Marketing has spent decades romanticising loyalty.
The language is emotional. Brands talk about love, trust, devotion, communities, advocates and fans. CRM teams build journeys designed to “nurture relationships”. Loyalty programmes promise recognition. Board decks refer to “retained customers” as if retention were a settled fact.
But customer behaviour is rarely that tidy.
Most customers are not loyal in the way brands want them to be. They are habitual, distracted, repertoire-driven, promotion-sensitive, context-dependent and frequently silent. They do not always leave with a complaint. They simply buy less often, split their category spend, experiment with alternatives, or disappear into another brand’s dataset.
That is why the evolution of loyalty and attrition models matters. From early Markov models of brand switching to the combined Hendry, Dirichlet and Negative Binomial Distribution frameworks, marketing science has steadily moved away from loyalty as a sentiment and toward loyalty as a behavioural probability.
This is not just an academic distinction. It changes how marketers should think about retention, churn risk, repeat purchasing, brand growth and the limits of CRM.
In a market where acquisition costs are high, switching is easy and consumer attention is unstable, the brands that win will not be those that believe in loyalty most loudly. They will be those that model it most accurately.
The loyalty myth
The traditional marketing idea of loyalty is seductive because it is simple. A customer buys a brand repeatedly, therefore the customer is loyal. A customer stops buying, therefore the customer has churned. A customer joins a loyalty programme, therefore the brand has strengthened the relationship.
All three assumptions are fragile.
Repeat buying can be driven by habit, availability, inertia, lack of alternatives, price promotions or category buying frequency. Churn can be formal, visible and contractual, as in telecoms or subscription software, but in most consumer categories it is hidden. Loyalty programme membership can indicate interest, but it can also indicate deal-seeking.
The deeper problem is that loyalty is often treated as a state. In reality, it is a process.
A grocery shopper may buy the same detergent several times, then switch because another brand is on promotion. A streaming subscriber may remain active for months, then cancel after a content lull. A fashion customer may buy twice in one season and then not return for a year. A banking customer may remain technically active while mentally disengaged.
In all these cases, the useful managerial question is not whether the customer is loyal. It is what the customer is likely to do next.
That is where stochastic models enter the picture.
Diffusion models explain adoption. Loyalty models explain survival.
The prompt asks for a comparative analysis of diffusion models in relation to loyalty and attrition. The distinction is important.
Diffusion models, such as Frank Bass’s A New Product Growth Model for Consumer Durables (1969), explain how adoption spreads through a market. They help marketers think about innovators, imitators, word-of-mouth effects and the shape of category growth. They are useful when the core question is: how will a product, technology or behaviour penetrate a population?
Loyalty and attrition models answer a different question: after customers enter the market or buy the brand, what happens next?
A diffusion model can show how quickly a subscription service might acquire users. A loyalty model shows whether those users stay. A diffusion model can forecast category adoption for a new product. A repeat-purchase model shows how often adopters come back and how they split purchases across competing brands.
The two perspectives are complementary. Diffusion without retention creates leaky growth. Retention without diffusion creates a stable but limited customer base. The commercial engine needs both: customer inflow and customer persistence.
That is why the evolution from adoption modelling to loyalty and attrition modelling is so significant. It moves marketing from “how do customers arrive?” to “how do customers behave over time?”
Markov models: loyalty as movement between states
Markov models were among the earliest formal tools used to model brand switching and repeat purchase behaviour.
The basic idea is straightforward. Customers move between states. A state might be “buys Brand A”, “buys Brand B”, “inactive”, “active”, “at risk” or “churned”. A Markov model estimates the probability that a customer moves from one state to another between periods or purchase occasions.
In brand switching analysis, the model might estimate the probability that someone who bought Brand A last time buys Brand A again next time, switches to Brand B, or leaves the category. In retention analysis, it might estimate the probability that an active customer becomes dormant, or that a dormant customer reactivates.
