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# Voting matters - Issue 12, November

## Ordered List Selection revisited

#### J Otten

Joe Otten is the author of a Windows program for the current ERS STV rules.

### 1. Problems with methods advocated in Voting Matters 9

I was struck by a comment by Hill[1] that Rosenstiel's alternative method to the use of constraints violated the principle that later preferences should not be allowed to count against earlier ones (I will refer to this as 'the principle' in this paper). This was because the method involved running repeated counts on the same vote profile, and thus a later preference may have its effect in one count when the fate of earlier preferences was still to be decided in a later count.

I realised that the same criticism could be levelled at the method I advocated for selecting an ordered list[2] (in this case of candidates for a party to offer at a European Parliament election conducted using a list system). It could also be levelled at the similar system proposed by Rosenstiel[3]. In each case multiple counts were used, and the result of one count could affect the result of another - by the use of a constraint in my case, or by overriding it in the other.

#### Example 1:

```    AC      2
BC     10
C       8
DC      6
```
This gives the following results:
```Vacancies    Results
1      C
2      AC
3      ABC
```
Both methods give the Result: CABD.

Suppose Rosenstiel's method was used, and those voting BC changed their vote to BDC, example 1 gives

```Vacancies    Results
1     C
2     AC
3     ABD
```
Now, C gets last place, and B and D are tied for third. The tie is broken by looking at first preferences, so D is third. Then there is a similar tie for second between A and B, so B is second. Result ABDC. Voters have improved the position of B by changing later preferences.

My method would still give the order CABD with example 2, but would violate the principle given a similar example.

Wichmann[4] suggests using the Meek keep factor for determining the ordering, and this case is not so immediately obvious, since only one count is held. The Meek algorithm does not allow later preferences to influence whether earlier preferences may be elected. However later preferences may affect the size of the keep factor for elected candidates, and so if this is used to order the candidates, the principle is violated. Electing 3, this gives ABCD in example 1 and ABDC in example 2.

### 2. Using the Orange Book method

The Orange Book simply suggests that if an order is required, the order in which candidates are elected during the count should be used. This seems, on face value, to be inadequate for selecting a long list of candidates, since the contest for the significant top places would be rather similar to a First Past the Post election, with a few candidates above the quota being given positions dependent only on the numbers of first preferences received. Newland himself, author of the early editions of the Orange Book, indicates in his Comparative Electoral Systems why he thinks this method is wrong, advocating a top-down method.

The method appears to rest on the assumption that it is the determination of the whole membership of the list that is the primary purpose of the election. That is not the case. The purpose is that however many seats the party wins, the people thus elected are those who were selected by an STV ballot with the appropriate number of vacancies. Thus in the Green Party, where no more than 1 seat was won in any region, the top of the list should be the AV winner (as indeed they were, since the Green Party used a top-down method.) The Liberal Democrats won 2 seats in some regions, so there the appropriate selection would be that of the top two candidates by an STV election with 2 vacancies.

The problem is that the number of seats that a party will win is unknown at the time of selection. However, it may be reasonable to guess at that number. The order of election (orange book method) would give the order of the candidates elected in the selection ballot, and the reverse order of exclusion could determine the order of later candidates. If a party wins 1 more or fewer seats, the distortion might not be that great.

This does not seem entirely satisfactory, but I cannot see how better can be done without abandoning the principle.

### 3. Abandoning the Principle

A great many articles in Voting matters have discussed the principle. Some have suggested that it might be relaxed, for example to allow Borda scores to be used to break ties[5]. Personally I think the only strong argument against Condorcet style election rules is that they violate the principle. Therefore if the principle must be lost, we may as well look at later preferences more freely and use an election rule more in keeping with Condorcet principles, and do better than any of the methods advocated in Voting matters 9.

It seems to me that a great many voters would welcome a substantial benefit to a second or third preference at the expense of a small risk to a first preference. STV does seem to rest on the assumption that the strength of a voter's support for their first preference is such that other considerations are overridden. While I don't think this assumption is true for very many voters (except perhaps for die-hard party loyalists), it is right for STV to make it. It is right because it makes the task of voting much easier. The voter does not need to assess how his or her use of later preferences might affect the fate of an earlier one. The principle encourages voters to indicate their true preferences.

