Setup: I was trying to simplify the following minimization problem
$$\min\limits_{x\in\mathcal{X}}\left\{\mathbb{E}\left[\max\limits_{k\in[m]}\left\{s_k \sum\limits_{i=1}^n \xi_i x_i\right\}\right]\right\}$$ where $\xi_i\sim$ Exp$(i)$, $s_k\geq 0$ are known constants, and $\mathcal{X}$ denotes the simplex in $\mathbb{R}^n$. I was able to reduce the problem to the following $$\min\limits_{x\in\mathcal{X}}\left\{\left(\sum\limits_{i=1}^n ix_i\right)^{-1}\sum\limits_{\mathcal{K}\in\mathcal{P}[m]} (-1)^{|\mathcal{K}|-1}\left(\sum\limits_{k\in\mathcal{K}} s_k\right)^{-1}\right\}$$ $$=\min\limits_{x\in\mathcal{X}}\left\{\left(\sum\limits_{i=1}^n ix_i\right)^{-1}S\right\}$$
where $\mathcal{P}[m]$ denotes the powerset of the set of the first $m$ natural numbers excluding the empty set and $S$ denotes the sum over the terms in the powerset.
Question: If $s_k=k$, then $S>0$ (by numerically checking) implying $x^*=e_n$ (and if $S<0$, we take $x^*=e_1$) trivially, but is there a general, simple combinatorial argument to see if this sum, $S$, is positive/negative based on our choice of $\{s_k\}$ or no?
Source: I read this paper by Nemirovski and Juditsky on a similar stochastic utility problem, and thought that instead of i.i.d normals, independent exponentials with varying rates and no shifts would make for a slightly less trivial problem to optimize.