A Markov chain models movement among a finite set of states. Its defining assumption is the Markov property: the next state depends on the present state, not the sequence of earlier states. With transition probabilities, it can forecast outcomes such as customer switching, market share, or movement between loan-delinquency categories. section 11.0 section 11.1
Start by defining states that are mutually exclusive and collectively exhaustive: each observation belongs to one state, and no possible condition is omitted. For a customer-switching model, the states can be the available carriers; a state-probability vector gives each carrier’s current share. section 11.2 section 11.2.2
Place period-to-period probabilities in a square transition matrix P. Entry Pᵢⱼ is the probability of moving from current state i to next state j; each row must account for all possible next states. Using a row vector πₜ for the current distribution, calculate πₜ₊₁ = πₜP. Repeat the multiplication for later periods, or use πₜ₊ₙ = πₜPⁿ. section 11.2.2 section 11.5
Forecasting carrier shares
Suppose four carriers have current shares [0.40, 0.35, 0.20, 0.05]. With the example transition matrix, their next-period shares are [0.394, 0.329, 0.217, 0.060]. The first share, 0.394, combines customers retained by the first carrier with customers switching to it: 0.40×0.85 + 0.35×0.08 + 0.20×0.10 + 0.05×0.12. One more multiplication gives [0.39012, 0.31454, 0.22803, 0.06731]. section 11.2.1 section 11.2.2
>>> import numpy as np >>> pi_0 = np.array([0.40, 0.35, 0.20, 0.05]) >>> P = np.array([ >>> [0.85, 0.06, 0.07, 0.02], >>> [0.08, 0.80, 0.10, 0.02], >>> [0.10, 0.10, 0.75, 0.05], >>> [0.12, 0.10, 0.08, 0.70] >>> ])
The NumPy listing defines the initial four-carrier state vector and its transition matrix; use them to reproduce the forecast. section 11.2.2
>>> pi_1 = np.dot(pi_0, P) >>> print(‘Market shares for period 1:’) >>> print(pi_1) Market shares for period 1: [0.394 0.329 0.217 0.06 ]
This listing multiplies the vector by the matrix and displays the one-period market-share forecast. section 11.2.2
For long-run interpretation, an equilibrium distribution stays unchanged after another transition: π* = π*P. An absorbing state is different: once entered, it cannot be left, making it useful for terminal outcomes such as default or churn. Before extending a forecast, check that its states still cover the system and that transition probabilities and the system’s makeup remain stable. A merger, new entrant, or changing switching pattern undermines the original projection. section 11.1 section 11.2.1 section 11.5
