I am analyzing some game that I am researching (in Game Theory), $n$ and $p$ are parameters of the game, and the strategy of the player can be identified by a variable $\ell \in \{ 0,...,n \}$ that the player can choose.
I am currently looking to find the optimal $\ell$, i.e. the one that will maximize the utility of the player.
Since that seemed too difficult, I decided to make things more simple by examining the change in utility from moving from strategy $\ell$ to strategy $\ell-1$, and I have found that this change is given by:
$$\frac{1}{\ell(\ell-1)} \left( \sum_{k=0}^{\ell-1} {n \choose k}p^{k}{(1-p)}^{n-k} k \right) ^ 2 - \\ \frac{1}{(n-\ell)(n+1-\ell)} \left( \sum_{k=\ell}^{n} {n \choose k}p^{k}{(1-p)}^{n-k} (n-k) \right) ^ 2 $$
My hypothesis is: This expression is positive if $\ell \geq np + 1$, and is negative if $\ell \leq np$.
In other words, I believe that $np$ should be the optimal value of $\ell$, and if $np$ is not an integer, then either $\lfloor np \rfloor$ or $\lceil np \rceil$. It makes sense theoretically, and also when I calculated the above expression for different values of $n$ and $p$, I indeed got such results.
I am looking for a way to mathematically prove it. Any hints, ideas and directions are welcome, e.g. mathematical/combinatorial identities that can be applied here. Thanks.