Finding the Threshold of Exclusion for all single seat and multi-seat scoring rules: Illustrated by results for the Borda and Dowdall rules

We provide a general analytic approach for calculating the Threshold of Exclusion (TE) for the single-seat and the multi-seat versions of all scoring rules, an infinitely large class of voting systems (Fishburn, 1973; Young, 1975; Saari, 1994, 1995). We offer specific results for two rules used for parliamentary elections at the national level: Borda (Black, 1958), used for national elections to special reserved seats for Hungarian and Italian ethnic minorities in Slovenia, and the unique and little known electoral system used for legislative elections on the Pacific Island state of Nauru, the Dowdall rule. When voters are required to provide a complete ranking of all candidates, we find that Borda does not, in general, operate as a majoritarian system in that a supermajority of roughly 2/3rd is required to guarantee electing a candidate of choice. In contrast, we find that, as district magnitude increases, the Threshold of Exclusion for the Dowdall rule tends to zero, in the same way as do list systems of proportional representation (PR). However, in contrast to the case for list PR rules, TE for Dowdall can still be relatively close to 12 for small district magnitudes.