Editing Droop quota (section)

== Common errors ==
There is a great deal of confusion among legislators and political observers about the correct form of the Droop quota.<ref name="dancisin-2013">{{Cite journal |last=Dančišin |first=Vladimír |date=2013 |title=Misinterpretation of the Hagenbach-Bischoff quota |journal=Annales Scientia Politica |volume=2 |issue=1 |pages=76}}</ref> At least six different versions appear in various legal codes or definitions of the quota, all varying by [[Fencepost error|one vote]].<ref name="dancisin-2013" /> The [[Electoral Reform Society|ERS]] handbook on STV has advised against such variants since at least 1976, as they can cause problems with proportionality in small elections.<ref name="mw-2007" /><ref name="newland-1980" />  In addition, it means that vote totals cannot be [[scale invariance|summarized into percentages]], because the winning candidate may depend on the choice of [[unit of measurement|unit]] or total number of ballots (not just their distribution across candidates).<ref name="mw-2007" /><ref name="newland-1980" /> Common variants of the Droop quota include:

<math>\begin{array}{rlrl}
    \text{Historical:} &&  \left\lceil \frac{\text{votes}}{\text{seats}+1} \right\rceil     
        &&\Bigl\lfloor \frac{\text{votes}}{\text{seats}+1} + 1 \Bigr\rfloor
        &&\Bigl\lfloor \frac{\text{votes}}{\text{seats}+1}\Bigr\rfloor + 1 \\
    \text{Accidental:} && \phantom{\Bigl\lfloor} \frac{\text{votes} + 1}{\text{seats} + 1} \phantom{\Bigr\rfloor}  
        && \phantom{\Bigl\lfloor} \frac{\text{votes}}{\text{seats}+1} + 1    \phantom{\Bigr\rfloor}    \\
    \text{Inadmissible:} && \left\lfloor \frac{\text{votes}}{\text{seats}+1} \right\rfloor
        && \left\lfloor \frac{\text{votes}}{\text{seats}+1} + \frac{1}{2} \right\rfloor
\end{array}</math>

A quota being "inadmissable" refers to the possibility that more could achieve quota than the number of open seats. However preventing such an occurrence is not necessary. The archaic rounded-off form of the Droop quota (votes/seats+1, plus 1, rounded down) was traditionally seen as needed in the context of modern fractional transfer systems, and it was believed that any smaller portion of the votes, such as exact Droop, would not work because it would be possible for one more candidate than there are winners to reach the quota.<ref name="dancisin-2013" /> s Newland and Britton noted in 1974, this is not a problem: if the last two winners both receive a Droop quota of votes, rules can be applied to break the tie, and ties can occur regardless of which quota is used.<ref name="mw-2007" /><ref name="newland-1980" /> Due to this misunderstanding, Ireland, Malta and Australia have used Droop's original quota - votes/seats+1, plus 1 - for the last hundred years.

The two variants in the first line come from Droop's discussion in the context of [[Thomas Hare (political reformer)|Hare]]'s STV proposal. Hare assumed that to calculate election results, physical ballots would be reshuffled across piles, and did not consider the possibility of fractional votes. In such a situation, rounding the number of votes up (or, alternatively, adding one and rounding down{{efn|The two are only different when the quotient produced by the number of votes divided by one more than the number of seats is exactly a whole number.}}) introduces as little error as possible, while maintaining the [[Electoral quota#Admissible quotas|admissibility of the quota]], by ensuring that no more can achieve quota than just the number of seats available.<ref name="dancisin-2013"/><ref name="droop-1881" />

=== Confusion with the Hare quota ===
The Droop quota is often confused with the more intuitive [[Hare quota]]. While the Droop quota gives the number of voters needed to mathematically guarantee a candidate's election, the Hare quota gives the number of voters represented by each winner in an exactly-proportional system (i.e. one where each voter is represented equally).

The confusion between the two quotas originates from a [[Off-by-one error|fencepost error]], caused by forgetting unelected candidates can also have votes at the end of the counting process. In the case of a single-winner election, misapplying the Hare quota would lead to the incorrect conclusion that a candidate must receive 100% of the vote to be certain of victory; in reality, any votes exceeding a [[Majority|bare majority]] are [[excess vote]]s.<ref name="droop-1881"></ref>