A conceptual primer on the single transferable vote – 5: Weighted inclusive Gregory method
In part 4, we presented the Gregory method, which removes random effects from STV. In that part, we continued with a decision made in part 3 to restrict the ballot papers which can contribute to a surplus distribution, yielding the exclusive Gregory method.
To review, in part 3, we made this decision – in a surplus distribution to examine only the last parcel of ballot papers received – for two reasons. Firstly, this was closer to the effect of Hare's original method. Secondly, this was more practical for handcounting, as the number of ballot papers to keep track of is minimised.
However, in the context of the Gregory method, there is no objective reason to prefer the exclusive method to the alternative.
Weighted inclusive Gregory method
Let's return to the example from part 4, where the ballots cast were:
Votes  Preferences 

21  D1 > D2 > F > C1 > C2 
19  D2 > D1 > F > C1 > C2 
40  C1 > C2 > F > D1 > D2 
6  C2 > C1 > F > D1 > D2 
7  F > D1 > D2 > C1 > C2 
6  F > D1 > C1 > C2 > D2 
1  F > D1 (and no further preferences) 
100  Total 
Return to the thought experiment, where each voter casts 1000 ballots, such that the standings after the exclusion of Fish Owner are:
Candidate  Votes  

Dog Owner 1  35 000  Elected 
Dog Owner 2  19 000  
Cat Owner 1  26 000  Elected 
Cat Owner 2  20 000  
Fish Owner  0  Excluded 
Total  100 000  
Quota^{1}  26 000 
As before, Dog Owner 1 has a surplus of 9000 votes. In part 4, we applied the exclusive approach, distributing the surplus from the 13000 transferable ballots received in the last parcel.
Let us now take the opposite approach – maximising the number of ballot papers to examine in the surplus distribution, therefore allowing the greatest number of voters to contribute to the result. While this makes handcounting more difficult, it is by no means prohibitive.
Thus, let us distribute the surplus by examining all of Dog Owner 1's 35000 votes (whether transferable or not).^{2} These comprise:
 21000 firstpreference votes for Dog Owner 1, with a next available preference for Dog Owner 2
 7000 votes received from Fish Owner's exclusion, with a next available preference for Dog Owner 2
 6000 votes received from Fish Owner's exclusion, with a next available preference for Cat Owner 2
 1000 votes received from Fish Owner's exclusion, which are now nontransferable
Applying analogous reasoning to part 4, if the 9000 votes to transfer are independently drawn from the 35000 ballots, we would expect that, on average, each individual voter contributing to Dog Owner 1's votes at this stage would have approximately $\frac{9000}{35000}$ × 1000 ≈ 257 of their 1000 ballots transferred to their next available preference.^{3}
Therefore, we can transfer 257 of each of those voter's ballots to their next available preferences. This would result in:
 257 × (21 + 7) = 7196 ballots being transferred to Dog Owner 2
 257 × 6 = 1542 ballots being transferred to Cat Owner 2
 257 × 1 = 257 ballots being set aside as exhausted as they are now nontransferable
 5 ballots of the surplus being disregarded due to rounding
The standings then become:
Candidate  Votes  

Dog Owner 1  26 000  Elected 
Dog Owner 2  26 196  Elected 
Cat Owner 1  26 000  Elected 
Cat Owner 2  21 542  
Fish Owner  0  Excluded 
Exhausted  257  
Lost to Rounding  5  
Total  100 000  
Quota  26 000 
Or, dividing the preceding figures by 1000 to obtain a table where the votes are expressed in terms of individual voters:
Candidate  Votes  

Dog Owner 1  26.000  Elected 
Dog Owner 2  26.196  Elected 
Cat Owner 1  26.000  Elected 
Cat Owner 2  21.542  
Fish Owner  0.000  Excluded 
Exhausted  0.257  
Lost to Rounding  0.005  
Total  100.000  
Quota  26.000 
In any case, Dog Owner 2 now meets the quota, becoming the final elected candidate.
This system, as opposed to the exclusive Gregory method described in part 4, is the weighted inclusive Gregory method (WIGM). WIGM is regarded by the Proportional Representation Society of Australia as one of the options in a gold standard STV system, and is now used for the Western Australia Legislative Council, and New South Wales local government elections.
Comparing exclusive Gregory with WIGM
Note that the result under WIGM, where Dog Owner 2 wins the final seat, is different to that obtained under the exclusive Gregory method, where Cat Owner 2 won the final seat. This is because the election is very close, and so to award either candidate the final seat is in some ways justifiable.
Is one system, exclusive or inclusive, inherently superior to the other? There is probably no single objectively correct answer. The philosophy of each is justifiable. The exclusive Gregory method takes an approach which is more convenient for handcounting, and which produces results closer to Hare's original method. The weighted inclusive Gregory method takes an approach which is less convenient for handcounting, and which produces results closer to the computerbased system we will introduce in the next part – which brings us to…
The next part
Consider the 6 voters who had voted ‘F > D1 > C1 > C2 > D2’, at the time Dog Owner 1 was elected. In this part, and in each previous part, we skipped over Cat Owner 1, as Cat Owner 1 had already reached the quota and been elected. We therefore said that they should not receive any further votes, as these would be wasted, and so considered Cat Owner 2 to be the next available preference.
In part 6, we explore a viable alternative approach.
Footnotes

As in part 4, for the purposes of this thought experiment, we ignore that if there were 100,000 votes the quota would actually be 25,001. ↩

Since some votes in this example are nontransferable, the maximally inclusive approach – examining the greatest number of votes – would suggest that we should also examine the nontransferable votes, and allow some of them to exhaust. This approach is used by some implementations of the weighted inclusive Gregory method, but other implementations stop short of this, and examine only transferable votes. ↩

The Australian Senate uses a different system, the unweighted inclusive Gregory method. This method is not mathematically sound and produces different results when ballot papers in the surplus have themselves come from a previous surplus distribution. For a detailed discussion, see this article. ↩