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# Voting matters - Issue 8, May 1997

## Some Council Elections

### Introduction

This paper is an analysis of some Council elections based upon computer simulation in a similar manner to two previous papers [1],[2]. The analysis starts with (five) result sheets, since they are the publicly available record of the elections. The first stage consists of using a computer program to produce a set of ballot papers which reproduces the result sheet (or gets very close to that). The second stage consists of running a number of experiments based upon elections which select a random subset of the ballot papers. The third stage is a further analysis of the results.

This paper is concerned with STV elections in which there are no 'party' affiliations. Hence the voting patterns are different from those which applied in the Irish elections analysed in the first reference. The identity of the actual council elections used for this study is not stated here, since this is irrelevant and could detract from the conclusions which are thought to be relevant for all elections for several seats in which there are no parties involved.

### Constructing ballot papers

Given a result sheet, then it is possible to construct a set of papers which would produce the same results. In producing such a set by hand, the obvious method is to work forward stage by stage. However if no transfers occur from candidate A (say), such a method will give preferences that, if A appears, stop at that point. In other words, preferences that are not required to produce the results as given in the result sheet are not given. Clearly, the voter will not necessarily do this, and more significantly, other algorithms may use subsequent preferences. Hence a more general means is required of producing ballot papers.

The program used in this study works as follows. The program computes transfer rates from A to B if candidate A was eliminated or had a surplus to transfer (and B was available for transfers). If no such transfer occurred, then an estimate is used based upon the first preferences for B.

Ballot papers are now constructed using a random number generator with an exact match for the first preferences. This set is then used as the starting point of an iterative process, working stage by stage, to obtain a very close fit to the actual result sheet. The program cannot necessarily obtain a perfect match when transfers of surpluses are involved. Experiments showed that the starting position which was dependent upon the seeds for the random number generator did not have a large effect on the accuracy of the final fit to the actual election.

### An example

To give a fuller explanation of the method of constructing ballot papers from a result sheet, we give a simple example. Consider an election in which the votes for electing one candidate from 4 was:
```10 AB
5 BCD
6 CDA
1 C
8 DAB
```
The result sheet from these ballot papers using Newland-Britton is:
```     Stage 1  Stage 2  Stage 3

A         10       10        0
B         11       11       21
C          7        0        0
D          8       14       14
Non-T      0        1        1
```
Since we are concerned with a council election without parties, we consider each candidate in the same way. We can judge the overall popularity of each candidate from the first preference votes. We now construct a matrix to represent the probability of X being followed by Y in any preference (X could clearly be the last preference given, so Y is allowed to be the Non-Transferable option). For instance, given candidate D, then the preference specified after D is assumed to be A, B or C in the ratio 10:11:7 (since these are the ratios of the votes on the first preferences).

We can make a better estimate of the transfer probabilities, since we do have a limited amount of information from the result sheet. In this case, for stage 2 in which C is eliminated, we know that the next preferences were either D or non- transferable in the ratio 6:1, respectively. Hence, we can adjust our matrix accordingly. For stage 3, in which A is eliminated, the transfers were entirely to B, but the papers could have had a preference to C which would have been ignored. This clearly reflects the adjustments made to the matrix. The final matrix, based upon one hundredths of a vote, in this case becomes:

```                    TO
FROM    NT     A    B    C    D
STRT     -   278  306  194  222
A      0     - 1000  194  222
B      0   278    -  194  222
C    143   278  306    -  857
D      0   278  306  194    -
```
The program now computes a trial set of ballot papers with an exact match on the first preferences, but using a pseudo-random number generator and the above matrix to produce the remaining preferences. Finally, adjustments are made to the papers to obtain a better match to the result sheet. The root mean square error is computed over the entries in the result sheet, which gives 0 in this case for the 3×5 entries, since we have a perfect match.

The ballot papers produced in this case (which depends upon the seeds used for the random-number generator) were:

```2 ABCD
7 ABDC
1 ACBD
4 BCDA
2 BDAC
1 BDC
1 BDCA
1 C
2 CDAB
4 CDBA
1 DABC
4 DBAC
1 DBCA
1 DCAB
1 DCBA
```
There are clearly many differences between the initial ballot papers and the above. However, since there are 64 ways of voting, it is quite unlikely that 10 ballot papers would be identical as with the initial papers (and in this sense, the final set must be regarded as more likely than the starting set). The construction method in this case gives very few papers with incomplete preferences, since the result sheet had few non-transferables.

