Evolution of STV PR
Specific STV methods vary in how the quota is calculated, how surplus votes are transfered, and how ties are treated (ties can arise in Find Winners and Elimination). We’ll discuss STV variations in a historical context, looking at how increasingly sophisticated methods have been developed. Our historical approach to the evolution of STV methods follows Nicolaus Tideman’s Collective Decisions and Voting Chapter 15 [Tideman 2006].
In 1819, Thomas Wright Hill published a system for electing a committee of The Birmingham (England) Society for Literacy and Scientific Improvement. Each voter cast a ballot for one candidate, with the ballot also identifying the voter. The votes were counted, and any candidate who received five or more votes was elected. If a candidate received more than five votes, excess ballots cast for that candidate were selected at random, and the voter was invited to cast his vote for another (as yet unelected) candidate. When no surplus votes remained, voters whose ballots were cast for unelected candidates were invited to vote again. The process repeated until all the empty seats were filled, or until there were fewer than five left-over ballots.
Wright’s son Rowland Hill used a somewhat similar method in an Australian municipal election. A list of candidates was presented, and each voter in turn wrote his name in a candidate’s column until a candidate had the number of votes required for election. Subsequent voters would, of course, vote for another (as yet unelected) candidate.
One problem with Rowland Hill’s system is that later voters have an advantage over earlier voters, in that they can use their vote to elect a second or third choice if, by the time they vote, their earlier choices have already been elected. In Thomas Hill’s system, voters who cast their votes in later stages have a similar advantage. We call this advantage “free riding”.
In 1856, Danish mathematician and politician Carl Andrae (who was unaware of the Hills’ systems) introduced the idea of a ranked ballot. The ballots were counted in random order for the first choice on each ballot that had not yet been elected. Unlike Hill’s system (and later STV systems), Andrae’s system did not allow for the reallocation of ballots originally counted for losing candidates. The system saw limited use in Denmark until 1915.
Ranked ballots solve the voting-order problem, in that voters make their selection without knowing how others have voted.
Thomas Hare, a London barrister, unaware of both Hill’s and Andrae’s work, described a system similar to Andrae’s in his 1859Treatise on the Election of Representatives: Parliamentary and Municipal. In The election of representatives parliamentary and municipal : a treatise (1865), he added the idea of successively eliminating the candidate with the fewest votes and transferring those ballots to the next (unelected and uneliminated) candidate on the ballot. This was the first system to include the fundamental elements of all future STV systems: ranked ballots with transfers of surplus votes and transfers from eliminated candidates.
Hare’s system, like Andrae’s, transfers surplus ballots at random. This works reasonably well for large numbers of ballots, but there is of course no absolute guarantee that the ballots chosen randomly are completely representative. This is a problem with all STV systems that transfer only whole votes.
In the previous systems, the threshold of election (quota) was the number of voters divided by the number of seats to be filled, with any remainder ignored. In STV systems that deal exclusively with whole (as opposed to fractional) votes, this is the largest number such that there are enough quotas to fill all the seats, and is known as the “Hare quota”, after Thomas Hare. Henry Droop argued that a lower threshold, approximately the number of voters divided by one more than the number of seats to be filled (with rounding that depends on the counting method) had several advantages. We describe these advantages elsewhere, but they were and are sufficiently compelling that virtually all STV systems today use some version of the Droop threshold.
Hare’s system, with Droop’s threshold, is essentially the system used today by Cambridge, Massachusetts.
The methods we’ve discussed so far choose surplus ballots at random to distribute to lower-choice candidates. With a large enough electorate, this isn’t a serious problem, but especially in smaller elections it raises a question of fairness, since those ballots chosen for surplus distribution have a measure of elective power that the ballots not so chosen do not. J B Gregory proposed an improvement in which, instead of choosing a number of ballots at random, we instead distribute all the ballots to their next choices, but count those ballots at a reduced weight. (Gregory actually gave each ballot an initial weight of 100 and then used whole-vote counting, but the effect is the same as transferring hundredths of votes.) In this way, every voter contributing to a surplus has a voice in how the surplus is distributed. There are variations on the Gregory method that differ in which ballots are deemed to be surplus, and how exhausted ballots are treated; we describe them elsewhere.
