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Customer Retention and Benford’s Law

Originally published May 16, 2005

Over the past few months we have looked at customer analytics, customer lifetime analysis and curious applications of logairthmic size laws such as Benford's Law for the purposes of data analysis. This month, I hope to tie these concepts together by exploring whether the application of Benford's Law to customer analysis might question the conventional wisdom associated with strategies for customer relationship management (CRM) programs. Clear your mind of preconceptions as you read Customer Retention and Benford's Law, and let me know what you think!

In last month’s article, we looked at how a large class of measurements associated with naturally occurring statistics reflected conformance with a logarithmic size law called Benford’s Law. The essence of the law provides a model for predicting the frequency of distribution of the initial numeric digits in data sets that:

  • Describe sizes of similar items or phenomena (such as populations, lengths and durations);
  • Are not assigned values (e.g., telephone numbers or social security numbers); and
  • Have no built-in maximums or minimums.

More to the point, we explored how logarithmic size laws reflect natural behaviors, and when put into the proper business context, could be integrated into information analysis for both data exploration and knowledge discovery purposes. In some of my previous articles, we have talked about customer lifetime value analytics, and this month I would like to tie these topics together by exploring how conclusions derived from the application of Benford’s law can be applied to customer lifetime analysis, retention and whether we should question the approaches taken to developing our customer relationship management (CRM) programs.

We can derive some basic inferences about data sets that observe Benford’s law:

  1. Within each order of magnitude, you are more likely to have smaller numbers than larger numbers (i.e., if all the numbers were between 0 and 99,  the probability of a measurement being between 10 and 20 is 30 percent, while the probability of the measurement being between 70 and 80 is a little under 6 percent).
  2. Increasing the size by a factor N will reduce the probability of occurrence by 1/N (for example, 10,000 is 1/10th as likely to appear as 1,000).
  3. The larger the magnitude, the longer it takes for the leftmost digit to switch over to the next consecutive digit (i.e., it takes longer for the leftmost digit of a 4-digit number to shift from a ‘1’ to a ‘2’ than it does for a 3-digit number).

With respect to durations, inference number 1 can be restated as “shorter durations are much more probable than longer ones.” Inference number 2 provides some guidance as to the relationship between orders of magnitude based on proportional size increases. Inference number 3 effectively says that the longer the duration, the more likely it is to get even longer.

Since the durations of a customer’s relationship with a company conform to the Benford’s criteria, applying Benford’s law to customer lifetimes yields some particularly interesting observations based on our inferences. The first is that based on the first two inferences, the customer attrition rate curve is effectively already predetermined—we should expect to have a lot more shorter-term customers than longer-term ones, and that the number of customers that will last for 20 months will probably be 1/10th the number of customers that last for 2 months. The second observation, based on inference number 3, is that the longer a party remains a customer, the more likely that party will continue to remain a customer. As a corollary, this also can be interpreted to say that the average customer lifetime for long-term customers grows for long-term customers.

But if our observations are true, then that certainly questions the purported effectiveness of any CRM program that attempts to reduce attrition, doesn’t it? Before casting doubt on anyone’s CRM program, it might be worthwhile to reevaluate the expectations in the context of what Benford’s law implies. On the positive side, taking these observations together, we might allow ourselves the flexibility of assessing the collective value of our customer pool by factoring in a lifetime value into a function of the how the current customer pool lies on the Benford curve, the expected attrition rates and the expected “time left” for each customer, even if we can’t predict the expected remaining lifetime value for any one particular customer.

The question, then, no longer should be how to retain all of your current customers, but rather to better understand return on investment associated with your approaches to customer retention. In other words, at each phase of a customer’s engagement, what is the appropriate amount of resources to invest in maintaining the relationship to maximize the “remaining lifetime value?”

This leads to some other possible good questions:

  • Are there any sentinel attributes or events associated with our long-term customers that can distinguish them early on in the relationship?
  • If we could identify long-term customers at an early stage, should we invest more resources in the relationship to ensure its long duration, or invest fewer resources because we already know the duration will be long?
  • Alternatively, if we could predict who the short-term customers are, should we adjust our customer relationship investment with those customers in order to maximize short-term profit?
  • At what point does the investment in retention pay off, and the increase in remaining lifetime value exceed the additional investment?
These, and other questions introduced by our little thought experiment might provide interesting insight into ways to gain competitive advantage by exploiting some simple mathematics. The challenge is to capture enough “mind-space” on this topic to enable some data analysis to test how closely longitudinal customer information conforms to logarithmic size laws. As a potential follow-up to this article, I would like to hear from any readers who are interested in participating in an analytical study of “Customer Lifetime Analysis and Logarithmic Size Laws.” If you are interested, please contact me at loshin@knowledge-integrity.com.

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