# Estimating sales transitions between competing products via optimal transport

## TL;DR

This paper introduces a method to estimate how customers switch between competing products using aggregated sales data without individual customer IDs.

## Contribution

The novel approach formulates sales transitions as an optimal transport problem with regularization terms for brand-switching costs.

## Key findings

- Estimated transitions matched real market changes, such as shifts during a liquor tax reform.
- In the coffee market, many customers moved to a newly launched brand.
- Proprietary data limited reproducibility, and individual customer tracking remains impossible without IDs.

## Abstract

In mature markets, where products are widely adopted, understanding how customers switch between competing products is crucial for companies to conduct effective marketing actions. However, due to privacy regulations, it is increasingly difficult to obtain point-of-sale (POS) data with individual customer identifiers (IDs). In this paper, we propose a method that estimates how sales shift between products using aggregated POS data without customer IDs. We formulate this as an optimal transport problem aimed at minimizing the total cost of brand-switching and introduce two regularization terms based on assumptions about sales transitions. We then solve the optimization problem with these regularizations using a projected gradient method.

We validated our approach on proprietary POS data from the Japanese beverage industry and found that the estimated transitions aligned with real market changes. For instance, during a liquor tax reform period, customers switched from products whose tax rates increased to those with lower rates. In the coffee market, many customers moved toward a newly launched brand. Although these results suggest that our method can capture market dynamics, the proprietary data limits reproducibility. In addition, the absence of customer IDs makes it impossible to track individual customer transitions. Incorporating such identifiers in future research could offer more deeper insights into consumer behavior.