When the long-run answer is just even

Question

A token hops among three lily pads A, B, C each second. The move depends only on the current pad: - From A: to A with 0.2, to B with 0.5, to C with 0.3. - From B: to A with 0.5, to B with 0.3, to C with 0.2. - From C: to A with 0.3, to B with 0.2, to C with 0.5. In the long run, what fraction of time does the token spend on pad B?

Accepted Answer

Each row above adds to 1, as it must (from any pad the token goes somewhere). Now look down the columns, which collect all the ways to arrive at a pad: - Arriving at A: 0.2 + 0.5 + 0.3 = 1. - Arriving at B: 0.5 + 0.3 + 0.2 = 1. - Arriving at C: 0.3 + 0.2 + 0.5 = 1. Every column also adds to 1. A chain whose rows and columns both sum to 1 is called doubly stochastic, and such a chain has a very special long-run distribution: it is perfectly even across all states. Why: suppose the long-run share of every pad were the same value u. The share flowing into any pad next second is u 	imes (	ext{t…

When the long-run answer is just even

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