Talk:Task:Define voting procedure for Community Council elections

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Variables

 * $$l$$: number of voters
 * $$m$$: number of candidates
 * $$n$$: number of seats
 * $$i$$: index of voters ($$1 \le i \le l$$)
 * $$j$$: index of candidates ($$1 \le j \le m$$)
 * $$k$$: index of seats (and rounds) ($$1 \le k \le n$$)

Parameters
A range for votes is selected by two limits, $$a_{max}$$ and $$a_{min}$$. Reasonable choices include: The effects of the voting system are generally independent of the limits selected, but a balanced range may be preferred by some voters to permit unknown candidates to be rated 0 with known, disliked candidates rated negative. Obviously this effect may be accomplished in an unbalanced range by rating unknown candidates $$\tfrac{a_{min} + a_{max}}{2}$$. Some calculations are simplified by using the unbalanced, normalized range, which makes it preferable from a numerical perspective.
 * $$a_{min}=0$$, $$a_{max}=1$$ (unbalanced, normalized)
 * $$a_{min}=-1$$, $$a_{max}=1$$ (balanced)
 * $$a_{min}=0$$, $$a_{max}=100$$ (unbalanced percentage; potentially useful if ratings were to be quantized)

A ballot from voter $$i$$ consists of $$m$$ ratings $$a_{ij}$$ for the $$m$$ candidates, such that $$a_{min} \le a_{ij} \le a_{max}$$.

Procedure
After all ballots are collected, $$n$$ rounds are held to choose the winners $$w_{k}$$, with one winner chosen per round. In the first round, the weighted ratings for each voter are initialized as $$b_{ij1}=a_{ij}$$. The weighted scores in each round are summed:

$$B_{jk}=\sum_{i=1}^{l}b_{ijk}$$

The highest-scoring candidate not yet elected wins; this is not simple to express formally:

$$w_k = \max ( B_{jk} \ni j \notin w_{1 \cdots k-1})$$

The weighted scores for each succeeding round are calculated by de-emphasizing ballots according to the portion in which they've already won:

$$b_{ij(k+1)}=\frac{a_{ij}}{1+\sum_{p=1}^{k}\tfrac{a_{iw_p}-a_{min}}{a_{max}-a_{min}}}$$

In the case of an unbalanced range ($$a_{min} = 0$$), $$b_{ij(k+1)}=\frac{a_{ij}}{1+\sum_{p=1}^{k}\tfrac{a_{iw_p}}{a_{max}}}$$, and in the normalized case ($$a_{max} = 1$$), $$b_{ij(k+1)}=\frac{a_{ij}}{1+\sum_{p=1}^{k}a_{iw_p}}$$.