$\theta_i (i=1,\ldots,N)$ are real numbers and we have $ \sum_{i=1}^N \theta_i = 1 $ For any $i\neq j$, $ \sum_{w\in W} \frac{f_i(w)}{\sum_{k=1}^N \theta_k f_k(w)} = \sum_{w\in W} \frac{f_j(w)}{\sum_{k=1}^N \theta_k f_k(w)} $
Here $W$ is a set, and for any $i$, $f_i()$ is a function that maps an element in $W$ to a scalar.
I guess there should be a closed form solution for $\theta_i (i=1,\ldots,N)$ (in terms of $f_i$ and $W$) but couldn't figure it out. Thank you!
Update
The equations above are what I get by applying the Karush–Kuhn–Tucker conditions to the following optimization problem:
Maximize $ \prod_{w\in W} \sum_{k=1}^N \theta_k f_k(w) $ subject to $ \sum_{i=1}^N \theta_i = 1\\ \forall i, \theta_i\geq 0 $