The failure of unweighted inclusive Gregory STV
The single transferable vote (STV) is a family of voting systems designed to achieve proportional representation. It is widely used in Australia in multiwinner elections, in both governments and private contexts.
Most STV systems in use in Australia fall into one of two categories: exclusive Gregory methods (used, for example, by the ACT Legislative Assembly), and inclusive Gregory methods. Within the category of inclusive Gregory methods, systems can be divided into weighted and unweighted inclusive Gregory methods.
The weighted inclusive Gregory method (WIGM) is regarded by the Proportional Representation Society of Australia as one of the options in a gold standard STV system, and is used for WA Legislative Council and NSW local government elections. The unweighted inclusive Gregory method is the method currently used to elect members of the Australian Senate.
The unweighted inclusive Gregory method is deeply flawed.^{1} To demonstrate this, we will consider a concrete example.
Worked example
Consider an election with 6 candidates – A1, A2, B1, B2, C1 and C2 – competing for 4 vacancies. Suppose the ballots are cast as follows:
Voters  Preferences 

1010  A1 > C1 > A2 > C2 
1900  B1 > C1 > B2 > C2 
920  A2 > A1 > C1 > C2 
770  B2 > B1 > C1 > C2 
400  C2 > C1 > B2 > B1 
Gregory methods of STV can be constructed from a model where each voter casts, say, 1000 votes, and transfers are made according to statistical expectation of how those votes would be distributed in a random transfer system.^{2}
In the 1st stage of counting, all ballot papers are allocated to their firstpreference candidates:^{3}
Candidate  Votes  Voters  

A1  1 010 000  1010  Elected 
A2  920 000  920  
B1  1 900 000  1900  Elected 
B2  770 000  770  
C1  0  0  
C2  400 000  400  
Total  5 000 000  5000  
Quota  1 000 001 
Candidates A1 and B1 meet the quota and are elected.
In the 2nd stage, B1's surplus of 899 999 votes will be distributed. If the 899 999 votes to distribute were selected randomly from B1's 1 900 000, each vote would have a $\frac{899999}{1900000}$ ≈ 0.473 chance of being selected. In other words, each B1 voter would expect on average 473 of their votes (out of 1000) to be transferred.
Using the Gregory method, we accordingly transfer 473 of each B1 voter's votes to their next preference, C1, and record these voters as now counting for C1:
Candidate  Transfers  Votes  Voters  

A1  1 010 000  1010  Elected  
A2  920 000  920  
B1  −899 999  1 000 001  0  Elected 
B2  770 000  770  
C1  +898 700  898 700  1900  
C2  400 000  400  
Lost to Rounding  +1 299  1 299  
Total  5 000 000  5000  
Quota  1 000 001 
In the 3rd stage, A1's surplus of 9999 votes will now be distributed. Using analogous reasoning, $\frac{9999}{1010000}$ ≈ 0.009, so we transfer 9 of each A1 voter's votes to their next preference, C1, and record those voters as now counting for C1:
Candidate  Transfers  Votes  Voters  

A1  −9 999  1 000 001  0  Elected 
A2  920 000  920  
B1  1 000 001  0  Elected  
B2  770 000  770  
C1  +9 090  907 790  2910  
C2  400 000  400  
Lost to Rounding  +909  2 208  
Total  5 000 000  5000  
Quota  1 000 001 
No surpluses remain to distribute, so in the 4th stage, C2 will be excluded. Their votes are transferred to their next preference, C1:
Candidate  Transfers  Votes  Voters  

A1  1 000 001  0  Elected  
A2  920 000  920  
B1  1 000 001  0  Elected  
B2  770 000  770  
C1  +400 000  1 307 790  3310  Elected 
C2  −400 000  0  0  Excluded 
Lost to Rounding  2 208  
Total  5 000 000  5000  
Quota  1 000 001 
C1 now reaches the quota and is elected.
In the 5th stage, C1's surplus of 307 789 votes will be distributed.
Let us briefly pause to consider the composition of C1's 1 307 790 votes at present:
 898 700 votes, accounting for 1900 voters (473 votes each), were received from B1's surplus
 9 090 votes, accounting for 1010 voters (9 votes each), were received from A1's surplus
 400 000 votes, accounting for 400 voters (1000 votes each), were received from C2's exclusion
In this situation, how C1's surplus is now distributed differs between the weighted and unweighted inclusive Gregory methods.
The election under WIGM
In the weighted inclusive Gregory method, the underlying principle is as previously described – we proceed on the basis that each of C1's votes should have an equal chance of being transferred in the surplus distribution. If the 307 789 surplus votes to distribute were selected randomly from C1's 1 307 790 total votes, each vote would have a $\frac{307789}{1307790}$ ≈ 0.235 chance of being selected.
Note that this does not mean that each voter can expect 235 of their votes to be transferred. For example, among the 1010 voters with a first preference for A1, they have only given C1 9 votes each – hardly enough to transfer away 235 each! These voters would each expect approximately 9 × 0.235 ≈ 2 of their votes to be transferred. Over the 1010 voters, this sums to 2020 votes to transfer to their next preference, A2.
Proceeding analogously, the 1900 voters with a first preference for B1 would each expect approximately 473 × 0.235 ≈ 111 of their votes to be transferred, for a total of 210 900 votes to transfer to their next preference, B2.
Finally, the 400 voters with a first preference for C2 would each expect 235 of their votes to be transferred, for a total of 94 000 votes to transfer to their next preference, B2.
Performing these transfers, we obtain:
Candidate  Transfers  Votes  Voters  

