Market share is the proportion of a market attributable to a product or provider. In a multistate model, each provider’s share represents the probability that a randomly selected consumer subscribes to that provider during a given period. section 11.2.1
Represent the shares as a state-probability vector. For four carriers, the initial vector is [0.40, 0.35, 0.20, 0.05], and its entries sum to 1, corresponding to 100% of the market. A transition matrix describes the probability of remaining with a provider or switching to another between periods. section 11.2.1 section 11.2.2
Forecasting carrier shares
Using the carrier vector and transition matrix with NumPy, compute the next period as pi_1 = dot(pi_0, P). The resulting shares are [0.394, 0.329, 0.217, 0.060]. Applying the matrix again gives period-two shares of [0.39012, 0.31454, 0.22803, 0.06731]. Multiplying the matrix by itself first produces the same period-two result. 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] >>> ])
Use the NumPy listing to construct the initial market-share vector and transition matrix before calculating future distributions. section 11.2.2
