Fair assortment planning

Many online platforms, ranging from online retail stores to social media platforms, employ algorithms to optimize their offered assortment of items (e.g., products and contents). These algorithms often focus exclusively on achieving the platforms' objectives, highlighting items with the highest popularity or revenue. This approach, however, can compromise the equality of opportunities for the rest of the items, in turn leading to less content diversity and increased regulatory scrutiny for the platform. Motivated by this, we introduce and study a fair assortment planning problem that enforces equality of opportunities via pairwise fairness, which requires any two items to be offered similar outcomes. We show that the problem can be formulated as a linear program (LP), called (FAIR), that optimizes over the distribution of all feasible assortments. To find a near-optimal solution to (FAIR), we propose a framework based on the Ellipsoid method, which requires a polynomial-time separation oracle to the dual of the LP. We show that finding an optimal separation oracle to the dual problem is an NP-complete problem, and hence we propose a series of approximate separation oracles, which then result in a 1/2-approx. algorithm and an FPTAS for Problem (FAIR). The approximate separation oracles are designed by (i) showing the separation oracle to the dual of the LP is equivalent to solving an infinite series of parameterized knapsack problems, and (ii) leveraging the structure of knapsack problems. Finally, we perform numerical studies on both synthetic data and real-world MovieLens data, showcasing the effectiveness of our algorithms and providing insights into the platform's price of fairness.