# The Mathematics of Irish Elections

Together with Gary McGuire, Department of Mathematics, National University of Ireland, Maynooth, I have written a draft for a paper on Mathematical Aspects of Irish Elections. The paper discusses the rules for the Irish elections. In addition, it discusses the problems with only considering the last received sub-parcel and gives a simple example of why STV is non-monotonic.

## The Last Received Sub-parcel Problem

In order to save time, only the last received sub-parcel is considered when transferring. Unfortunately, this could cause fundamental distortion. Surprisingly, the only place we have found a discussion of this is the book Irish Voters Decide by Sinnott.

In order to understand this, we have constructed some hypothetical examples. Suppose County X is a three-seater with 39996 votes and hence a quota of 10000, and that the first preference vote is given as in the table. Assume that voters for candidate A all give their second preference to candidate B and their third preference to candidate C. The surplus from A will then first push B over the quota, and is then transferred to C who also is elected.

 C 1 C 2 C 3 A 12000 12000 12000 B 9001 11001 11001 C 9000 9000 10001 D 9995 9995 9995

There are two things to notice. First of all, the same sub-parcel of votes in involved in electing all three candidates. In a sense, the votes cast for A become “transferable super-effective”! Secondly, this happens regardless of the preferences of B. Even if all first preference votes for B have D as their second preference, there will be zero transfers to D, since we only consider the sub-parcel transferred from A. However, if we look at all of B's votes, as much as 82% of them could have a next preference for D, so D could have received 819 votes instead of 0 if all of B's votes were taken into account!

We will now consider a variation of this. Suppose that all of the votes for E have A as their second preference, B as their third preference, and C and their fourth preference. When E is eliminated, the transfers will then first elect A, then B and finally C. Again, this happens regardless of the preferences of A, B and C. Suppose that D has wide cross-party appeal, and that all the voters for A, B and C have D as their second preference. The only problem is that the 5% who vote for E do not like D. So D is the first choice of 25% and the second choice of 70% and still fails to get elected! Notice how the current system totally ignores the fact that 70% have D as their second preference, while the ballots for E are “transferable super-effective”! In fact, in this example the only ballots where further preferences are considered are the votes for E.

 C 1 C 2 C 3 C 4 A 9994 12000 12000 12000 B 9001 9001 11001 11001 C 9000 9000 9000 10001 D 9995 9995 9995 9995 E 2006 - - -

Let us consider a real life example where the transfer patterns could be complex. After count 5 in Dublin Central Kehoe from Sinn Féin was ahead of Fitzpatrick from Fianna Fáil before the distribution of the surplus from Costello from Labour. However, the papers that brought Costello over the quota, were all from the elimination of Mitchell from Fine Gael. Are first preference Labour voters more likely to have a preference for Sinn Féin that Fine Gael voters? Kehoe had received 332 transfers when Mitchell was eliminated, compared to 523 for Fitzpatrick. However the Fine Gael voters who first transferred to Labour then transferred 707 to Fianna Fáil compared to 361 to Sinn Féin. What would have happened if all Costello's votes had been considered?

With electronic voting, there is no longer any reason for this kind of simplifications, and votes can be transferred based on all the preferences from all the votes.

## STV is Non-monotonic

One odd aspect of STV is that it is non-monotonic. That means that getting more votes may prevent you from being elected. This is discussed extensively in the literature on voting. We will give a simple example that shows that STV is non-monotonic.

Three Fianna Fáil candidates are fighting for the two remaining seats in county Z. When FF3 is eliminated, the transfers are evenly split among FF1 and FF2 and FF1 is elected. But FF2 does not like FF1, and demands a recount. The recount shows that two FF1 ballots had been placed among the FF2 ballots, so at the second to last count FF1 actually has 7999 and FF2 4000. But when FF2 is eliminated, it turns out that all the votes have FF3 as their next preference, and FF3 is elected!

 FF1 FF2 FF3 7997 4002 4001 9998 6002
 FF1 FF2 FF3 7999 4000 4001 7999 8001

We do not know if anything like this has ever happened in Ireland, but Irish elections are often close and mathematically surprising things do happen frequently. We challenge you to keep your eyes open! The world is filled with exciting mathematics all around you!

Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

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