The strength of Markov models is that they make loyalty dynamic. Customers are no longer classified once and forgotten. They are seen as moving through behavioural states.
This is useful for managers because it creates transition-based thinking. A retention strategy is not just about “saving customers”. It is about reducing the probability of movement from active to inactive, increasing the probability of movement from trial to repeat, or improving the chance that a lapsed customer returns.
In telecoms and subscription markets, Markov models have been used to estimate switching behaviour and long-run market share. Studies such as Ansah-Narh and colleagues’ work on subscribers’ brand switching in Ghanaian network services show how Markov chains can be applied to predict switching behaviour and ergodic market share.
More recent papers, including Jiaqi Liang’s Analysis of Application of Markov Chain in Consumer Behavior Prediction (2024), continue to position Markov chains as useful tools for modelling consumer transitions and supporting marketing strategy.
But Markov models have limits.
Classic Markov models often assume that the next state depends mainly on the current state. They can underplay deeper customer heterogeneity: some people are naturally heavier category buyers, some are more variety-seeking, some are more loyal, and some are more promotion-responsive. A transition matrix may capture observed movement, but it may not fully explain why customers move differently.
That limitation matters because loyalty is not only about switching. It is also about buying frequency.
NBD: the forgotten foundation of repeat purchase
The Negative Binomial Distribution, or NBD, is central to understanding repeated buying.
The NBD is used to model purchase incidence: how often customers buy in a category over a period of time. Its importance lies in recognising heterogeneity. Customers do not all buy at the same rate. A few buy frequently. Many buy occasionally. Some barely buy at all.
This sounds obvious, but the consequences are profound.
If one customer buys a category ten times a year and another buys once, the first customer has ten opportunities to repeat, switch, respond to promotions or be counted as loyal. The second customer has one. Without modelling purchase frequency, marketers can easily misread loyalty.
Morrison and Schmittlein’s Generalizing the NBD Model for Customer Purchases: What Are the Implications and Is It Worth the Effort? (1988) is a key contribution in this tradition. It addresses the modelling of customer purchase counts and the implications of extending the NBD framework.
The NBD helps explain why many customer bases are highly uneven. Heavy buyers dominate volume, but light buyers dominate headcount. That distinction is essential for brand strategy.
A marketer looking only at repeat purchase rates may conclude that a brand has a small group of highly loyal customers. A marketer using NBD thinking may see that the brand’s sales are concentrated among heavy category buyers, while the larger growth opportunity lies in increasing penetration among light and occasional buyers.
This is one of the reasons loyalty strategy can become misleading. It over-focuses on heavy buyers because they are visible, measurable and responsive. But brand growth often depends on reaching many light buyers as well.
The Dirichlet model: the great demystifier of loyalty
The Dirichlet model is one of marketing science’s most important, and most inconvenient, contributions.
Developed by Goodhardt, Ehrenberg and Chatfield, The Dirichlet: A Comprehensive Model of Buying Behaviour (1984, with later editions and extensions) combines two components: the NBD for category purchase incidence and a Dirichlet-multinomial structure for brand choice.
In simpler terms, it models how often people buy the category and how they allocate those purchases across brands.
Its power lies in showing that many observed loyalty patterns can be explained without assuming deep psychological attachment. Repeat buying, brand switching, duplication of purchase, repertoire behaviour and differences in loyalty between large and small brands often follow predictable statistical regularities.
This creates a very different view of brand loyalty.
Large brands tend to have more buyers and slightly more loyal buyers. This is the well-known “double jeopardy” pattern: smaller brands suffer twice, because they have fewer customers and those customers buy them slightly less often. The implication is that loyalty is strongly related to penetration. Brands do not usually grow by turning small numbers of customers into exclusive devotees. They grow by increasing their buyer base.
