NBD-Dirichlet Model

Double jeopardy is the empirical pattern that small brands are penalized twice: they have far fewer buyers than large brands, and those buyers also buy them slightly less often and are slightly less loyal. The NBD-Dirichlet model predicts this as a near-mathematical consequence of brand size under near-random brand choice. When households choose among brands roughly in proportion to stable preferences with little true loyalty, a brand with a small share simply gets fewer occasions from most households, which lowers both its penetration and the average frequency and loyalty of its buyers. The practical implication is profound: differences in loyalty across brands of different sizes are usually normal and structural, not evidence that big brands have better loyalty programs or that small brands have failed.

In an unpartitioned market, the duplication-of-purchase law says that the proportion of brand A's buyers who also buy brand B is roughly proportional to brand B's penetration, regardless of which brand A is. The NBD-Dirichlet model formalizes this expectation. To detect partitioning, an analyst compares observed brand-to-brand duplication against the model's predicted duplications. If a subset of brands shares customers much more than the model predicts, while sharing less with brands outside the subset, that excess duplication signals a genuine submarket or partition, for example diet versus regular variants, or premium versus value tiers. Thus the Dirichlet acts as the 'no partition' null hypothesis, and systematic deviations from it are the evidence of structure.

The model is built for stationary, competitive, frequently purchased categories, classically consumer packaged goods like detergents, cereals or beverages, analyzed from consumer or household panel data that records repeated category purchases and the brands chosen over a period. It assumes the market is roughly in equilibrium (no strong growth, decline or structural upheaval), that brand choice is close to zero-order with limited true loyalty, and that there is no strong segmentation in buying timing. It is not designed for durable or infrequently purchased goods, for markets in rapid transition, or for situations dominated by promotions and price dynamics. When these assumptions hold, the model fits strikingly well; when they do not, the misfit itself can be informative about what is unusual in the market.