Markov Chain Steady-State Market Share

Stochastic Processes · Medium · Free problem

Two companies, $A$ and $B$, compete for customers. In each period, a fraction of each company's customers may switch to the other. Company $A$ retains $80\%$ of its customers and loses

0\%$ to $B$. Company $B$ retains $70\%$ of its customers and loses $30\%$ to $A$.

The initial market shares are $s_A^{(0)}$ and $s_B^{(0)}$ with $s_A^{(0)} + s_B^{(0)} = 1$.

  1. Write down the transition matrix $P$ and the market share update rule.
  2. Find the steady-state (long-run) market shares $\pi_A$ and $\pi_B$.
  3. Do the steady-state shares depend on the initial market shares? Why or why not?

Hints

  1. Model this as a Markov chain. What is the transition matrix, and what equation does the steady-state distribution satisfy?
  2. At equilibrium, the flow of customers from $A$ to $B$ must equal the flow from $B$ to $A$. Set $0.2\,\pi_A = 0.3\,\pi_B$.
  3. Use the constraint $\pi_A + \pi_B = 1$ along with the balance equation to solve for both shares. Then argue why the initial distribution does not matter.

Worked Solution

How to Think About It: This is a textbook Markov chain problem. Customers switch between companies according to fixed probabilities each period, so market shares evolve as a matrix multiplication. The steady state is the left eigenvector of the transition matrix with eigenvalue

$ -- it is the distribution that, once reached, does not change. The key practical insight: for an ergodic (irreducible, aperiodic) chain, the steady state is unique and independent of where you start.

Quick Estimate: Company $A$ loses

0\%$ per period, $B$ loses $30\%$. Since $B$ leaks faster, you would expect $A$ to end up with a larger share. Roughly, the steady state should balance the flows: $0.2 \pi_A = 0.3 \pi_B$, giving $\pi_A / \pi_B = 3/2$, so $\pi_A = 0.6$, $\pi_B = 0.4$.

Approach: Set up the balance equations from the transition matrix and solve.

Formal Solution:

Part 1 -- Transition matrix:

The transition matrix (rows are "from", columns are "to") is:

$P = \begin{pmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{pmatrix}$

Market shares update as:

$\mathbf{s}^{(t+1)} = \mathbf{s}^{(t)} P$

where $\mathbf{s}^{(t)} = (s_A^{(t)},\, s_B^{(t)})$ is a row vector.

Part 2 -- Steady-state shares:

At steady state, $\boldsymbol{\pi} = \boldsymbol{\pi} P$, which gives:

$\pi_A = 0.8\,\pi_A + 0.3\,\pi_B$ $\pi_B = 0.2\,\pi_A + 0.7\,\pi_B$

Both equations reduce to the same condition:

$0.2\,\pi_A = 0.3\,\pi_B$

Combined with $\pi_A + \pi_B = 1$:

$\pi_A = \frac{0.3}{0.2 + 0.3} = \frac{3}{5} = 0.6, \quad \pi_B = \frac{0.2}{0.2 + 0.3} = \frac{2}{5} = 0.4$

Part 3 -- Independence from initial shares:

Yes, the steady state is independent of the initial market shares. This is because the chain is ergodic: it is irreducible (any state can reach any other with positive probability) and aperiodic (the diagonal entries are positive, so the chain can stay in a state). For ergodic Markov chains, the stationary distribution is unique, and the chain converges to it from any starting distribution.

Answer: The long-run market shares are $\pi_A = 0.6$ (60%) and $\pi_B = 0.4$ (40%), regardless of the initial market shares.

Intuition

The steady state of a Markov chain is where inflows and outflows balance for every state. For two companies, this balance condition is simple: the rate at which $A$ loses customers ($0.2 \pi_A$) must equal the rate at which it gains them ($0.3 \pi_B$). The company with the lower churn rate ends up with a larger market share -- which makes intuitive business sense. The independence from initial conditions is the remarkable property of ergodic Markov chains: no matter how lopsided the starting position, the long-run equilibrium is fully determined by the transition probabilities. This is why Markov chain models are so popular in finance for modeling credit ratings, customer behavior, and regime-switching -- the long-run distribution tells you where things settle regardless of today's snapshot.

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