That conclusion is uncomfortable for loyalty-led marketing. It challenges the idea that brands can easily engineer deep devotion through CRM, content or community. It suggests that, in many categories, customers are repertoire buyers who spread their purchases across a set of acceptable brands.
Sharp, Wright and Goodhardt’s Purchase Loyalty is Polarised into Either Repertoire or Subscription Patterns (2002) is useful here. It distinguishes between repertoire markets, where consumers buy from several brands over time, and subscription markets, where consumers tend to maintain a single provider until switching. This matters because loyalty strategy in a grocery category should not look like loyalty strategy in mobile contracts or insurance.
The Dirichlet model’s message is not that loyalty is irrelevant. It is that loyalty must be understood in relation to category structure.
Hendry models: the adaptive bridge
The Hendry model family sits in a more pragmatic tradition of modelling repeat purchasing and brand switching. It was developed for commercial market modelling and often used to forecast how new products, marketing activity and competitive dynamics affect repeat buying and share.
Where the Dirichlet model is famous for identifying stable empirical regularities, Hendry-type models are more intervention-oriented. They aim to incorporate the effects of awareness, trial, repeat purchase, switching and marketing activity.
This makes them relevant to managers because real markets are not always stable. Launches, promotions, distribution shifts, packaging changes, price moves and competitive campaigns can alter behaviour. A model that only describes equilibrium may not be enough when a brand is trying to forecast the impact of action.
The strength of Hendry-style approaches is that they connect behavioural modelling to commercial planning. They can help estimate trial, repeat, switching and share trajectories, particularly for new products or changing markets.
The weakness is that such models can become data-hungry and assumption-heavy. If the inputs are weak, the model may give a misleading sense of precision.
Still, in the evolution of loyalty and attrition modelling, Hendry frameworks represent an important bridge between aggregate market regularities and managerial simulation.
Comparing the models
Each model family answers a different question.
Markov models ask: how do customers move between states?
They are useful for modelling brand switching, churn states, reactivation paths and transition probabilities. Their managerial value lies in making retention dynamic.
NBD models ask: how often do customers buy?
They are useful for understanding purchase frequency, customer heterogeneity and the concentration of volume among heavy buyers. Their value lies in preventing marketers from confusing frequency with loyalty.
Dirichlet models ask: how do customers split category purchases across brands?
They are useful for understanding repertoire buying, duplication of purchase, double jeopardy and expected loyalty patterns. Their value lies in grounding brand strategy in empirical regularities.
Hendry models ask: how do trial, repeat, switching and marketing interventions shape future share?
They are useful for forecasting the effects of launches, competitive moves and marketing activity. Their value lies in managerial simulation.
No model is “best” in isolation. The better question is which behavioural problem a marketer is trying to solve.
For packaged goods, where customers frequently buy across multiple brands, NBD-Dirichlet thinking is particularly powerful. For telecoms or subscriptions, where customers are more likely to remain with one provider until switching, Markov and attrition models may be more useful. For new product launches, diffusion and Hendry-style repeat models can work together. For CRM, churn prediction and customer lifetime value, Markov logic can be combined with survival models, machine learning and probabilistic purchase models.
The important lesson is that loyalty is multi-dimensional. It involves purchase frequency, brand choice, switching probability, category structure and time.
What these models reveal about attrition
Attrition is often misunderstood because businesses want it to be visible. A customer cancels. A user unsubscribes. A buyer closes an account. A contract ends.
But in many categories, attrition is invisible. The customer simply stops buying.
This is the classic non-contractual problem. A retailer does not know whether a customer has churned after 60 days of silence. A fashion brand does not know whether a customer who bought last winter is gone, waiting, or just out of season. A grocery brand rarely knows whether a household has defected entirely or merely bought another brand on the last trip.
Markov models help by estimating transitions between active, dormant and lost states. NBD-based models help by estimating whether the observed buying pattern is consistent with normal low-frequency behaviour or genuine attrition. Dirichlet models help by clarifying whether switching is abnormal or simply part of repertoire buying.
