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# Voting matters - Issue 2, September 1994

## Sequential STV

#### I D Hill

David Hill is a retired statistician. He is a member of Council and Chairman of the Technical Committee of the ERS.

The Meek system for counting an STV election overcomes most of the troubles encountered in using older systems designed for counting by hand, but the problem of premature exclusion remains. Premature exclusion of a candidate occurs when someone is the lowest because hidden behind another who, in the end, is also not going to succeed. If A, who would otherwise have been elected, fails because B stood and was elected instead, it is bad luck for A but there is nothing disturbing about it in principle. If, however, A fails because B stood, but then B does not get in either, that is disturbing.

Exclusion of the lowest candidate, when an exclusion is necessary, is the trouble. After all, if the so-called first past the post is not necessarily the right person to elect, then neither is the last past the post necessarily the right one to exclude. Is there some other way of handling things that would do better? What is needed is a mechanism to discover initially which candidates have some hope of election and which have virtually none, and to get rid of the 'no-hopers' at the start of the count. Others cannot then suffer from their presence.

Let the election be to fill k seats from n candidates, and let m = n - k. Sequential STV then consists of a number of main-phases and sub-phases, each being an STV election for k seats but with varying selections of candidates. The STV elections are preferably conducted using Meek-style counting but other rules could be used.

Main-phase 1. All n candidates, but instead of dividing into elected and excluded, divide them into probables and others respectively. Set all n candidates to unmarked.

Sub-phase 1.1. The k probables plus any other one candidate. Set the winners to marked.

Sub-phase 1.2. The same k probables plus any other one candidate not yet tested. Set any unmarked winners to marked.

etc.

Sub-phase 1.m. The same k probables plus the last candidate not yet tested. Set any unmarked winners to marked.

If at any sub-phase there is a tie that has to be settled using random selection, then all k + 1 of the candidates involved are set to marked.

Main-phase 2. All marked candidates, dividing into probables and others. If the resulting set of probables is the same as a previous set, those candidates are elected and the process finishes. Otherwise reset all n candidates to unmarked and continue.

Sub-phases 2.1 - 2.m. As 1.1 - 1.m but using the new probables.

Main-phase 3. As main-phase 2.

etc. etc.

It may be noted that anyone getting a quota of first preferences on the original count is, in fact, certain to be elected in the end, but to be classified for the time being as probable does no harm.

The process must terminate because there is only a finite number of sets of k that can be formed from n. Usually it will terminate with two successive main-phases showing the same set of k probables. In that case the result is firmly established. If, however, the two showing the same set are not successive it will mean that the system is cycling in Condorcet-paradox style. In that case it may be that a better rule could be devised than taking the first set to occur twice but it has to be recognised that a totally satisfactory answer is impossible.

Each candidate is given a fair chance by being tested against each new set of probables and since each sub-phase consists of only k + 1 candidates for k seats, exclusion is never necessary during the sub-phases so the 'exclude the lowest' rule is not operative there.

### Example

With 5 candidates for 2 seats, suppose the votes
```104 AEBCD
103 BECDA
102 CEDBA
101 DEBCA
3 EABCD
3 EBCDA
3 ECDBA
3 EDBCA
```
It is evident that E is a strong candidate, in that if any one of A, B, C or D were to withdraw, E would be the first elected. Yet under simple STV the first action is to exclude E, and B and C are elected. Under sequential STV we find
```Phase   Candidates  Winners  Probables  Marked
1      ABCDE       BC       BC
1.1    BCA         BC                  BC
1.2    BCD         BC
1.3    BCE         BE                  E
2      BCE         BE       BE
2.1    BEA         BE                  BE
2.2    BEC         BE
2.3    BED         BE
3      BE          BE       BE
```
B and E are consequently elected. It will be noted that some elections may be repeats of ones already done (main-phase 2 and sub-phase 2.2 in the above example are both repeats of sub-phase 1.3). The result may of course merely be copied down without actually repeating any calculations.

### Should it be used?

If any scheme is to be adopted to get rid of (or at least to ease) the problem of premature exclusion, I believe that this is about as good as can be devised. Yet, after much consideration, I do not recommend it for general use, because it breaks the rule, which simple STV always obeys, that a voter's later preferences ought not to interfere with that voter's earlier preferences.

The following example to demonstrate this trouble is derived from those that Douglas Woodall devised to prove his 'impossibility' theorem. Let there be 3 candidates for 1 seat and votes

``` 25 A
17 BC
16 C
```
```Phase  Candidates  Winner  Probable   Marked
1      ABC         A        A
1.1    AB          A                  A
1.2    AC          C                  C
2      AC          C        C
2.1    CA          C                  C
2.2    CB          B                  B
3      BC          B        B
3.1    BA          A                  A
3.2    BC          B                  B
4      AB          A        A
```
So A is elected. But if the A voters had put in C as a second preference, we get
``` 25 AC
17 BC
16 C

Phase Candidates  Winner   Probable  Marked
1     ABC         A        A
1.1   AB          A                 A
1.2   AC          C                 C
2     AC          C        C
2.1   CA          C                 C
2.2   CB          C
3     C           C        C
```
and C is elected. So the A voters have failed to elect A because they gave C as a second preference.

Even if this is a rare event, it still means that we cannot assure voters that their later preferences cannot upset their earlier preferences. I believe that this is too high a price to pay. There is not much point in reducing the frequency of one type of fault if, in doing so, you introduce another fault as bad.

### Only one seat

The system is really intended, as is STV in general, for situations where there are several seats to be filled, but it can also be used in place of Alternative Vote for a single seat. Trying it out on many examples suggests that, for realistic voting patterns, it is almost certain to elect the Condorcet winner if there is one, but artificial examples can be devised to demonstrate that there is no guarantee that it will do so.

For example, let there be 4 candidates for 1 seat and votes

```98 ADCB
98 CDBA
99 BDAC
3 ACBD

Phase Candidates  Winner  Probable  Marked
1     ABCD        A        A
1.1   AB          B                 B
1.2   AC          A                 A
2     ABD         B        B
2.1   BA          B                 B
2.2   BC          C                 C
2.3   BD          D                 D
3     BCD         C        C
3.1   CA          A                 A
3.2   CB          C                 C
3.3   CD          D                 D
4    ACD          A        A
```
So A is elected, even though D would be the Condorcet winner (for the results of AD, BD and CD are all D). It should be emphasised, though, that this is not likely in practice but only with carefully devised artificial examples.

### Acknowledgement

I acknowledge that, since I first produced this scheme, Dr David Chapman has produced an almost identical scheme entirely independently.
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