Researching a small project on what would happen if the 2014 European Parliament election had been carried out using the proposed new seat allocation for the 2019 election, I came across an interesting quirk of the electoral system in Slovakia.

In 2014, Slovakia elected 13 MEPs. The turn out was dismally low – 13% of registered voters – with the interesting result that the MOST-HÍD party got an MEP elected with only 32,708 votes, lower than any other political party in the EU.

Eight political parties got more than the 5% of the vote needed to qualify them for seats in the European Parliament; their support was as follows.

The seats were allocated as follows. (I should add a word of appreciation here for the excellent website of the Slovak state statistics agency, which has full details.)

The quota (the Republic Electoral Number, in Slovak legalese) is calculated by adding the total votes for all parties that cross the 5% threshold, and dividing by one more than the number of seats to be allocated. That's 439,522 votes, divided by 14 (one more than Slovakia's 13 seats in the European Parliament), giving a Republic Electoral Number of 31,394.

Each party's vote is divided by the quota, and they get the whole number of seats produced by that calculation. Any seats left unallocated after that first round are given to the party or parties with the highest remainder after the division has been carried out. In 2014, 12 seats were allocated in the first round of counting, and the thirteenth went to the SDKÚ–DS coalition, because its remainder was 753 votes more than KDH’s (ow – a small margin!). That gives the following outcome.

The proposal currently on the table for the 2019 European Parliament election would give Slovakia an extra seat. If 14 seats rather than 13 had been allocated to Slovakia in 2014, the result would have been somewhat different. The Republic Electoral Number would have been 439,522 divided by 15 rather than by 14, with a result of 29,301. That doesn't change who would have got the seats in the first round of counting, but it would have left two rather than one to be allocated by largest remainder. And in fact, those two seats would have gone to different parties – SMER and KDH, rather than the SDKÚ–DS coalition which actually won the remaining seat in 2014, but whose remainder would have been 1340 votes shy of KDH’s in this scenario.

In this case, the two candidates who would have been elected were Katarína Neveďalová of SMER (who would have taken the place originally won by Maroš Šefčovič, who became Slovakia's European Commissioner after the election) and Ján Hudacký of the KDH; Ivan Štefanec of the SDKÚ–DS would not have been elected, as he had just 1,483 individual votes less than his running mate Eduard Kukan.

That would have meant an extra seat for the Party of European Socialists in the European Parliament, as that is the group with which SMER is affiliated. Both KDH and SDKÚ–DS are affitialited with the European People's Party, so the change of seats allocation between them would not have affected the EPP's tally of MEPs.

It seems paradoxical that a party's individual number of seats can decrease when the total number of available seats increases. This is a risk of the largest remainder system used in Slovakia, and most countries instead allocate seats by the D'Hondt or Saint-Lagüe counting systems, where the quota is recalculated at every step in such a way that no party can lose out if the number of seats is increased. And this is where we start getting into Arrow's Theorem