Summability criterion: Difference between revisions

{{wikipedia|Summability criterion}}

 

This is important for elections with many voting jurisdictions to be able to practically transmit their vote totals for tabulation. Summability is important to be able to report real-time combined vote totals in an understandable way. Some non-summable methods require that the individual ballot images are are transmitted to a centralized counting location to find the combined result.

Back in 2009, [[English Wikipedia]] stated that the criterion was stated as follows:<ref>An article titled "[[wikipedia:Summability criterion|Summability criterion]]" was deleted from [[English Wikipedia]] in 2009.<nowiki><ref></nowiki>[[English Wikipedia]] AfD for "Summability Criterion": https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Summability_criterion

 

Strictly speaking, a method is kth-order summable if an election involving <math>V</math> voters and <math>c</math> candidates can be stored in a data structure (a summary) that requires <math>O(\log(V) \cdot c^k)</math> bits in total, where there exists a summation operator that takes any two such summaries and produces a third for the combined election, and the election method itself can use these summaries instead of ballot sets to produce the same results. This definition closes the obvious loophole of using a few very large numbers to store more data than would otherwise be permitted.

 

 

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*the [[Contingent vote]]

*Borda-elimination ([[Baldwin's method|Baldwin]]<ref>{{Cite journal|last=Hogben|first=G.|date=1913|title=Preferential Voting in Single-member Constituencies, with Special Reference to the Counting of Votes|url=http://rsnz.natlib.govt.nz/volume/rsnz_46/rsnz_46_00_005780.html|journal=Transactions and Proceedings of the Royal Society of New Zealand|series=|volume=46|issue=|pages=304–308|via=}}</ref> and [[Nanson's method|Nanson]]<ref name="sumNanson">{{Cite journal|last=Nanson|first=E. J.|date=1882|title=Methods of election|url=https://archive.org/details/transactionsproc1719roya/page/197|journal=Transactions and Proceedings of the Royal Society of Victoria|volume=19|pages=197–240|via=}}</ref>)

*[[Majority Judgment]]

*[[MCA|MCA-IR]], and some forms of [[MCA|MCA-AR]]

*[[STAR voting]]<ref>{{Cite news|title=Compare STAR and IRV - Equal Vote Coalition|url=https://www.equal.vote/star-vs-irv#simplicity|work=Equal Vote Coalition|access-date=2018-11-12}}</ref>

||

*[[Benham's method]]

==Examples==

 

 

 

=====Positional methods=====

For example, with [[Score voting]], a voter who votes A:10 B:6 C:3 D:1 is treated as giving a 10 to A, a 6 to B, etc. Comparisons across different score scales can be made by dividing the score by the max score (i.e. instead of a 6, treat it as a 6/10=0.6, etc.) so that a voter who scores a candidate a 3 out of 5 and a voter who scores a candidate a 6 out of 10 can have their scores treated and counted the same without any issues.

 

{{Main|Pairwise counting}}

Some voting methods, such as [[STAR voting]] are precinct-summable using voter's pairwise preference order alongside the total score received for each candidate.

 

 

In [[Schulze method|Schulze]] and many other summable [[Condorcet method|Condorcet methods]], each vote is equivalent to a two-dimensional array referred to as a pairwise matrix. If candidate A is ranked above candidate B, then the element in the A row and B column gets a 1, while the element in the B row and A column gets a 0. The pairwise matrices for all the votes are summed, and the winner is determined from the resulting pairwise matrix sum. The precincts' matrices may be added together to get the matrix for the whole electorate, just like a precinct's voters' matrices may be added together to get the matrix for that precinct.

|+

!

!B

!C

|-

| ---

|1

|1

|-

|B

| ---

|1

|-

|C

|0

| ---

|1

|-

|0

|0

Suppose, for example, that the number of candidates is ten.

 

*For [[Schulze method|Schulze]], a 10×10 matrix is needed (although only 10x9=90 data values are actually kept).

*In an [[IRV]] system, however, each precinct would need to send a list of ten numbers, the number of first-place votes for each candidate. The central system would then return to each precinct a candidate to eliminate. Each precinct would then return the first-place votes for each of the nine remaining candidates, and receive another candidate to eliminate. This would be repeated at most 9 times. This is more than the others.

In first-order summable election systems, adding new ballots to the count (say, ballots that were found after the initial count, or late absentee ballots, or ballots that were initially ruled invalid) is as simple as "summing" the original result with the newly-found ballots. Under non-summable systems, though, finding new ballots means all ballots might possibly need to be recounted. This is not a big problem for computer recounts, but manual recounts can be extremely time-consuming and expensive.

 

 

 

There's no one extension of the summability criterion for multiple winners, because the summability criterion only considers two variables (the number of voters and the number of candidates), whereas there's one more variable for a multi-winner method: the number of seats. One possible extension of the summability criterion is as follows:<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2011-July/092996.html|title=Weighted voting systems for proportional representation|website=Election-methods mailing list archives|date=2011-07-24|last=Munsterhjelm|first=K.}}</ref>

 

 

 

 

Unless otherwise specified, these results pertain to strong summability.

Most [[block voting]] methods that are based on summable single-winner methods are also of the same degree of summability in the multi-winner case, and strongly so.

 

 

[[Ebert's method]] is summable in <math>O(c^2)</math> for any number of seats.<ref>{{cite web | title=proof that sum of squared loads is precinct-summable | date=2020-01-14 | url=https://forum.electionscience.org/t/possible-improvement-to-pamsac-gets-rid-of-cfat/566/19 | access-date=2020-04-29}}</ref>

Summability focuses on the amount of data that has to be captured, but not necessarily the amount of work required to capture it. For example, when doing [[pairwise counting]], an election featuring a ballot that ranks a candidate last requires as many marks to count as if the same ballot had been cast without the last-ranked candidate. Yet in practice, the vote-counters must still take some time to check that that candidate is indeed last-ranked, meaning some work is done even while no data was produced.

 

Summability focuses to a large extent on the number of data value types, not just the amount of data overall that has to be captured. This can make a difference in certain cases; for example, the regular [[pairwise counting]] approach only requires <math>n^2-n</math> data value types to be captured for all ballots, whereas the [[Negative vote-counting approach for pairwise counting]] requires <math>n^2</math> value types. This is because the latter not only records preferences in each pairwise matchup, but also the number of ballots ranking each candidate. Yet, depending on implementation, the negative counting approach actually has the same upper bound on number of data values to capture as the regular approach, and in practice could require fewer.

 

Most sequential [[Cardinal PR]] require less two-way communication and/or centralized counting work than most other [[PR]] methods.

 

 

*[[Voting system]]

*[[Condorcet Criterion]]

*[[Generalized Condorcet criterion]]

*[[Participation criterion]]

 

<references />

 

==See also==

 

*[[Monotonicity criterion]]

*[[Condorcet Criterion]]