### Five real elections

The results of running the program for the five elections are given in Table 1. The result sheets were from the application of Newland-Britton. A very close fit was obtained in all cases. The entry Next gives the difference in the number of votes between the last candidate elected and the next highest. This figure is also divided by the number of votes to give a numeric indication of how close the choice of the last elected candidate is. For election B, the result was very close since this difference was a mere 14 votes (from 8739, ie 0.16%). In performing both Newland-Britton and Meek upon the ballot papers constructed by the program, only one result was obtained which was different from the actual result. For election B, Meek produced a result different from the actual election, but this is hardly surprising, due to the closeness of the final candidate elected.

### The experiment

The experiment concerns the influence of candidates with no realistic hope of being elected upon the result. With the UKCC analysis, it was observed that such no-hopers had a bigger influence with Newland-Britton rules than with Meek. In this case, 100 elections were run by selecting 200 ballot papers at random (repeated five times for each actual election). For these 100 elections, both Newland-Britton and Meek were run. The second row in Table 2 gives the number of times out of the 100 that the results from Newland-Britton and Meek were different. For the 500 elections the result was different for 88 cases, which implies that 4% of the candidates were treated differently.

Table 1: Five Council Elections

The first row in Table 2 gives the number of candidates which were never elected in any of the 100 elections, called no-hopers. It would seem that this is not an unreasonable definition of those that have no chance of election, since we know that the number of first-preference votes is not always a good indication.

Table 2: Results of simulations

The 88 elections in which Newland-Britton/Meek gave a different result were now re-run with the no-hopers eliminated. The results of this are recorded in Table 2 in the rows with indented titles. In all but one case, the difference between the two algorithms was just one candidate. However, the result of the re-runs is somewhat confusing except for the simple case in which the elimination of the no-hopers makes no difference. The result in the table are classified as follows:

• No change. In this case, the elimination makes no difference and hence these cases are not supportive of either Newland-Britton or Meek.
• Revert to Meek. In this case, the result from Meek does not change, but that for Newland-Britton changes to that of Meek. Such a case is taken as supporting the use of Meek.
• Revert to Newland-Britton. This is the exact opposite of the previous case and is taken as supporting the use of Newland-Britton.
• Meek unchanged. In this case, the result for Meek does not change, but that for Newland-Britton does (but not to that of Meek). This case is regarded as supporting the use of Meek.
• Both change. In this case, both change to a different result. This is obviously not supportive of either algorithm.
• Invert both. In this case, the results of both algorithms change to the previous result of the other one! Clearly not indicative of either Newland-Britton or Meek.
• Other. None of the above, and again not supportive of either algorithm.
The overall count from the above classification is that 56 cases are neutral, 27 support Meek and 5 support Newland-Britton.

### Conclusions

In appears that realistic ballot papers can be computed from the result sheets. However, it is difficult to validate this process, since at the moment, actual ballot papers are not available from real elections of any size. I would like to appeal for such ballot papers, perhaps in computer format, since such papers could be made available without revealing the source which surely would be satisfactory once the period of elected candidates had finished. All the election data obtained so far is for small elections for which the study above could not be applied.

The first result from this study is that Newland-Britton and Meek produce a different result for about 4% of the seats. The observed rate for the Irish elections in 1969 was 2.8% (3 out of 143) and for 1973 was 4.9% (7 out of 143). The difference between 1969 and 1973 is due to a decline in the party voting and hence is consistent with a figure of 4% given in this study.

Does a difference of 4% matter between two STV algorithms? Obviously, it is reasonable to say this is insignificant against a difference of around 30% when STV is compared to First Past The Post. On the other hand, for the Electoral Reform Society, it is surely unsatisfactory to have such differences. Unfortunately, resolving this issue, as we are all aware, is not easy.

The remaining result is that Meek has more indicative cases in its support than Newland-Britton by about 5 to 1 in the above experiment. Does this matter? Surely, a key advantage of STV is that candidates can enter without upsetting the result if they have no realistic chance of being elected. Providing other hurdles for candidates seems against the spirit of democracy.

### References

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