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Voting matters - Issue 16, February 2003

Proportionality Revisited

B A Wichmann


The issue of proportionality in the last article[1], raised two problems in my mind which are addressed here.

A flaw

Consider the hypothetical case of an STV election in the UK, in which there is a United Kingdom Independence Party (UKIP) candidate together with a Tory candidate. A Tory voter who is on the Europhobic wing of the party could well decide to give his/her first preference to the UKIP candidate. On the other hand, if the Tory candidate was also Europhobic, then the voter would surely place his/her first preference with the Tory. In other words, the first preference votes for the Tory and UKIP cannot reasonably be analysed in isolation.

Of course, this issue is not specific to the Tory party -- the same problem could arise with a Socialist Party candidate standing against a New or Old Labour candidate.

I conclude from this that an analysis of party support based upon first preferences alone is doomed to failure.


In this section, we set aside the flaw noted above, and analyse the issue of proportionality from just one point of view: the granularity imposed by the size of the constituencies. If a constituency elects 4 members, then it is clear that strict proportionality could only be obtained if each party had a multiple of 25% of the first preference votes. Obviously, there will always be a mismatch between the first preference votes and the proportion of candidates elected.

As an example, we consider the 1997 Irish General election[2]. The 166 seats for the Dáil are from 41 constituencies having 3, 4 or 5 seats each. In this analysis, we consider three categories for the first preference votes: those of Fianna Fáil (FF), those for Fine Gael (FG) and the others. It can reasonably be said that the 'others' does not represent a party, but if strict proportionality is obtained for FF and FG, then the others as a single group will also be represented proportionally. We return to this problem later.

Kestelman[3] considers several measures of proportionality. Here, we consider some of those measures as applied to each individual constituency and compare this with the actual result. The measures used here are the Loosemore-Hanby Index, Gallagher Index of Disproportionality, Sainte-Laguë Index and the Farina Index (all taken from the above paper).

Given a specific index, then one can determine the number of seats for each party which would give the closest fit with respect to that index. In fact, all the indices give the same result with one exception: the Sainte-Laguë Index gives a different result for the Dublin Central constituency. Ignoring this isolated value we have the table as follows:

Constituency         Actual   Best    Fit(%) Comparison

Carlow-Kilkenny      (2,2,1) (2,2,1) 13.998   =
Cavan-Monaghan       (2,2,1) (2,2,1)  8.850   =
Clare                (3,1,0) (2,1,1)  7.452   FF to Other
Cork East            (2,2,0) (2,1,1) 16.773   FG to Other
Cork North-Central   (3,2,0) (2,1,2) 12.473   Two changes
Cork North-West      (2,1,0) (2,1,0) 24.912   =
Cork South-Central   (3,2,0) (2,2,1) 11.923   FF to Other
Cork South-West      (1,2,0) (1,1,1) 20.608   FG to Other
Donegal North-East   (2,0,1) (1,1,1) 17.801   FF to FG
Donegal South-West   (1,1,1) (1,1,1) 12.710   =
Dublin Central       (2,1,1) (2,0,2) 17.771   FG to Other
Dublin North         (2,1,1) (1,1,2) 16.756   FF to Other
Dublin North-Central (2,1,1) (2,1,1)  4.487   =
Dublin North-East    (2,1,1) (2,1,1) 19.113   =
Dublin North-West    (2,0,2) (2,1,1) 15.808   FG to Other
Dublin South         (2,2,1) (2,1,2) 11.999   FG to Other
Dublin South-Central (2,1,1) (1,1,2) 13.301   FF to Other
Dublin South-East    (1,1,2) (1,1,2)  4.042   =
Dublin South-West    (2,1,2) (1,1,3) 12.192   FF to Other
Dublin West          (2,1,1) (1,1,2) 11.492   FF to Other
Dun Laoghaire        (2,2,1) (1,2,2) 11.226   FF to Other
Galway East          (2,2,0) (2,1,1)  7.923   FG to Other
Galway West          (2,1,2) (2,1,2) 10.324   =
Kerry North          (1,1,1) (1,1,1) 19.729   =
Kerry South          (1,0,2) (1,0,2) 18.479   =
Kildare North        (1,1,1) (1,1,1)  9.214   =
Kildare South        (1,1,1) (1,1,1)  8.464   =
Laoighis-Offaly      (3,2,0) (3,1,1) 13.281   FG to Other
Limerick East        (2,1,2) (2,1,2)  9.015   =
Limerick West        (1,2,0) (1,1,1)  4.945   FG to Other
Longford-Roscommon   (2,2,0) (2,1,1) 15.181   FG to Other
Louth                (2,1,1) (2,1,1) 12.575   =
Mayo                 (2,3,0) (2,3,0) 14.288   =
Meath                (3,2,0) (2,2,1)  3.803   FF to Other
Sligo-Leitrim        (2,2,0) (2,1,1) 15.211   FG to Other
Tipperary North      (2,0,1) (1,0,2) 24.890   FF to Other
Tipperary South      (1,1,1) (1,1,1) 11.361   =
Waterford            (2,1,1) (1,1,2) 14.951   FF to Other
Westmeath            (1,1,1) (1,1,1) 15.218   =
Wexford              (2,2,1) (2,2,1)  3.036   =
Wicklow              (2,1,2) (1,1,3) 13.758   FF to Other

