Representing Customer Loyalty and Transfer With Matrices

Three companies A, B and C shared a fixed number of companies. Customers switch companies, as shown in the table.
Start Month 1 Gains Losses Start Month 2
From From From To To To
Company Customers A B C A B C Customers
A 200 0 35 25 0 20 20 220
B 500 20 0 20 35 0 15 490
c 300 20 15 0 25 20 0 290
In moth 1 A, loses 20 customers to B and 20 to C, B loses 35 to A and 15 to C, and C25 to A and 20 to B. The table below shows the probability of gaining (or retaining) customers
A B C
A 0.8 0.1 0.1
B 0.07 0.9 0.03
C 0.083 0.067 0.85
The equilibrium market share will the solution to  
\[(a,b,c)= \left| \begin{array}{ccc} 0.8 & 0.1 & 0.1 \\ 0.07 & 0.9 & 0.03 \\ 0.083 & 0.067 & 0.85 \end{array} \right| (a,b,c)\]
,
The solution is  
\[a=273, \; b=454, \; c=273\]
 

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