Probabilities of majority and minority violation in proportional representation

Apportionment methods are used in proportional representation systems for rounding electoral vote proportions to integer numbers of seats in parliament. Many criteria are known to judge this adjustment process and to decide whether a specific method is suitable for practical use. As in general it is not possible to fulfill all desired criteria together, it is necessary to consider probabilities of violated criteria for specific apportionment methods. This paper demonstrates how to calculate the probability of such violations for the important majority and minority criteria by means of a recently developed geometric-combinatorial approach. Results are given for several popular apportionment methods.