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I've researched voting systems lately. I haven't encountered this idea, so I'd be grateful if somebody shows books/articles that include it or something similar.

Take a voting result: $$\begin{array}{c|c|c} & \text{Votes}& \text{Ideal seats} \\\hline Keskusta &640428 &45.9886 \\ Kokoomus &616841 &44.2948 \\ SDP &594194 &42.6685 \\ Vasemmistoliitto &244296 &17.5427 \\ Vihreät &234429 &16.8341 \\ KD&134790 &9.6792 \\ RKP &126520 &9.0853 \\ PerusS &112256 &8.0610 \\ SKP &18277 &1.3125 \\ Senioripuolue &16715 &1.2003 \\ Itsenäisyyspuolue &5541 &0.3979 \\ Sinivalkoiset &3913 &0.2810 \\ Liberaalit &3171 &0.2277 \\ Köyhien Asialla &2521 &0.1810 \\ KTP &2007 &0.1441 \\ STP &1764 &0.1267 \\ IKL &821 &0.0590 \\ Yhteisvastuu &164 &0.0118 \\ \text{(small lists)} \\ \hline Total &2771236 &198.0961 \\ \end{array}$$

The seats are obtained by multiplying the votes by 199 (total seats) and dividing by 2771236 (total votes). They would add up to 199 if all the minuscule lists were included too.

$Quota\ rule$ means that the actual seats should be rounded up or down, but not any further. So, KD and RKP should receive either 9 or 10 seats. This is also called $satisfying\ the\ fair\ share$.

This brings an idea that it is a binary choice: either the upper or lower value is chosen. $Hamilton\ method$ uses this directly: the upper values are given to parties with largest fractional parts until the total is reached. The rest get the lower values.

This can be extended to other methods. We define $frunction\ box$ that gets an input of a fractional number. It has the whole part $w$ and fraction part $f$ (can be expressed with a few decimals for the comparisons). The whole stays intact, but we can take any function of $w$ and $f$ to define a pseudofraction. Hence the name. These pseudofractions are compared and the maximal ones get the upper values $w+1$ until the total is reached. The rest get $w$.

So, usual voting methods would get these functions:

Hamilton (aka Vinton, Hare-Niemeyer): $f$

Jefferson (D'Hondt): $\frac{w+f}{w+1}$

Webster (Sainte-Laguë): $\frac{w+f}{2w+1}$

Huntington–Hill: $\frac{w+f}{\sqrt{w(w+1)}}$

Dean: $\frac{(w+f)(2w+1)}{2w(w+1)}$

Adams:$\frac{w+f}{w}$

Huntington–Hill, Dean and Adams can't handle $w=0$, so the data should perhaps be prepared: if a party or state is short of the one seat, it is given one and excluded from the next round where the rest of the seats are recalculated.

Then some fictional examples:

Big Bullies: $w$ (favours the large parties)

Small Change: $-w$ (favours the small parties)

If the pure method satisfies quota in a certain election, we lose nothing. It even gets easier: for example in Jefferson and Webster we don't have to calculate all those comparison numbers, only one per party. But in the rare cases it violates quota, this "wrapper" stops it. Other anomalies may be introduced because of that, but not necessarily. So we have already diminished the probability of error.

This can also be extended to two dimensions (or more): for example divide by districts and parties. I'll post more if there is interest, but in practice Balinski's biproportional method is more refined for that.

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Not sure what the question is here, but how would you handle the Huntington–Hill method, which is the method actually used to apportion representatives in the US Congress and which involves the geometric mean?

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  • $\begingroup$ That is a little problematic, because it assumes that each already has at least one - it can't handle zero. That's why I skipped it. It's also known for very blatant quota violations. $\endgroup$ Commented Mar 30, 2018 at 17:21
  • $\begingroup$ Added it, and because I had to explain the special requisite, also added Dean and Adams that work similarly. While I was at it, I boldened my need (not actual question). $\endgroup$ Commented Mar 30, 2018 at 21:12

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