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Loyalty tiers

Linear algebraMedium

Each month a customer sits in tier $1$ , $2$ or $3$ and moves according to the transition matrix $P = 0.6 0.2 0 0.4 0.5 0.4 0 0.3 0.6$ , where the entry in row $i$ , column $j$ is the probability of moving from tier $i$ to tier $j$ . What is the long-run fraction of months spent in tier $2$ ? Give a fraction in lowest terms.

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Hint

The chain only moves between neighbouring tiers, so in the long run the flow across each boundary balances.

Solution

The stationary vector $π = (π_{1}, π_{2}, π_{3})$ solves $π P = π$ , and for a chain on a line this reduces to balance across each boundary.
Between tiers $1$ and $2$ : $0.4 π_{1} = 0.2 π_{2}$ , so $π_{2} = 2 π_{1}$ .
Between tiers $2$ and $3$ : $0.3 π_{2} = 0.4 π_{3}$ , so $π_{3} = 0.75 π_{2} = 1.5 π_{1}$ .
The shares sum to $1$ , so $π_{1} \times (1 + 2 + 1.5) = 1$ , giving $π_{1} = \frac{2}{9}$ and $π_{2} = \frac{4}{9}$ .

Answer: $\frac{4}{9}$

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