In this paper, we are concerned not with the limiting case of the ERS rules electing candidates without support, but with other large elections in which some candidates are elected despite having less than half the quota. In such situations, it might appear that the ERS rules might elect the 'wrong' person. Unfortunately, it is not easy to devise a means of determining the 'right' choice. Here we use random ballot papers with some characteristics of a real election.
The data from the last UKCC election is impressive: 129 candidates for 7 seats with 62,216 ballot papers. The election is conducted to the ERS rules assisted by Rosenstiel's program.
Mr Wadsworth of ERBS has kindly given me the information above and also a print-out from the Rosenstiel program which gives for the seven elected candidates:
First Stage Votes
Preference when when
Candidate Votes/ elected elected/
Quota Quota
A 112.8% 1 100.0%
B 17.3% 122 47.7%
C 19.8% 121 53.2%
D 21.0% 121 50.5%
E 16.0% 123 34.9%
F 11.1% 123 36.4%
G 11.0% 123 42.6%
The concern here is that since one candidate was elected on only about one
third of the votes that had to be retained by the most popular candidate,
can one be sure the correct choice was made? The result of that particular
election is not being questioned, but the choice of algorithm for
elections of this type.
To determine the feasibility of using these programs on a PC (personal computer) for elections like UKCC, a program was written to construct a large number of random ballot papers. Of course, real ballot papers were not available, and even if they were, the data preparation problem would be formidable.
At this point, a major problem arose. Both Meek and the ERS computer programs allow for the storage of the complete set of preferences. If this information is written to temporary disc storage, then the programs will run quite slowly. However, the total storage for UKCC-like elections is around 8Mbytes, which is only just within the reach of current PCs. The obvious solution was to undertake modifications to both programs to take advantage of the fact that only a small fraction of the total possible number of preferences would be specified. In fact, the modification to Meek was very easy and undertaken, but that for ERS (which is much more complex in computer terms) was too difficult. In any case, both programs were successfully run with random data on my home computer.
The conclusion from this study was that running the Meek or ERS rules on a modern PC would be possible for large elections. However, program modifications would be desirable to ensure that the programs kept within system limits. It was also observed that both programs produced result files which were excessive in size (and too big to print with convenience). With the preferences kept in main store, the time both programs took to execute was limited by the speed of processing; moreover, it was linear in the number of ballot papers. The time taken for the programs on my home computer was about 500 seconds per 10,000 papers for Meek and about ten times faster than that for the ERS rules. These times are clearly minor compared with the data preparation overheads in undertaking such counts.
For this part of the study we use simulated data with only 1,000 papers, rather than the 62,000 that were actually recorded for UKCC. The reason for this reduction is to save on the computer time required, since many elections must be analysed (in fact, 100 elections were used). However, to have a realistic chance of determining the effect of using either algorithm, it is clear that the ballot papers must adhere to some of the characteristics of the real data.
The method used to construct the papers was to use a random number generator, but to use some of the characteristics of the UKCC election to determine the distribution functions used. The two major parameters are the popularity of each candidate and the length of each ballot paper. We can estimate the popularity of each candidate in the real election by means of their (known) number of first preference votes. Hence the popularities of the candidates in the simulated elected were adjusted so that the leading candidate had more than the quota of first-preference votes, candidates numbered 2 to 20 had reducing popularity of 95% of the previous candidate, and the remaining candidates had a constant popularity of 95% of the 20th candidate. The reason for this constant tail is that if the 95% rule was carried on, it was observed that the lower candidates had virtually no votes at all.
The distribution of the length of preferences chosen was as follows: For those expressing a single preference: 8.0% of the papers; for two preferences, 8.7%; for 3: 9.4%; for 4: 10.1%; for 5: 10.9%; for 6: 11.6%, for 7: 12.3%, and for 8 to 11 preferences: 7.2%. This distribution increases linearly to 7, the number of candidates then drops to a constant amount.
We can now compare a randomly produced set of papers with those above from UKCC. In this case with random ballots, the quota becomes 1000/8=125, instead of 62216/8=7777 for the real election. The table entries below and for the comparative table for UKCC are expressed in proportion to the quota to give directly comparable data.
First Stage Votes
Preference when when
Candidate Votes/ elected elected/
Quota Quota
A 129.6% 1 100.0%
B 16.8% 119 53.3%
C 10.4% 121 46.1%
D 13.6% 121 46.9%
E 12.0% 121 45.3%
F 8.8% 121 44.1%
G 9.6% 121 42.2%
The pattern is clearly similar. We need not be concerned about minor
differences, since the study is of elections of this general type. To
generate each set of ballot papers merely requires as input the three
integer seeds for the random number generator. In consequence, all the data
presented here which is based upon a set of 100 elections can be recomputed
from 300 integers. The seeds for the election in the above table were 1, 1
and 18.
We now consider the results of each of these tests:
Meek 559.0 (±18.8); ERS 482.6 (±13.7); Meek/ERS 1.159 (±0.031); where the range represents two standard deviations. Hence Meek consistently gives 16% more non-transferable votes.
Meek Elects ERS Elects
129 candidates [S]+A [S]+B
20 candidates [S]+A [S]+A
8 candidates [S]+A [S]+A
Here [S] represents a set of six candidates and A and B are different
candidates not in the set [S]. In other words, ERS reverts to the Meek
result when the no-hope candidates are removed, and this reversion is
retained when only 8 candidates are considered. This is clearly strong
evidence that the full election using the ERS rules produces the 'wrong'
result.
Of course, this study only relates to elections with a large number of candidates. It can hardly be considered a criticism of Newland and Britton, since it is doubtful whether they ever conducted an election of the size considered here.