Fair Assortment Planning

Abstract

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 tend to prioritize the platforms' short-term goals by featuring items with the highest popularity. However, this practice can then lead to too little visibility for the rest of the items, making them leave the platform, and in turn hurting the platform's long-term goals. Motivated by that, we introduce and study a fair assortment planning problem, which requires any two items with similar merits (popularities) to be offered similar visibility. 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 two approximate separation oracles: a 1/2-approx. algorithm and an FPTAS. The approximate separation oracles result in a polynomial-time 1/2-approx. algorithm and an FPTAS for the original problem (fair) using the Ellipsoid method. Further, they 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) taking advantage of the structure of the parameterized knapsack problems. Finally, we conduct a case study using the MovieLens dataset, demonstrating the efficacy of our 1/2-approx. and FPTAS fair Ellipsoid-based assortment planning algorithms.

Beyond submodularity: a unified framework of randomized set selection with group fairness constraints

“…In other words, the assortment that maximizes revenue consists of the l products with the highest revenues (r i + t∈[m] z t • 1 i∈Vt ), where l ∈ [n]. As a result, FairMax(z, F ) (problem (12)) can be solved optimally in polynomial time by examining at most n potential assortments. Thus, the combination of Theorem 1 and the ability to solve FairMax(z, F ) optimally in polynomial time (i.e., ρ = µ = 1) implies that an optimal solution for P.0 exists.…”

“…Remark: A recent work by [12] proposed an ellipsoid-based method for the assortment planning problem that includes pairwise fairness constraints. In page 14 of [12] (the October 28th, 2022 version), they discussed the case where all items have uniform revenues. They showed that for this special case, their separation oracle is to maximize the summation of a non-negative monotone submodular function and a (not necessarily positive) modular function.…”

“…We will now explore another frequently utilized notation of fairness, which relies on achieving parity in pairwise utility between groups. This type of notation has been employed in diverse scenarios, such as recommendation systems [3], assortment planning [12], ranking and regression models [23], and predictive risk scores [19]. Under this notion, it is expected that groups will experience comparable levels of utility.…”

“…The following proposition follows immediately from Theorem 3 and inequality (16). Remark: A recent work by [12] proposed an ellipsoid-based method for the assortment planning problem that includes pairwise fairness constraints. In page 14 of [12] (the October 28th, 2022 version), they discussed the case where all items have uniform revenues.…”

“…This approach offers more flexibility in attaining group fairness. Our framework is general enough to encompass various existing studies on achieving group fairness through randomization, such as those examined in [2,12,29].…”