A Condorcet divisor method proportional representation procedure is presented that is a variant of Nicolaus Tideman’s Comparison of Pairs of Outcomes by Single Transferable Vote (CPO-STV) and Shultz STV but requires the determination of fewer candidate set comparisons than either. The method will produce the same result as a party list election that uses the same divisor method provided that each voter votes their party’s list. The procedure is a Condorcet variant of the procedure presented in the February 2011 issue of Voting Matters.
For an N-seat election, one primary election electing N candidates must be performed for each set of N + 1 candidates. For example, for a two-seat election involving candidates A,B,C and D, primary elections for the candidate sets ABC, ABD, ACD and BCD are held. For each of these primary elections, the winning set and its priority over loosing sets is determined by the following procedure (the method is presented for the d’Hondt divisor method but is easily generalized to other divisor methods.): Step 1. Every candidate in the N+1 primary sub-election candidate set is hopeful and every candidate not in that set is excluded. The seat value of every ballot is set to zero. Step 2. The priority, PC, for each hopeful candidate C that is the topmost hopeful candidate on at least one ballot is determined from PC = VC/(SC+1) where VC is the total number of ballots where C is the topmost hopeful candidate and SC is the sum of the seat values of ballots where C is the topmost hopeful candidate. The candidate with the highest priority is elected. If the total number of elected candidates is N, the count is ended and the N elected candidates are declared the winning candidate set of the primary with its priority over losing sets equal to the priority of the Nth elected candidate. Otherwise, if candidate C is elected, the seat value for each ballot that contributed to electing C is increased to (SC+1)/VC. Repeat Step 2 until N candidates are elected. Each loser set from a primary contains the candidate from the primary candidate set that is not in the winning set plus N-1 additional candidates from the winning set. For a two-seat election in which AB is the winning set of the primary candidate set ABC, AC and BC are the loser sets. Only the priority of the winning set for each primary is calculated. The method determines the priorities of fewer relations than Shultz STV but still elects the Condorcet winner candidate set if there is one since the Condorcet winner candidate set cannot be a losing set. Once every primary election has been held, winning set > losing set relations are then elected from highest priority to lowest. However, if electing a relation would violate transitivity then that relation is excluded instead of elected. In practice, only loosing sets that are the winning set of at least one primary election need be considered. When every relation has been elected or excluded, the highest ranked candidate set is declared the elected candidate set. An example with a Condorcet cycle is the two-seat election presented in Election 1. Election 1 7 A B C D 6 B C D A 5 C D A B 4 D A B C Primary ABC 11 ABC 6 BCA 5 CAB AB > AC and AB > BC. Priority: 8.5 Primary ABD 7 ABD 6 BDA 9 DAB AD > AB and AD > BD. Priority: 8 Primary ACD 7 ACD 11 CDA 4 DAC CD > AC and CD > AD. Priority: 7.5 Primary BCD 13 B C D 5 C D B 4 D B C BC > BD and BC > CD. Priority: 9 The winning sets are AB, AD, CD and BC. Since a candidate set must be a winning set in at least one primary to win the election, only relations involving winning sets need be considered. The relevant candidate relations are BC > CD. Priority: 9 AB > BC. Priority: 8.5 AD > AB. Priority: 8 CD > AD. Priority: 7.5 AB > CD. Priority: 6.5 Transitivity can be preserved by electing relations in priority order that preserve transitivity and excluding those that do not. When the three highest priority relations are elected, they produce the transitive candidate set order AD > AB > BC > CD. The next highest priority relation CD > AD is excluded since the higher priority relations have determined that AD > CD. According to this procedure, candidates A and D are elected. -Ross Hyman
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