New Perspectives on Customer 'Death' Using a Generalization of the Pareto/NBD Model

Abstract:     
Several researchers have proposed models of buyer behavior in noncontractual settings which assume that customers are "alive" for some period of time, then become permanently inactive.The best-known such model is the Pareto/NBD, which assumes that customer attrition (dropout or "death") can occur at any point in calendar time. A recent alternative model, the BG/NBD, assumes that customer attrition follows a Bernoulli "coin-flipping" process that occurs after every purchase occasion. While the modification results in a model that is much easier to implement, it means that heavy buyers have more opportunities to "die."

In this paper, we develop a model with a discrete-time dropout process tied to calendar time. Specifically, we assume that every customer periodically flips a coin" to determine whether she "drops out" or continues as a customer. For the purchasing while "alive" component, we maintain the assumptions of the Pareto/NBD and BG/NBD models. This periodic death opportunity (PDO) model allows us to take a closer look at how assumptions about customer death influence model t and various metrics typically used by managers to characterize a cohort of customers. When the time period after which each customer makes his or her dropout decision (which we call periodicity) is very small, we show analytically that the PDO model reduces to the Pareto/NBD. When the periodicity is longer than the calibration period, the dropout process is "shut off" and the PDO model converges to the NBD model. By systematically varying the periodicity between these limits, we can explore the full spectrum of models between the "continuous-time death" Pareto/NBD and the naive "no death" NBD.

In covering this spectrum, the PDO model performs at least as well as either of these models; we show this theoretically and our empirical analysis demonstrates the superior performance of the PDO model on two datasets. We also show that the different models provide significantly different estimates of both purchasing-related and death-related metrics for both datasets, and these differences can be quite dramatic for the death-related metrics. As more researchers and managers make managerial judgments that directly relate to the death process, we assert that the model employed to generate these metrics should be chosen carefully.

Keywords: Customer-base analysis, Pareto/NBD, BG/NBD, customer attrition

Working Paper Series