Industry: Retail Product: ModelRisk Application: Quantifying brand loyalty and share-of-wallet under competitive switching
A retailer's category team reported a clean headline for its flagship brand: a 41% share-of-wallet in the target segment, comfortably ahead of two rivals. The number came from the standard approach — take the average brand-switching behaviour, build one transition matrix, solve for the long-run share. It read like a winning position, and the team planned next year's marketing spend as if 41% were a fact.
Share-of-wallet is not a fact; it is the steady state of a competitive switching process whose parameters are estimated, not known. Every quarter a shopper either repeats their last brand or switches, and the rates at which they do so are uncertain. Carry that uncertainty through the Markov chain and the same brand looks both stronger and more fragile than 41% suggests: it is the single largest brand 73.7% of the time, but it holds an outright majority (more than 50% of wallet) only 9.3% of the time, with a share that ranges from a P10 of 32.6% to a P90 of 49.7%. "We lead" and "we dominate" are different claims, and the point estimate could not tell them apart.
We model the category as a three-brand Markov chain — Us, Rival A, Rival B — where each shopper occupies a brand state and, on each purchase occasion, either repeats or switches according to a 3×3 transition matrix. Long-run share-of-wallet is the chain's stationary distribution, the share each brand settles to once switching reaches equilibrium.
The transition rates are uncertain, so each row of the matrix is drawn from a Dirichlet distribution — the natural choice for a set of probabilities that must sum to one. The priors encode the believed behaviour and its confidence: our brand has a mean repeat rate near 59%, Rival A stays with itself about 53% of the time while leaking roughly 29% of its defectors to us, and Rival B is similar. A single matrix is drawn per trial and shared across all shoppers in that trial — a market-regime common factor, so a "sticky market" draw lifts everyone's repeat rate together rather than averaging away.
The resulting distribution of our steady-state share has a mean of 41.0%, a median of 40.8%, a P10 of 32.6% and a P90 of 49.7%. The deterministic answer built from the mean matrix is also 41.0% — and that exact agreement is the trap. The point estimate gets the centre right and the risk completely wrong: it cannot show that the upside tail barely scrapes a majority while the downside reaches into the low thirties.
A 41% point share invites a binary reading — "we're the leader" — that the distribution refuses to support. Leading the category and owning it are separate questions with very different answers. Our brand is the largest of the three in 73.7% of trials, which justifies the "category leader" claim. But crossing 50% of total wallet happens in only 9.3% of trials. A strategy that assumes the brand can act like a dominant incumbent — dictating terms, under-investing in defence — is betting on a 9% outcome while the brand actually lives in a contested plurality.
Plotting all three brands as cumulative curves shows why the segment is genuinely competitive rather than won. Our mean share is 41.0% (P10 32.6%, P90 49.7%), Rival A sits at 31.5% (P10 23.8%, P90 39.6%), and Rival B at 27.4% (P10 19.6%, P90 35.8%). The curves overlap heavily: there are plausible draws in which Rival A's share exceeds our own. The gap between us and the field is real but thin, and a deterministic 41% versus 31% versus 27% reading hides how often the order could be contested.
If the goal is to move from "leading" to "owning," the lever is our own repeat-purchase rate — the probability a buyer stays with us next time. Sweeping that rate from 40% to 85% and re-solving the chain at each level converts a fuzzy ambition into a target.
At a 55% repeat rate the chance of an outright majority is effectively 0% (mean share 38%); at 70% it climbs to 33% (mean share 48%); at 85% it reaches 100% (mean share 65%). The crossover — where a majority becomes more likely than not — is a repeat-purchase rate of 72.2%. That is the concrete retention bar the brand would have to clear to convert plurality into dominance, and it is a far more actionable target than "improve loyalty."
With several uncertain switching cells, the tornado isolates the ones our share-of-wallet actually turns on.
Our own repeat rate dominates, swinging share across a 12.4pp range (from 35.1% to 47.5%) between its P10 and P90 — defending existing buyers matters more than anything else. Winning defectors from Rival A is next at 8.6pp, then losing buyers to Rival A at 8.4pp and converting Rival B's defectors at 7.9pp. Rival A's own stickiness is the weakest lever at 5.6pp. The strategic read is unambiguous: a point of retention on our own base is worth more than a point of conquest from either rival, and Rival A — not the smaller Rival B — is the competitor whose two-way flow with us matters most.
The category team stopped treating 41% as a position and started treating it as a contested distribution:
Monte Carlo turns brand loyalty from "what is our share?" into "how often do we lead, how rarely do we dominate, and what repeat rate closes the gap?" — and on this segment the honest answer was a brand that leads three years in four but rules barely one year in ten.