This is critical because not every silence is churn.
A light buyer may have a long natural interpurchase interval. A heavy buyer who suddenly stops is more concerning. A repertoire buyer switching occasionally is normal. A subscription customer cancelling is structurally different.
Marketing needs models that distinguish these cases.
The AI retention boom and its blind spot
Recent industry coverage has moved strongly toward retention, personalisation and AI-led customer management.
Adobe’s 2025 customer loyalty coverage frames loyalty as a sustained relationship built through trust, repeat interactions and positive experiences. SAP Emarsys’ Customer Loyalty Index 2025 and its retention playbook focus on the need for structured engagement, personalisation and retention strategies. McKinsey’s 2025 article on “next best experience” argues that AI can power more relevant customer interactions and improve customer lifetime value through predictive analytics and personalisation.
The direction is commercially understandable. Brands want to identify the next best offer, message, channel and timing for each customer. They want to prevent churn before it happens. They want to lift repeat purchase through better orchestration.
But there is a risk: AI can rediscover old truths badly.
A churn model trained on behavioural data may predict defection, but without understanding category buying patterns it may confuse light buyers with lost customers. A personalisation engine may over-target heavy buyers because they are easiest to activate. A loyalty programme may reward customers who would have bought anyway. A retention campaign may spend money preventing churn that was never likely.
This is why the older models still matter. Markov, Dirichlet and NBD frameworks provide behavioural discipline. They remind marketers to ask whether observed behaviour is unusual, or simply expected given purchase frequency and category structure.
Modern AI can improve prediction. But without marketing science, it can misinterpret behaviour.
The managerial consequences
For marketers, the practical implications are substantial.
First, do not treat loyalty as an attitude alone. Attitudes matter, but behaviour is the evidence. A brand can have strong affinity and weak repeat buying. It can also have low emotional salience but strong habitual purchase.
Second, separate purchase frequency from brand loyalty. Heavy buyers will naturally appear more loyal because they buy more often. Light buyers are not necessarily disloyal. They may simply have fewer category occasions.
Third, understand the category structure. Repertoire markets and subscription markets behave differently. A loyalty strategy copied from one category to another may fail.
Fourth, distinguish switching from attrition. A customer buying another brand once has not necessarily defected. In many categories, switching is normal. The issue is whether the probability of future purchase has materially changed.
Fifth, use retention investment selectively. Not every at-risk customer is worth saving. Not every inactive customer is lost. Not every loyal customer needs an incentive. Modelling should guide intervention economics.
Sixth, do not expect CRM to overturn category laws. CRM can improve timing, relevance and service. It cannot magically turn repertoire buyers into exclusive buyers at scale.
Seventh, measure incrementality. A retention campaign that reaches high-probability repeat buyers may look successful while creating little incremental value. Models should identify not only who is likely to buy, but whose behaviour can be changed.
Why this matters for brand growth
The biggest strategic tension is between loyalty and penetration.
Many businesses over-invest in loyalty because existing customers are easier to identify, target and measure. Retention dashboards feel more controllable than broad brand-building. Loyalty programmes produce data. CRM campaigns produce immediate response.
But the Dirichlet tradition suggests that growth often depends heavily on penetration: getting more people to buy the brand at least occasionally. Loyalty tends to move with scale, not independently of it.
This does not mean retention is unimportant. It means retention should be understood realistically.
For small brands, trying to manufacture extreme loyalty may be less effective than increasing mental and physical availability. For large brands, protecting repeat buying matters, but the brand still needs broad reach. For subscription businesses, reducing churn can be essential, but acquisition quality and onboarding may matter as much as late-stage save campaigns.
The models do not eliminate strategy. They discipline it.
The future: combined models, not model tribes
The evolution of loyalty and attrition modelling points toward combination rather than replacement.
Markov models bring transition logic.
NBD models bring purchase frequency and heterogeneity.