Nonetheless, if the price of the principle is reducing a contest to near equivalence to First Past the Post, I believe that price is too high. I suggest the next question is how may we reap the benefits of the information the principle denied us. In the one vacancy election, systems which violate the principle may benefit by being able to guarantee the election of the Condorcet winner if there is one. I seek now to generalise this benefit to the election of an ordered list.

### 4. Generalising Condorcet principles to multiple vacancies

Hill[6] describes the complexity that can arise when trying to generalise the concept of a Condorcet top-tier to a multiple vacancy election. However, if we are considering a list selection then we are not simply looking for a subset of all the candidates, but adding them one at a time to a list. This simplifies the problem somewhat. Also for the purposes of simplicity I shall refer to Condorcet to mean any single- winner rule satisfying the Condorcet Criterion. The manner in which cycles should be resolved is not a significant concern here; nor is whether Meek or ERS97 rules are used, although computer counting will be necessary.

The method which follows builds an ordered list from the top down. It, like Condorcet, does not use exclusions at all, but considers at every stage, all possible pairs of candidates for the next position to see if one beats all the others. Like STV, votes are retained by elected candidates so they have less or no influence on later positions.

The top position is elected by Condorcet (call this candidate P).

For every pair, X and Y of other candidates, we must determine which is preferred to the other for the second place. We calculate the result of an STV election between P, X and Y for 2 places (other candidates being withdrawn). This calculation determines whether X is preferred to Y or vice versa. We read off the support for X and Y after any surplus for P has been redistributed and this completes one element in the Condorcet result square. (Normally it is only of interest which of X or Y is elected in this election. However the magnitude of the difference in support will be relevant if a cycle-breaking method needs to be employed.) The calculation is repeated for all other pairs of candidates, not including P (or at least for as many pairs as are necessary to determine the winner). Call the candidate thus elected to position 2 Q.

We need to repeat this exercise for position 3, 4, 5, etc, and we now have more than one elected candidate. Each time we perform an STV count including all the elected candidates, PQR..., and a pair of unelected candidates X and Y, and no others, giving one element of the Condorcet result square as before. We then repeat this for every pair of unelected candidates, and add our new Condorcet-style winner to the list.

Applied to Example 1, the result tables look like this:

```(+ values imply row candidate beats column candidate)
Condorcet (6 AV counts between 2 candidates)
A      B      C      D
A            +2    -12      +6
B     -2            -6      -6
C    +12     +6             +4
D     -6     +6     -4
```
Position 2: (3 STV counts with 3 candidates, C and two others)

AvB: C has a surplus of 2, which is non-transferable - A 12, B 10

AvD: C has a surplus of 6, which is non-transferable - A 12, D 6

BvD: C has no surplus - B 10, D 16

```       A     B      D
A           +2     +6
B     -2           -6
D     -6    +6
```
A is elected to position 2

Position 3: (1 STV count with all candidates)

BvD: A has a surplus of 3, which goes 0.5 to C and 2.5 to D - B 10, D 8.5

```        B     D
B           +1.5
D     -1.5
```
B is elected to position 3

Result: CABD

Changing the 10 votes from BC to BDC as before (example 2) creates a cycle:

```Position 1:
A      B      C       D
A           +2     -12      -4
B     -2            -6      -6
C    +12    +6             -16
D     +4    +6     +16
```
D is the Condorcet Winner and is elected to position 1.

Position 2:

AvB: - A 12, B 10

AvC: - A 12, C 8 (D is guarded, so A is not elected)

BvC: D has a surplus of 4 which goes to C (strictly 3.96 with ERS97) - B 10, C 14

```      A      B      C
A           +2      +4
B     -2            -4
C     -4    +4
```
A is elected to position 2

Position 3:

BvC: - B 10, C 8.5 (C and D are guarded, so B is not elected)

```      B       C
B           +1.5
C   -1.5
```
A is elected to position 3.

Result: DABC

D and C have swapped places, as is reasonable given the change of votes from BC to BDC.

Instead of using a usual cycle-breaking rule, an alternative would be to combine the election for the position in question with the following one, elect two, and then go back to the first, where there are now only 2 candidates to choose from, so there can be no cycle. (This would be a normal STV election for the top two. Alternatively we could consider every possible triple, but this may lead to further cycles.)

