If we number the states as 1: Win; 2: Loss; 3: Draw, we get the transition matrix from the given table:
This is the one-step transition matrix. To obtain the long-term behavior using technology, we need to compute the matrix P. The rows of P are given by the solution of the following system of equations:
where P is the above transition matrix. This gives the following system of linear equations:
|x + y + z||=||1|
|-0.6x + 0.3y + 0.3z||=||0|
|0.4x - 0.6y + 0.3z||=||0|
|0.2x + 0.3y - 0.6z||=||0|
Solving the above system gives x = 1/3, y = 10/27, and z = 8/27. P is now the 33 matrix whose rows are [ 1/3 10/27 8/27 ]. These three solutions are also the fraction of times England can expect to win, lose, and draw in the long term.
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