Combinatorial Assortment Optimization
Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a choice model in which each consumer selects a utility-maximizing bundle subject to a private valuation function, and study the complexity of the resulting optimization problem. Our main result is an exact algorithm for k-additive valuations, under a model of vertical differentiation in which customers agree on the relative value of each pair of items but differ in their absolute willingness to pay. For valuations that are vertically differentiated but not necessarily k-additive, we show how to obtain constant approximations under a “well-priced” condition, where each product’s price is sufficiently high. We further show that even for a single customer with known valuation, any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even when the valuation is succinctly representable.
We would like to thank Aviad Rubinstein for pointing out an improvement on Theorem 7.
- 1.Agrawal, S., Avadhanula, V., Goyal, V., Zeevi, A.: A near-optimal exploration-exploitation approach for assortment selection. In: Proceedings of the 2016 ACM Conference on Economics and Computation, EC 2016, pp. 599–600. ACM, New York (2016). https://doi.org/10.1145/2940716.2940779
- 4.Cai, Y., Daskalakis, C., Weinberg, S.M.: Optimal multi-dimensional mechanism design: reducing revenue to welfare maximization. In: Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, FOCS 2012, pp. 130–139. IEEE Computer Society, Washington (2012). https://doi.org/10.1109/FOCS.2012.88
- 6.Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: Proceedings of the Forty-second ACM Symposium on Theory of Computing, STOC 2010, pp. 311–320. ACM, New York (2010). https://doi.org/10.1145/1806689.1806733
- 8.Desir, A., Goyal, V.: Near-optimal algorithms for capacity constrained assortment optimization. Technical report, Department of Industrial Engineering and Operations Research, Columbia University (2015)Google Scholar
- 10.Haghpanah, N., Hartline, J.: Reverse mechanism design. In: Proceedings of the Sixteenth ACM Conference on Economics and Computation, pp. 757–758. ACM (2015)Google Scholar
- 11.Kleinberg, J., Mullainathan, S., Ugander, J.: Comparison-based choices. In: Proceedings of the 2017 ACM Conference on Economics and Computation, EC 2017, pp. 127–144. ACM, New York (2017). https://doi.org/10.1145/3033274.3085134
- 12.Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. In: Proceedings of the 3rd ACM Conference on Electronic Commerce, EC 2001, pp. 18–28. ACM, New York (2001). https://doi.org/10.1145/501158.501161