Beyond Exposure: Optimizing Ranking Fairness with Non-linear Time-Income Functions

Ranking is central to information distribution in web search and recommendation. Nowadays, in ranking optimization, the fairness to item providers is viewed as a crucial factor alongside ranking relevance for users. There are currently numerous concepts of fairness and one widely recognized fairness concept is Exposure Fairness. However, it relies primarily on exposure determined solely by position, overlooking other factors that significantly influence income, such as time. To address this limitation, we propose to study ranking fairness when the provider utility is influenced by other contextual factors and is neither equal to nor proportional to item exposure. We give a formal definition of Income Fairness and develop a corresponding measurement metric. Simulated experiments show that existing-exposure-fairness-based ranking algorithms fail to optimize the proposed income fairness. Therefore, we propose the Dynamic-Income-Derivative-aware Ranking Fairness algorithm, which, based on the marginal income gain at the present timestep, uses Taylor-expansion-based gradients to simultaneously optimize effectiveness and income fairness. In both offline and online settings with diverse time-income functions, DIDRF consistently outperforms state-of-the-art methods.

💡 Research Summary

The paper addresses a critical gap in fair ranking research: most existing provider‑side fairness metrics equate a provider’s utility with the amount of exposure an item receives, ignoring that real‑world income often depends on additional contextual factors such as time, location, or sudden events. To remedy this, the authors introduce the concept of Income Fairness, which requires that the expected income earned by a provider be proportional to the relevance of the provider’s items. They formalize both an instantaneous version (for a single ranking) and an amortized version (across a sequence of rankings), defining precise mathematical conditions for proportionality between cumulative income and cumulative relevance.

To evaluate the shortcomings of exposure‑based fairness, the authors construct synthetic time‑income functions on public learning‑to‑rank datasets (e.g., MQ2007, Yahoo! LTR). These functions can be periodic (e.g., hour‑of‑day effects for restaurant recommendations) or event‑driven (e.g., breaking‑news spikes). Experiments show that algorithms designed to equalize exposure—such as linear‑programming‑based methods, policy‑gradient approaches, and differentiable Plackett‑Luce optimizers—fail to achieve the proposed income‑fairness metric, often producing large income disparities despite balanced exposure.

The core contribution is the Dynamic‑Income‑Derivative‑aware Ranking Fairness (DIDRF) algorithm. DIDRF operates in both offline (known relevance) and online (unknown relevance) settings and jointly optimizes user‑side effectiveness (measured by cumulative NDCG) and provider‑side income fairness. The key insight is to use the marginal income gain at the current timestep, i.e., the derivative of the income function with respect to exposure, as a guiding signal. By applying a second‑order Taylor expansion, the otherwise complex objective is approximated as a linear combination of user‑effectiveness gradients and income‑fairness gradients. Concretely, the overall loss is

L = λ * (−cNDCG) + (1‑λ) * IncomeFairnessLoss,

where IncomeFairnessLoss penalizes deviations from the proportionality condition. The gradient with respect to each item’s score incorporates the term ∂v_t/∂E_i, the derivative of the time‑dependent income function evaluated at the current exposure. This derivative directly captures how much additional income would be generated by moving the item one position higher, allowing the optimizer to prioritize moves that improve both relevance and income fairness.

In offline experiments, DIDRF consistently outperforms state‑of‑the‑art exposure‑fairness baselines across a range of synthetic income functions. Gains are especially pronounced for non‑linear, sharply changing income curves, where the derivative information is crucial. In online simulations, user clicks are modeled with position‑biased examination probabilities; the system updates relevance estimates and income statistics in real time. DIDRF maintains amortized income fairness while achieving comparable or slightly higher cumulative NDCG than baselines, demonstrating that fairness does not have to come at the expense of relevance.

The authors also conduct extensive ablation studies. Varying the trade‑off parameter λ reveals a smooth efficiency‑fairness frontier, with λ≈0.7 offering a balanced sweet spot. Sensitivity analysis shows that DIDRF remains stable even when the income function exhibits abrupt spikes, thanks to the second‑order approximation that prevents gradient explosion. Computationally, DIDRF adds only the cost of evaluating the income derivative, which is negligible compared to the overall ranking pipeline, and it fits naturally into feedback‑loop frameworks that reuse historical exposure and income data.

Overall, the paper makes three major contributions: (1) a formal definition and measurement framework for income fairness that captures time‑dependent, non‑linear provider utility; (2) a thorough empirical demonstration that exposure‑fairness methods are insufficient under realistic income models; and (3) a novel, theoretically grounded algorithm (DIDRF) that leverages income derivatives to jointly optimize relevance and fairness. By moving beyond exposure as a proxy, this work opens new avenues for fair ranking in time‑sensitive domains such as news feeds, real‑time advertising, and location‑based recommendation services.