A1  1 000 001  0  Elected  
A2  +2 020  922 020  1930  
B1  1 000 001  0  Elected  
B2  +304 900  1 074 900  3070  Elected 
C1  −307 789  1 000 001  0  Elected 
C2  0  0  Excluded  
Lost to Rounding  +869  3 077  
Total  5 000 000  5000  
Quota  1 000 001 
Candidate B2 now meets the quota, and is elected to the final vacancy, completing the count.
Sanity check
A key feature of most STV systems is achieving Droop proportionality for solid coalitions (DroopPSC). DroopPSC says that if a solid coalition has at least n Droop quotas worth of votes, then it should be able to elect at least n candidates. It is this feature that allows STV to achieve proportional representation.
Note that the 1900 B1 voters, 770 B2 voters and 400 C2 voters all prefer each of B1, B2, C1 and C2 to either of A1 or A2. They are therefore a solid coalition of 1900 + 770 + 400 = 3070 voters. Since the Droop quota in this election is 1001, this coalition has just over 3 Droop quotas worth of votes, and so should be able to elect 3 candidates out of B1, B2, C1 and C2. This is exactly what is achieved in the weighted inclusive Gregory method.
The issue with unweighted inclusive Gregory
Now let's see how C1's surplus of 307 789 votes is distributed under the unweighted inclusive Gregory method. Recall the composition of C1's 1 307 790 total votes at the time of election:
 898 700 votes, accounting for 1900 voters (473 votes each), were received from B1's surplus
 9 090 votes, accounting for 1010 voters (9 votes each), were received from A1's surplus
 400 000 votes, accounting for 400 voters (1000 votes each), were received from C2's exclusion
In the weighted inclusive Gregory method, the underlying principle is that each of C1's votes should have an equal chance of being transferred in the surplus distribution. The unweighted inclusive Gregory method does away with this principle, and proceeds on the inexplicable basis that each voter should have the same number of votes transferred in the surplus distribution.^{4}
The unweighted inclusive Gregory method says that because 1900 + 1010 + 400 = 3310 voters have contributed votes to C1, each voter should have $\frac{307789}{3310}$ ≈ 92 of their votes transferred to their next preferences.
That this calculation is meaningless should be selfevident. Most obviously, the 1010 voters with a first preference for A1 have each contributed only 9 votes to C1. How can they now each transfer 92 votes from C1 to further preferences? If we conducted this process using physical ballot papers, giving each voter 1000 each, the absurdity would become obvious at this point where we must transfer more ballot papers than physically exist.
The unweighted inclusive Gregory method invites us to ignore this glaring problem, and simply proceed to credit the candidates with a number of votes suggested by these figures. For the 1010 voters with a first preference for A1, a total 1010 × 92 = 92 920 votes are (somehow) transferred to their next preference, A2.
For the 1900 voters with a first preference for B1, 1900 × 92 = 174 800 votes are transferred to their next preference, B2.
For the 400 voters with a first preference for C2, 400 × 92 = 36 800 votes are transferred to their next preference, B2.
Performing these transfers, we obtain:
Candidate  Transfers  Votes  Voters  

A1  1 000 001  0  Elected  
A2  +92 920  1 012 920  1930  Elected 
B1  1 000 001  0  Elected  
B2  +211 600  981 600  3070  
C1  −307 789  1 000 001  0  Elected 
C2  0  0  Excluded  
Lost to Rounding  +3 269  5 477  
Total  5 000 000  5000  
Quota  1 000 001 
As a consequence of the preceding mathematical perversity, note that if one follows the flow of A1 voters' votes, they transferred a total of 9090 votes to C1, and yet were able to transfer out a whopping 92 920 votes from C1 during C1's surplus distribution. Where did these extra 83 830 votes come from? The unweighted inclusive Gregory method has no answer to this question.
Regardless, it is now candidate A2 who meets the quota, and who is elected to the final vacancy, completing the count.
Violation of Droop proportionality
Recall from earlier that the 3070 combined B1, B2 and C2 voters are a solid coalition with over 3 Droop quotas of votes, and so, in a proportional voting system meeting DroopPSC, should be able to elect at least 3 candidates out of B1, B2, C1 and C2. However, using the unweighted inclusive Gregory method, this has not been the case, and the coalition has elected only 2 of their candidates.
The potential for the unweighted inclusive Gregory method to violate DroopPSC is a major, practical failure, and calls into question the extent to which it can really be called a proportional voting system at all.
Summary
The unweighted inclusive Gregory method espouses an untenable basis for distributing surplus votes, which – unlike all other Gregory methods in use – has no meaningful concrete analogue. As a consequence, it can violate DroopPSC and fail to produce proportional representation.
The weighted inclusive Gregory method does not suffer from this fault. Neither does the exclusive Gregory method, random transfer methods, or the Meek method. It is a problem unique to the unweighted inclusive Gregory method.
The unweighted inclusive Gregory method is deeply flawed, and should not be used.
Footnotes

See also: Miragliotta N. Determining the result: transferring surplus votes in the Western Australian Legislative Council. Perth: Western Australian Electoral Commission; 2002. https://www.elections.wa.gov.au/sites/default/files/content/documents/Determining_the_result.pdf ↩

See my conceptual primer on STV. ↩

In this article, we use terminology consistent with the abovelinked conceptual primer on STV, which better illustrates the conceptual basis for the Gregory method. Realworld rules for Gregory methods tend to use different terminology, such that their ‘ballot paper’ corresponds with our ‘voter’, and their ‘vote’ corresponds with 1000 of our ‘votes’. This does not affect the conclusions of the article. ↩

Realworld rules for the unweighted inclusive Gregory method phrase this in a different way, along the lines of ‘calculate the transfer value by dividing the surplus by the total number of ballot papers’. This obfuscates the problem behind a layer of abstraction, the ‘transfer value’, but the end result is exactly the same. ↩