WINE 2018: Web and Internet Economics pp 218-231

# Combinatorial Assortment Optimization

• Nicole Immorlica
• Brendan Lucier
• Jieming Mao
• Vasilis Syrgkanis
• Christos Tzamos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

## Abstract

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.

## Notes

### Acknowledgment

We would like to thank Aviad Rubinstein for pointing out an improvement on Theorem 7.

## References

1. 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).
2. 2.
Alptekinolu, A., Semple, J.H.: The exponomial choice model: a new alternative for assortment and price optimization. Oper. Res. 64(1), 79–93 (2016)
3. 3.
Bront, J.J.M., Méndez-Díaz, I., Vulcano, G.: A column generation algorithm for choice-based network revenue management. Oper. Res. 57(3), 769–784 (2009).
4. 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).
5. 5.
Caro, F., Gallien, J.: Dynamic assortment with demand learning for seasonal consumer goods. Manag. Sci. 53, 276–292 (2007)
6. 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).
7. 7.
Davis, J.M., Gallego, G., Topaloglu, H.: Assortment optimization under variants of the nested logit model. Oper. Res. 62(2), 250–273 (2014).
8. 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
9. 9.
Feige, U.: On maximizing welfare when utility functions are subadditive. SIAM J. Comput. 39(1), 122–142 (2009).
10. 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. 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).
12. 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).
13. 13.
Leme, R.P.: Gross substitutability: an algorithmic survey. Games Econ. Behav. 106, 294–316 (2017). . http://www.sciencedirect.com/science/article/pii/S0899825617301884
14. 14.
Méndez-Díaz, I., Miranda-Bront, J.J., Vulcano, G., Zabala, P.: A branch-and-cut algorithm for the latent-class logit assortment problem. Discrete Appl. Math. 164, 246–263 (2014).
15. 15.
Rusmevichientong, P., Shen, Z.J.M., Shmoys, D.B.: Dynamic assortment optimization with a multinomial logit choice model and capacity constraint. Oper. Res. 58(6), 1666–1680 (2010)
16. 16.
Talluri, K., van Ryzin, G.: Revenue management under a general discrete choice model of consumer behavior. Manag. Sci. 50(1), 15–33 (2004)
17. 17.
Ulu, C., Honhon, D., Alptekinolu, A.: Learning consumer tastes through dynamic assortments. Oper. Res. 60(4), 833–849 (2012)

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Nicole Immorlica
• 1
• Brendan Lucier
• 1
• Jieming Mao
• 2
Email author
• Vasilis Syrgkanis
• 1
• Christos Tzamos
• 3
1. 1.Microsoft ResearchBostonUSA