A conceptual primer on the single transferable vote – 4: Exclusive Gregory method
In part 3, we discussed a refinement to random transfer STV, noting that even in refined form it is still subject to random effects. In this part, we introduce a method which eliminates randomness from STV completely.
For the sake of illustration, let's again slightly adjust the votes in the Suburbia Pet Owners' Association election, such that upon the distribution of Dog Owner 1's surplus votes, received from Fish Owner's exclusion, the choice of ballots to transfer will affect the result:
| 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 |
Illustrating the problem with random transfers
Let's return to the moment in part 3 where, upon Fish Owner's exclusion, Dog Owner 1 is elected:
| Candidate | Votes | |
|---|---|---|
| Dog Owner 1 | 35 | Elected |
| Dog Owner 2 | 19 | |
| Cat Owner 1 | 26 | Elected |
| Cat Owner 2 | 20 | |
| Fish Owner | 0 | Excluded |
| Total | 100 | |
| Quota | 26 |
Dog Owner 1 now has a surplus of 35 − 26 = 9 votes, and as decided in part 3, the 9 ballots to transfer will be drawn from the transferable ballots received in the last parcel from Fish Owner's exclusion. In this example, they are:
- 7 ballots with a next available preference for Dog Owner 2 (F > D1 > D2 > C1 > C2)
- 6 ballots with a next available preference for Cat Owner 2 (F > D1 > C1 > C2 > D2, skipping over C1 who has already been elected and needs no more votes)
Clearly, the selection of which ballots to transfer will change the result. In the extreme cases:
- If 7 ballots are transferred to Dog Owner 2 and 2 to Cat Owner 2, Dog Owner 2 will reach the quota and be the final elected candidate.
- If 3 ballots are transferred to Dog Owner 2 and 6 to Cat Owner 2, Cat Owner 2 will reach the quota and be the final elected candidate.
In effect, the third winner in the election would be decided by random chance. This would generally be regarded as an unsatisfactory outcome.
A thought experiment
Suppose, for a moment, that instead of casting 1 ballot, each voter casted 1000 ballots. We might imagine that the standings that this point would then be:
| 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 | |
| Quota1 | 26 000 |
In this thought experiment, Dog Owner 1 has a surplus of 9000 votes. 9000 ballots will be transferred from the 13000 transferable ballots received in the last parcel: 7000 with a next available preference for Dog Owner 2, and 6000 with a next available preference for Cat Owner 2.
Now, if the 9000 votes to transfer are independently randomly drawn from the 13000 transferable ballots, we would expect that approximately $\frac{7000}{13000}$ × 9000 ≈ 4846 of those selected would have a next available preference for Dog Owner 2, while $\frac{6000}{13000}$ × 9000 ≈ 4154 would have a next available preference for Cat Owner 2. Any substantial deviation from this would be statistically improbable.
Put another way, each individual voter accounting for the 13000 transferable ballots would expect approximately $\frac{9000}{13000}$ × 1000 ≈ 692 of their 1000 ballots to be transferred to their next available preference on average.
Removing random effects
We can now apply this observation to distribute Dog Owner 1's surplus without involving any random selection. For each individual voter accounting for the 13000 transferable ballots, we simply transfer 692 of their 1000 ballots to their next available preferences.
In this way, 692 × 7 = 4844 ballots are transferred to Dog Owner 2, and 692 × 6 = 4152 ballots are transferred to Cat Owner 2. Due to rounding, 4 ballots of the surplus remain unaccounted for, which we will disregard. The standings then become:
| Candidate | Votes | |
|---|---|---|
| Dog Owner 1 | 26 000 | Elected |
| Dog Owner 2 | 23 844 | |
| Cat Owner 1 | 26 000 | Elected |
| Cat Owner 2 | 24 152 | |
| Fish Owner | 0 | Excluded |
| Lost to Rounding | 4 | |
| Total | 100 000 | |
| Quota | 26 000 |
As there are no further surpluses to distribute, and fewer candidates have reached the quota than there are vacancies, Dog Owner 2 must now be excluded, leading eventually to the election of Cat Owner 2 as the sole remaining candidate.
Modern presentation of the Gregory method
If we divide the figures in the preceding table by 1000, we obtain a table where the votes are expressed in terms of numbers of individual voters:
| Candidate | Votes | |
|---|---|---|
| Dog Owner 1 | 26.000 | Elected |
| Dog Owner 2 | 23.844 | |
| Cat Owner 1 | 26.000 | Elected |
| Cat Owner 2 | 24.152 | |
| Fish Owner | 0.000 | Excluded |
| Lost to Rounding | 0.004 | |
| Total | 100.000 | |
| Quota | 26.000 |
This presents an alternative way of thinking about the above process of surplus distribution. In random transfer STV, we transferred a fraction of the surplus votes, each at full value. In this new system, we instead instead transferred every surplus vote, each at fractional value: Each of the 13 transferable ballots was transferred at value 0.642. This method of fractional transfers is now known as the Gregory method, after its inventor J B Gregory.
The notion of transferring fractions of votes is perhaps the concept thus far whose implications are the most difficult to grasp. However, as the above discussion demonstrates, it is no more advanced in principle than giving each voter 1000 ballots, and distributing surpluses according to the expected average outcome.
In particular, it remains the case that each voter has had equal voting power – whether you conceptualise the process as giving each voter 1000 ballots, or giving each voter a single vote which is transferred in fractions.
Revisiting an earlier choice?
In part 3, we made the choice during a surplus distribution to consider only the last parcel of votes received. In this part, we applied the same rule, yielding the exclusive Gregory method. This, however, is not the only choice we could have made. In part 5, we will explore the effect of a different choice in yielding the inclusive Gregory method.