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We define several different thresholds for election methods by consideringdifferent scenarios, corresponding to different proportionality criteria thathave been proposed by various authors. In particular, we reformulate thecriteria known as DPC, PSC, JR, PJR, EJR in our setting. We considermulti-winner election methods of different types, using ballots with unorderedlists of candidates or ordered lists, and for comparison also methods usingonly party lists. The thresholds are calculated for many different electionmethods. The considered methods include classical ones such as BV, SNTV and STV(with some results going back to e.g. Droop and Dodgson in the 19th century);we also study in detail several perhaps lesser known methods by Phragm\'en andThiele. There are also many cases left as open problems.

Thresholds quantifying proportionality criteria for election methods