Fair re-ranking aims to promote long-tail items and enhance diversity within groups in information retrieval. While previous research on online fairness-aware re-ranking has shown promising outcomes, our comprehensive evaluation of online fair re-ranking methods over 20 settings reveals significant performance disparities among existing methods. To uncover the root causes of these inconsistencies, we reformulate fair re-ranking within an attentional market framework governed by a Walrasian Equilibrium, where the fairness is treated as a taxation cost. This market-based formulation is then coupled with manifold optimization, demonstrating that seeking this equilibrium is equivalent to performing gradient descent on a specific ranking manifold constructed by the market. Different re-ranking settings induce distinct manifold geometries, and these intrinsic geometric differences dictate the gradient landscapes and optimization trajectories. We propose ManifoldRank, an efficient online fair re-ranking algorithm. ManifoldRank adjusts gradients to align with the ranking manifold, considering various contextual settings. On the supply side, it incorporates a gradient adjustment based on different fairness requirements, accounting for associated costs. On the demand side, it empirically predicts an additional gradient adjustment term derived from the ranking scores. By integrating these two gradient adjustments, ManifoldRank effectively balances fairness and accuracy. Experimental results across multiple datasets confirm ManifoldRank's effectiveness.

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