Fair Assortment Planning

TL;DR

Fair Assortment Planning introduces a pairwise fairness constraint into online assortment optimization, formulating Problem FAIR as an LP over randomized assortments. The authors show that optimal fairness can be achieved with a polynomially sized support, and they develop a dual-ellipsoid framework requiring a polynomial-time beta-approximate separation oracle for the subproblem (SUB-DUAL). They provide a 1/2-approximation method and a fully polynomial-time approximation scheme (FPTAS) by transforming SUB-DUAL into a family of parametric knapsack problems, analyzing well-behaved intervals, and employing adaptive partitioning and dynamic programming. The methods yield near-optimal solutions with provable guarantees, and numerical studies on synthetic data and MovieLens demonstrate favorable trade-offs between revenue and fairness, including practical runtimes and manageable numbers of assortments to randomize over. Overall, the work offers a principled, computable approach to enforcing equality of opportunity in assortment planning with real-world relevance and insights into the price of fairness.

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 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.

Figures (5)

• Figure 1: Evolution of $\textsc{kp}(W,K)$ with $W$.
• Figure 2: Boxplots of expected revenue across algorithms under different fairness levels ($\delta$). The box spans the first to third quartile, with a median line. The dashed line represents average optimal revenue without fairness constraints, and the red star marks the median optimal solution to Problem \ref{['eq:problem:fair']}.
• Figure 3: Price of fairness in the MovieLens case study. Left: platform's marketshare obtained under different fairness parameters $\delta$. The dashed line is the optimal marketshare attainable in the absence of fairness. Right: Number of sets to randomize over under different fairness parameters $\delta$.
• Figure 4: Visibility received by each movie under different $\delta$. The movie indices are decreasing based on qualities.
• Figure 5: We solve Problem \ref{['eq:problem:fair']} for $n=10$ and $K \in [n]$. Each item $i$ has $r_i {\sim} \text{ i.i.d. } \textsc{Unif}([0,1])$, and $w_i = \exp(\beta \cdot r_i + \theta_i)$, where $\beta$ is price sensitivity, $\theta_i \sim \text{ i.i.d. } \textsc{Unif}([0.4,1])$ for $i\leq m$ and $\textsc{Unif}([0.2,0.4])$ for $i>m$, where $1\leq m\leq \lceil n/4 \rceil$ are highly attractive items. We set $q_i = w_i$ for all $i$. For each $(K, m, \beta)$, we generate $50$ instances and compute Problem \ref{['eq:problem:fair']}'s objective when being $0$-fair w.r.t. different item outcomes. In markets with lower price sensitivity (Figure \ref{['fig:visibility-vs-sellability-low']}), fairness based on item visibility leads to higher platform revenue, while in markets with higher price sensitivity, fairness based on item revenue is more profitable (Figure \ref{['fig:visibility-vs-sellability-high']}).

Theorems & Definitions (18)

• Proposition 1: Randomization over at most $\mathcal{O}(n^2)$ sets
• Theorem 1: NP-Completeness of Problem \ref{['eq:subdual']}
• Theorem 2: $\beta$-Approx. Algorithm for Problem (FAIR)
• Proposition 2
• Theorem 3: Problem \ref{['eq:subdual']} as an infinite series of knapsack problems
• Lemma 1: $1/2$-Approx. Solutions to Problem \ref{['eqn:relaxed-knapsack']} caprara2000approximation
• Definition 1: Well-behaving interval
• Lemma 2
• Theorem 4: $1/2$-Approx. Algorithm
• Lemma 3