This procedure is a synthesis of STV and Condorcet. At each position a Condorcet-winner is added to the list, once votes cast for already-elected candidates have been discounted (reduced in value) in the manner of STV. It is not vulnerable to the exclusion of potential winners with few first preferences.

It could also form the basis for a synthesis of STV and Condorcet for unordered elections, although this would be a solution looking for a problem as regular STV is available here. Seeking to elect n candidates we could apply the STV rule to every subset of n+1 of the candidates and see which n were able to beat off any individual challenger. As Hill[6] says, the subset of n with this property may not exist, or may not be unique. However the generalised Condorcet method above could be adapted in such cases to arbitrate between competing sets of candidates, or to provide a result where there appears to be none.

### 5. Summary of examples

```
Ex 1    Ex 2
Repeated count rules:
Rosenstiel
/Bottom Up Overriding (R):
CABD   ABDC
Otten
/Top Down Constrained (O):
CABD   CABD
Top Down Overriding (TDO):
CABD   CABD
Bottom Up Constrained (BUC):
CABD   ABDC
One count rules:
Wichmann Meek (2 places) (WM2):
CABD   CABD
Wichmann Meek (3 places) (WM3):
ABCD   ABDC
Orange Book (1 place) (OB1):
CABD   CABD
Orange Book (2 places) (OB2):
ACBD   ACBD
Orange Book (3 places) (OB3):
ABCD   ABDC
Generalised Condorcet rule:
Generalised Condorcet (GC):
CABD   DABC
```
I have not described the last two repeated count rules - they are hybrids of the Rosenstiel and Otten rules, which might be called Bottom Up Overriding and Top Down Constrained respectively. It is worth noting that BUC, like GC, does not use exclusions, (candidates already allocated to lower positions are withdrawn before the start of the next count) but with different results.

What are the best results? CABD seems to be a clear favourite for example 1. With example 2, the elementary conflict is that if the electorate were to be represented by one person, the best person (from an AV point of view) would be C, and if it were to be three, the best people would be A, B and D. Rules which take greater care over the top end of the list (O, TDO, WM2, OB1) therefore place C highly and those which concentrate on the bottom (WM3, R, BUC, OB3) place C low. Notably WM3 and OB3 place C low even in example 1.

We have, it seems, not entirely escaped from the consideration in point 2, of needing to know what position on the list is the crucial one. If it is believed that a particular position on the list, say 4th, is the key one, an STV count for 4 winners could be followed by BUC to fill the top 3 and O to fill the positions from 5 down (or R and TDO respectively).

As to be expected GC succeeds in finding the Condorcet winner D in Example 2, who is not found by any of the other methods. Obviously this is an example of my choosing, and I have no doubt that other examples may show GC generating inferior results.

### 6. Conclusions

I have described three broad approaches to ordered list selection, all of which are unsatisfactory. The methods used by the Green Party and Liberal Democrats violate the principle, but fail to take advantage of the information this releases. The Generalised Condorcet method uses this information but also violates the principle. The orange book method, used as described here, may lead to a severely distorted result if the guess is wrong.

While the methods described in section 1, appear for the moment to be the most practical solution to the question of ordering, the fact that counts for differing numbers of candidates frequently produce inconsistent results undermines their credibility.

A significant source of these inconsistencies is changes in early exclusions or the order of exclusions and in which parcel of papers elects a candidate, resulting from the higher or lower quota. (Meek should be less vulnerable to two of these effects.) While my generalised Condorcet method conceals any comparable inconsistencies that might be present, the fact that it eliminates exclusions altogether, means that it should be robust against exclusion-related effects.

The disadvantages are greater complexity and probably a more frequent violation of the principle that later preferences should not count against earlier ones. It will also require considerably more computer time than the alternatives, which may be an issue with a very large election, particularly if Meek is used. It would not be desirable to adopt a rule that then had to be abandoned for very large elections.

I do not at this point advocate that a generalised Condorcet method is adopted. However, I think the idea has its merits, and I do believe the question of ordering demands further consideration. While a single rule may not be appropriate for all circumstances, it should be possible to narrow the field somewhat from that in section 5.

### References

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