Dirichlet models bring category-level regularities and brand choice structure.
Hendry models bring intervention and forecasting orientation.
Machine learning brings high-dimensional prediction.
Survival models bring time-to-event analysis.
Customer lifetime value models bring economic prioritisation.
The future is not choosing one model and declaring victory. It is assembling the right modelling stack for the decision.
A retailer might use NBD logic to understand purchase frequency, Markov states to classify active and dormant customers, machine learning to predict response, and CLV to decide whether intervention is worth the cost.
A subscription brand might use Markov transition models for lifecycle states, survival analysis for churn timing, and experimentation to measure which retention actions actually change outcomes.
A packaged goods brand might use Dirichlet benchmarks to understand expected loyalty patterns, then use media and distribution strategy to increase penetration.
The managerial challenge is not technical sophistication for its own sake. It is behavioural clarity.
The final point: loyalty is not owned. It is rented.
The most useful lesson from decades of loyalty and attrition modelling is also the least sentimental.
Customers do not belong to brands. Their future behaviour is probabilistic.
They may buy again. They may switch. They may lapse. They may return. They may split their spend. They may remain technically active but economically irrelevant. They may appear lost and then reappear after a long interval.
Marketing’s job is not to pretend this uncertainty does not exist. It is to manage it.
The contribution of Markov, Hendry, Dirichlet and NBD models is that they replace vague loyalty language with behavioural structure. They show that repeat purchasing can be modelled, switching can be normal, attrition can be inferred, and loyalty can be understood without mythology.
That is a more mature view of marketing.
It is also a more useful one.
Because once loyalty is seen as a probability rather than a promise, the question changes. It is no longer “how do we make customers loyal?” It becomes “how do we increase the probability, frequency and value of future buying behaviour?”
That is the question modern marketing should be organised around.
References
Ansah-Narh, T., et al. (2013). “Prediction of Subscribers’ Brand Switching Behaviour and Ergodic Market Share of Network Service Providers in Ghana Using Markov Chain Model.”
Bass, F. M. (1969). “A New Product Growth Model for Consumer Durables.” Management Science.
Ehrenberg, A. S. C. (1959). “The Pattern of Consumer Purchases.” Applied Statistics.
Goodhardt, G. J., Ehrenberg, A. S. C., & Chatfield, C. (1984). “The Dirichlet: A Comprehensive Model of Buying Behaviour.” Journal of the Royal Statistical Society. Series A.
Goodhardt, G. J., Ehrenberg, A. S. C., & Chatfield, C. (2006). The Dirichlet: A Comprehensive Model of Buying Behaviour.
Graham, C. (2012). “Brand Loyalty. Plus ça change…? Using the NBD-Dirichlet Parameters to Interpret Long-Term Purchase Incidence and Brand Choice.”
Liang, J. (2024). “Analysis of Application of Markov Chain in Consumer Behavior Prediction.”
Morrison, D. G., & Schmittlein, D. C. (1988). “Generalizing the NBD Model for Customer Purchases: What Are the Implications and Is It Worth the Effort?” Journal of Business & Economic Statistics.
Sharp, B., Wright, M., & Goodhardt, G. (2002). “Purchase Loyalty is Polarised into Either Repertoire or Subscription Patterns.” Australasian Marketing Journal.
Schmittlein, D. C., Morrison, D. G., & Colombo, R. (1987). “Counting Your Customers: Who Are They and What Will They Do Next?” Management Science.
Adobe for Business. (2025). “From Crowded Markets to Committed Customers: 2025 Insights to Improve Customer Loyalty.”
McKinsey & Company. (2025). “Next Best Experience: How AI Can Power Every Customer Interaction.”
SAP Emarsys. (2025). “Customer Loyalty Index 2025, Global Edition.”
SAP Emarsys. (2025). “The Retention Rate Playbook: 8 Proven Strategies for 2025 and Beyond.”