The content of the table is best explained by taking an entry: say Waterford, with 4 seats. The Actual and Best entries give the seats in the order (FF, FG, Other). The Best entry is computed according to all the indices apart from the isolated result already noted. The Fit% figures are calculated from the formula:

Fit% = Sqrt(Sigma(S%-V%)2), which is related to the Gallagher index.

The last column gives the comparison between the actual and best entries in seats. For Waterford, a single change in the actual result by a FF seat becoming an Other seat would produce the 'best' result.

One can see from this result that 18 constituencies would remain unchanged if they gave the best fit to first preference proportionality. The major difference is that the two major parties have gained over the others -- the best fit giving 56 seats in the Dáil for 'others' against the actual number of 35.

Two constituencies are different from the others. In the case of Cork North-Central, a two seat change is needed from the actual result to get the best fit. The reason for this is a high level of transfers from the other candidates to the two major parties. The case of Donegal North-East is special because the difference in the actual and best does not involve an increase in the 'other' seats. The reason for this was a significant transfer from FG to FF in the actual election when an FG candidate was still available for transfers.

As would be expected, there is a wide variation in the Fit entries. Also, the Fit values decrease with increased constituency seats: an average of 15.7% for 3-seats, 12.8% for 4-seats and 10.7% for 5-seats.

The under-representation of the Other group is to be expected as many of those candidates are excluded early in the count with many transfers to the major parties (as well as to non-transferables). This effect clearly indicates the dubious nature of grouping all the parties other than the major two into one.

The conclusion from this analysis seems to be that there is little loss in proportionality due to the natural granularity of the STV system. The lack of proportionality compared to the first preferences is caused by the vote transfers. There is a capital T in STV.

In addition to the above analysis of granularity, the same data reveals a very close correlation between the indices used. This is gratifying, since they are clearly supposed to be measuring the same property. However, the correlations can be represented approximately in a graph as follows in which the indices are indicated by their initials and the distance between them increases with a lack of correlation. From this it appears that the Loosemore-Hanby Index is centrally placed which reinforces Kestelman's support for that index.

Correlation graph


  1. I D Hill, What sort of proportionality? Voting matters, Issue 16, pp5-6. 2003
  2. 28th Dáil General Election, June 1997, Election Results and Transfer of Votes. The Stationery Office, Dublin 1998.
  3. P Kestelman, Quantifying Representativity. Voting matters. Issue 10. pp7-10. March 1999.

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