Context
Let say $S\in\mathbb{N}$ is the number of votes cast and $N\in\mathbb{N}$ is the number of seats in the parliament. We may assume $N\leq S$. The votes are distributed among $K$ political parties, so there exists $v_{1},\dotsc,v_{K}\in\mathbb{N}$ such that $\sum_{j=1}^{K}v_j=S$, where $v_j$ is the number of votes the $j$-th political party received.
We say that the $j$-th party is perfectly represented if its vote share equals its seat share, i.e. $\frac{e_j}{N}=\frac{v_j}{S}$, where $e_j\in\{0\}\cup\mathbb{N}$ is the number of seats the $j$-th party received. So, if $\frac{e_j}{N}>\frac{v_j}{S}$ the party is overrepresented, and, analogously, if $\frac{e_j}{N}<\frac{v_j}{S}$ the party is underrepresented. We may assume $\sum_{j=1}^{K}e_j=N$.
We want to define a measure of the over (under) representation of a party, one posibility is looking at the number
\begin{equation} a_j:=\frac{e_j}{N}-\frac{v_j}{S}\text{, } \end{equation}
so, if $a_j<0$, the $j$-th party is underrepresented and if $a_j<a_i$, the $i$-th party is more overreprented than the $j$-th one. Another possibility will be \begin{equation} q_j:=\frac{\frac{e_j}{N}}{\frac{v_j}{S}}\text{, } \end{equation} so, if $q_j<1$, the $j$-th party is underrepresented and if $q_j<q_i$, the $i$-th party is more overrepresented than the $j$-th one.
My question
Are these two measures compatible? In other words, if $a_j< a_i$, then $q_j<q_i$ and vice versa? Or, at least \begin{equation} \arg \min a_j=\arg\min q_j\text{?} \end{equation} Is it even true that $a_j=a_i$ if and only if $q_j=q_i$?.
I have been trying to prove this seemingly elementary facts for quite some time using elementary methods, but none of them seems to be fruitful. Any idea?
