Generating all possible rankings when merging sorted lists with rank instead of raw score information per element

Question

Let $\{e_1, e_2, ..., e_n\}$ be the set of elements. Let $\{s_1(e_1), s_1(e_2), ..., s_1(e_n)\}$, $\{s_2(e_1), s_2(e_2), ..., s_2(e_n)\}$ be the scores of elements after applying two ranking functions $s_1$, $s_2$. Let $\{w\cdot s_1(e_1) + (1-w)\cdot s_2(e_1), w\cdot s_1(e_2) + (1-w)\cdot s_2(e_2), ..., w\cdot s_1(e_n) + (1-w)\cdot s_2(e_n)\}$, $w\in[0, 1]$, be the final scores after the combination of individual scores. I am interested in the set of all potential rankings of the elements that can be constructed this way (considering the final scores above for all $w\in[0,1]$). Can I generate the same set of final rankings when ignoring the raw scores after the application of functions $s_1$, $s_2$ and relying on linear combination of the corresponding ranks per list instead? I.e. using $\{v\cdot r_1(s_1(e_1))) + (1-v)\cdot r_2(s_2(e_1)), v\cdot r_1(s_1(e_2)) + (1-v)\cdot r_2(s_2(e_2)), ..., v\cdot r_1(s_1(e_n)) + (1-v)\cdot r_2(s_2(e_n))\}$, $v\in[0, 1]$, where $r_1$ and $r_2$ are the ranks per list according to scorings $s_1$ and $s_2$.