Long-run share of biking commutes

Question

A commuter travels by car, bus, or bike each day, and today's choice depends only on yesterday's: - From car: to car 0.8, to bus 0.1, to bike 0.1. - From bus: to car 0.3, to bus 0.6, to bike 0.1. - From bike: to car 0.2, to bus 0.2, to bike 0.6. In the long run, what fraction of days does the commuter bike?

Accepted Answer

We want the steady shares (\piC, \piB, \piK) for car, bus, bike that stop changing day to day. Each share equals its total inflow. Bike: arrivals are from car (0.1), from bus (0.1), and stay (0.6): \piK = 0.1\,\piC + 0.1\,\piB + 0.6\,\piK \implies 0.4\,\piK = 0.1\,(\piC + \piB). The car and bus shares together are \piC + \piB = 1 - \piK, since the three add to 1. Substitute: 0.4\,\piK = 0.1\,(1 - \piK) = 0.1 - 0.1\,\piK \implies 0.5\,\piK = 0.1 \implies \piK = 0.2. So the commuter bikes on one fifth of days, or 20\%. Takeaway. If a state is entered from the others with the same probability, i…

Long-run share of biking commutes

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