The variation knows as the “Weighted Inclusive Gregory Method” (WIGM) is the most widespread of STV systems today, and is used with minor variations by Ireland and Scotland, and in the US for internal Green Party elections. It was also the system recommended, but not approved, for use in British Columbia.
For integer-based STV methods, the Droop threshold is calculated as the integer portion of 1+ballots/(seats+1).
When Robert Newland and Frank Britton produced a detailed set of STV counting rules for the UK’s Electoral Reform Society, based on a variation of the Gregory method, they observed that the exact Droop threshold of ballots/(seats+1) produced better results. This threshold is sometimes called the Newland-Britton (or NB) threshold (or quota) to distinguish it from the integral Droop threshold, but is more often treated simply as a variation on the Droop threshold.
In the STV methods described so far, the details of surplus vote transfers depend on the order in which candidates are elected or eliminated. This is a problem because it can motivate voters to vote strategically—that is, in a manner other than their actual preferences. In 1969, Brian Meek published a pair of papers in which he described a system that observes two principles:
Principle 1. If a candidate is eliminated, all ballots are treated as if that candidate had never stood.
Principle 2. If a candidate has achieved the quota, he retains a fixed proportion of every vote received, and transfers the remainder to the next non-eliminated candidate, the retained total equalling the quota.
The resulting method, known as “Meek’s method”, is impractical to count without a computer, but it avoids nearly all the shortcomings of previous methods, including a form of free riding to which the Gregory family of methods is vulnerable.
In 1983, C H E Warren proposed a variation on Meek’s method that differs in a detail of surplus transfer. There is no consensus on which method is superior, but fortunately they tend not to differ much in their results.
Meek’s method is used for some local elections in New Zealand.
STV methods have a property known as “later no harm”. That is, voters can rank candidates lower (later) than their favorite without any concern that so doing will make the election of their favorite less likely. A consequence of such a guarantee, though, is that when there are no immediate winners, STV can fail to find “everybody’s second choice”. A variety of approaches have been suggested to try to avoid the problem of “sure losers” preventing the election of other candidates with broad but not first-choice support while minimizing the violation of conventional STV’s later-no-harm property.
Three of these proposed approaches are composite methods. That is, they produce their results by using a conventional STV method more than once, typically many times.
David Hill first proposed Sequential STV in 1994, refining it with Simon Gazeley in 2002 and again in 2005. In an election to fill nseats, the general idea is to find a set of n candidates such that those n are elected from each set of n+1 consisting of those n and one other. Thus every rejected candidate has been fairly tested without interference from others who are not elected.
Such a set is not guaranteed to exist, and when it does not, Sequential STV tries to find the set of n candidates that comes closest to doing so. On the other hand, it is possible for there to be more than one such set of n; in such a case, Sequential STV will find one of them.
In the case of a single seat to be filled, Sequential STV finds the Condorcet winner if there is one.
In 1995, Nicolaus Tideman proposed a composite method called CPO-STV (CPO for “comparison of pairs of outcomes”). Tideman writes:
The comparison of two outcomes in CPO-STV proceeds somewhat like plain vanilla STV, beginning with the calculations of a quota. However, it is then necessary to list every set of potential winners and then compare these sets two at a time. In each such comparison of two sets of potential winners, each vote is allocated to the first candidate in that voter’s ranking that is in at least one of the two sets being compared. However, votes are transferred only from those candidates (if any) that have more than quota and are in both sets. The count is finished as soon as all such surpluses have been transferred. The result is given by the difference between the sum of the votes of the candidates in one set and the sum of the votes of candidates in the other set. When all pairs of outcomes have been compared in this fashion, the winning set is the set, if there is one, that beats all other sets in these head-to-head comparisons. If there is none, the winner is the set whose worst loss is least bad.
As Tideman points out, CPO-STV may not be computationally practical in the general case.
Schulze STV addresses, at least in part, the issues of vote management and free riding. See Wikipedia Schulze STV article for details.