Brand-Switching Markov Model

It assumes that the brand a consumer buys next depends only on the brand bought most recently, not on the entire prior purchase history. This is a strong simplification: real consumers exhibit longer memory, variety-seeking, and responsiveness to price and promotion that a first-order chain ignores. The assumption is most defensible in stable, frequently purchased categories where recent purchase is a good summary of state. When it fails, analysts move to higher-order chains, latent-class mixtures of chains, or covariate-driven choice models that carry richer history forward, such as logit models with a loyalty variable.

The steady-state, or stationary, distribution is the share vector pi* that does not change when multiplied by the transition matrix: pi* equals pi* times P, with the entries summing to one. It is obtained by solving that linear eigenvector system. Intuitively it is the long-run equilibrium the market drifts toward if the current switching pattern persists, regardless of where shares start. Comparing the steady state to today's shares signals whether a brand's loyalty and the inflows it attracts will let it grow or force it to decline over time. It exists and is unique when the chain is irreducible and aperiodic.

Both describe how past purchasing shapes future brand choice, but they do it differently. A Markov model summarizes history compactly in a transition matrix and ignores marketing covariates within a transition. A scanner-panel logit, like Guadagni and Little's, models each purchase as a function of price, promotion, and an explicit loyalty variable that smooths past purchases, capturing both history and the marketing mix. The Markov model is the simpler diagnostic and conceptual precursor; the logit is the richer, covariate-driven successor used when you need to quantify how price and promotion drive switching rather than just describe it.