Stationary Market Shares in a Three-Channel Distribution System
A direct insurer tracks households among three distribution channels: agency (A), broker (B) and direct (D). Annual transition probabilities are estimated from three years of policyholder data and are stable: from A, 80% stay in A, 15% move to B and 5% move to D; from B, 10% move to A, 70% stay in B and 20% move to D; from D, 5% move to A, 15% move to B and 80% stay in D. Current market shares are A = 40%, B = 35% and D = 25%. The direct channel's head argues that because 80% of direct customers stay and 20% of broker customers switch to direct each year, direct will soon exceed 50% of the market. The CFO is skeptical and asks for the long-run stationary distribution and a two-year projection. No new entrants, no exits, and a fixed total market are assumed. The company is considering a major investment in direct-channel capacity based on the head's claim.
1. Find the stationary distribution pi = (pi_A, pi_B, pi_D) by solving pi P = pi with pi_A + pi_B + pi_D = 1. Show the equations and the solution.
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The transition matrix is P = [[0.80, 0.15, 0.05], [0.10, 0.70, 0.20], [0.05, 0.15, 0.80]].
The stationarity equations pi P = pi give: -0.20 pi_A + 0.10 pi_B + 0.05 pi_D = 0 0.15 pi_A - 0.30 pi_B + 0.15 pi_D = 0 0.05 pi_A + 0.20 pi_B - 0.20 pi_D = 0 plus pi_A + pi_B + pi_D = 1.
From the second equation, 0.15(pi_A + pi_D) = 0.30 pi_B, so pi_A + pi_D = 2 pi_B. From the first equation, multiply by 20: -4 pi_A + 2 pi_B + pi_D = 0, so pi_D = 4 pi_A - 2 pi_B. Substitute into pi_A + pi_D = 2 pi_B: pi_A + 4 pi_A - 2 pi_B = 2 pi_B 5 pi_A = 4 pi_B pi_B = 1.25 pi_A. Then pi_D = 4 pi_A - 2(1.25 pi_A) = 1.5 pi_A. The sum is pi_A + 1.25 pi_A + 1.5 pi_A = 3.75 pi_A = 1, so pi_A = 0.2667, pi_B = 0.3333 and pi_D = 0.4000.
The long-run stationary distribution is therefore 26.67% agency, 33.33% broker and 40.00% direct. Direct does not reach 50%.
2. Project the shares after two years from the current vector and compare with the stationary distribution. Does the direct channel reach 50%?
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Current vector: pi0 = (0.40, 0.35, 0.25).
After one year, pi1 = pi0 P: A: 0.40 x 0.80 + 0.35 x 0.10 + 0.25 x 0.05 = 0.320 + 0.035 + 0.0125 = 0.3675. B: 0.40 x 0.15 + 0.35 x 0.70 + 0.25 x 0.15 = 0.060 + 0.245 + 0.0375 = 0.3425. D: 0.40 x 0.05 + 0.35 x 0.20 + 0.25 x 0.80 = 0.020 + 0.070 + 0.200 = 0.2900.
After two years, pi2 = pi1 P: A: 0.3675 x 0.80 + 0.3425 x 0.10 + 0.29 x 0.05 = 0.2940 + 0.03425 + 0.0145 = 0.34275. B: 0.3675 x 0.15 + 0.3425 x 0.70 + 0.29 x 0.15 = 0.055125 + 0.23975 + 0.0435 = 0.338375. D: 0.3675 x 0.05 + 0.3425 x 0.20 + 0.29 x 0.80 = 0.018375 + 0.0685 + 0.2320 = 0.318875.
After two years the shares are about A = 34.3%, B = 33.8% and D = 31.9%. Direct is rising from 25% toward its long-run 40%, but it is still far below 50%. The head's claim is not supported by the transition matrix: even in the limit, direct reaches only 40%, and it would take many years to approach that level.
3. Assess the investment decision. What assumptions could change the conclusion, and what linear-algebra property guarantees convergence?
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The stationary distribution shows that direct will not exceed 50% under the current transition probabilities. A major capacity investment based on the head's claim would be premature. The decision should instead be based on sensitivity analysis: what happens if direct retention rises from 80% to 90%, or if broker-to-direct switching rises from 20% to 30%? Those changes could push the stationary direct share above 50%, but they are assumptions, not current data.
The convergence is guaranteed because P is a stochastic matrix and is primitive: some power of P has all positive entries, so the Markov chain is irreducible and aperiodic. The largest eigenvalue is 1, and the stationary distribution is the corresponding left eigenvector. The second-largest eigenvalue controls the speed of convergence; if it is close to 1, the shares move slowly. The current second eigenvalue is not extremely close to 1, but the two-year projection already shows gradual movement.
The main qualitative risks are competitive response, channel conflict, changes in customer behavior and new entrants. The insurer should not invest on the raw 80% retention statistic alone. It should model scenarios, track transition probabilities over time, and set a decision threshold that requires the direct stationary share to exceed 50% under plausible assumptions, not just under the most optimistic one.