A Sequential Recommendation and Selection Model

Posted: 17 Sep 2019 Last revised: 23 Oct 2020

See all articles by Ningyuan Chen

Ningyuan Chen

University of Toronto at Mississauga - Department of Management; University of Toronto - Rotman School of Management

Guillermo Gallego


Pin Gao

School of Data Science, The Chinese University of Hong Kong, Shenzhen

Anran Li


Date Written: September 11, 2019


We propose a sequential recommendation-selection model where a seller recommends sets of products to consumers over multiple stages. consumers are heterogeneous in the patience levels, characterized by a certain number of stages that a consumer is willing to go through when making a purchase. Consumers view the products stage by stage. If a consumer can find a satisfactory product before exhausting her patience, she will purchase the product and leave the system immediately. Otherwise, the consumer stays till the last stage within her patience level but ends up without purchasing. The seller’s objective is to maximize his expected overall revenue by optimizing the recommendation sequence or the products’ prices. We note that the seller can learn the consumers’ patience levels as well as their utilities through the recommendation process, and thus can adjust his future recommendations accordingly. However, a static sequential recommendation strategy would suffice. Therefore, we derive a set of results:

1) For the pure recommendation order problem, the optimal solution possesses a sequential revenue-ordered property, which can be efficiently discovered by dynamic programming. We also find that a crude heuristic – only offering one set of products at a single stage – will earn a tight 50% of the optimal revenue.

2) In the single-leg dynamic capacity control problem, the optimal recommendations admit an inclusion property.

3) The optimal pricing policy under a fixed recommendation order is unique, which can be efficiently found by a binary search.

4) However, the joint recommendation and pricing problem is NP-hard, while recommending all products only at a single stage and optimizing their prices accordingly will earn a tight 88% of the optimal revenue.

Our results also characterize the reason that the assortment in stores is always same on different date in the following setting: A store provides one assortment on each date. Consumers make sequential decisions on consecutive dates, but consumers who first visit the store on different date may have different market sizes and different distributions of patience levels. The results are robust even when the market sizes and distributions of patience levels are unknown.

Keywords: Satisficing Choice Rule, Patience Levels, Sequential Recommendations, Sequential Selections

Suggested Citation

Chen, Ningyuan and Gallego, Guillermo and Gao, Pin and Li, Anran, A Sequential Recommendation and Selection Model (September 11, 2019). Available at SSRN: https://ssrn.com/abstract=3451727 or http://dx.doi.org/10.2139/ssrn.3451727

Ningyuan Chen

University of Toronto at Mississauga - Department of Management ( email )


University of Toronto - Rotman School of Management ( email )

105 St. George st
Toronto, ON M5S 3E6

Guillermo Gallego

HKUST ( email )

Clearwater Bay
Kowloon, 999999
Hong Kong

HOME PAGE: http://https://seng.ust.hk/about/people/faculty/guillermo-gallego

Pin Gao (Contact Author)

School of Data Science, The Chinese University of Hong Kong, Shenzhen ( email )

Anran Li

LSE ( email )

Houghton Street
London, WC2A 2AE
United Kingdom

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