I have a system of equations of a form:
$$ \forall_{i \not = j} P_{ij} = a_i b_j \frac{c_i}{c_i + c_j} $$
Where i and j iterate over some set of indexes, P are some given probabilities such that $\Sigma P_{ij} = 1$, and I want to find a set of solutions for a, b and c. Is there any useful way to characterize that set? Or even to show that the system has a solution? I am looking for any clue or starting point here.