Margines zwycięstwa kandydata i mandaty zagrożone w systemie d'Hondta-Jeffersona

The inspiration for writing this article was a relatively high number of invalid ballots in Poland’s local government election in 2014. We wanted to answer the following question: what might happen to a seat won in the d'Hondt-Jefferson electoral system if the invalid ballots were originally valid. We were interested in over-voted ballots, where one choice could be originally marked by a voter and other choices added by a biased commission. Would the number of such invalid ballots be potentially high enough to change the election result, if they were assigned to other parties as valid votes? Fundamental for weighing the influence of invalid ballots on the results of a contested election is the margin of victory of the winning candidate. The margin of victory is easy to calculate for a single-member district election method but in a d’Hondt-Jefferson method, it is complex and requires simulation of the whole system, by adding additional votes to different parties. We propose a method of operating the candidate’s margin of victory as the smallest number of additional votes needed by other competing parties to take that seat. For every seat, we introduce the indicator l as the ratio of the margin of victory and the number of invalid ballots. The indicator l allows for some assessment if the seat is being threatened by invalid ballots. The higher the value of the indicator, the more vulnerable the seat is. At a value of l lower than one, the seat is safe, as invalid ballots in any configuration, may not give it to other parties. We demonstrate the calculation of the margin of victory and the indicator l on the example of Poland’s local government elections to county councils conducted in 2014, where the seats are distributed by the d'Hondt-Jefferson method. Almost 17% of the votes were invalid at that time. Since there were no statistics on over-voted ballots we have assumed all invalid ballots could be originally valid and might be cast to any party. That drove us to a conclusion that 47% of all mandates were threatened. The defined margin of victory and the indicator l may be of practical use in resolving electoral protests by the courts, as they give a glimpse of the likelihood of changing the election outcome.