the training-user population and can be used for unseen users. Experiments on real-world recommendation datasets show that the sampled dual mixture preserves the fair-ranking objective while reducing the storage required at deployment by up to 2,596$\times$ relative to a user-coupled mixture baseline.