# Optimal Subscription Planning for Digital Goods

50 Pages Posted: 6 Nov 2019

See all articles by Saeed Alaei

## Ali Makhdoumi

Fuqua School of Business; Massachusetts Institute of Technology (MIT)

## Azarakhsh Malekian

University of Toronto - Rotman School of Management; Massachusetts Institute of Technology (MIT) - Electrical Engineering and Computer Science

Date Written: October 27, 2019

### Abstract

We consider a media service provider that gives customers access to digital goods by either means of subscription fees or rentals. In our model, different types of users repeatedly use a platform over a period of time. The rate of usage varies across different customers. There are also multiple item types on the platform and the value of an item to a user depends on both the user type and the item type. The design of the platform's subscription planning comprises selecting a subscription fee for each set of goods and also selecting the rental price of each good. Before the beginning of the subscription period, given the subscription planning, users decide which set of goods to subscribe to (if any). During the subscription period, a user pays zero price to use an item if she has subscribed to a set that includes this item and pays its rental price, otherwise. Our first main result establishes the sufficient and necessary condition for the optimality of grand subscription-- offering rental prices together with a single subscription set that includes all items. Our second main result shows that without this condition, there exist subscription fees proportional to the cardinality of each set of goods that achieves $\frac{1}{4 (\log 2m+ \log n)}$ of the optimal revenue where $n$ and $m$ are the number of user types and item types, respectively. Finally, we show that this approximation is essentially tight by proving that no polynomial time algorithm can achieve a better than $\Omega(\frac{1}{\log n})$ of the optimal revenue.

Keywords: Subscription planning, digital goods, approximation algorithms, pricing

Suggested Citation

Alaei, Saeed and Makhdoumi, Ali and Malekian, Azarakhsh, Optimal Subscription Planning for Digital Goods (October 27, 2019). Available at SSRN: https://ssrn.com/abstract=3476296 or http://dx.doi.org/10.2139/ssrn.3476296