I came across this issue while trying to combine multiple probability distributions into a single distribution which approximates them all simultaneously. This boils down to maximizing this expression $$ S = \sum_i \frac{N_i p_i^i}{\sum_j N_j p_i^j} $$ in terms of the unknowns $N_1, \dots, N_t$, $p_1, \dots, p_t$. Here $p_i \in [0,1]$ and $N_i \geq 0$ for all $i$.

It is easy to see that $S \leq t$ (because the denominator term $\sum_j N_j p_i^j \leq N_i p_i^i$. Are there any tighter bounds available?

Thanks for the help