Evaluations: Summary (Party-List)

Party-List CHPV Elections

Properties and features of CHPV that are highlighted in the preceding sections of this chapter are summarised below:

• Stick diagrams are used as visual evaluation tools in the analysis of party-list CHPV elections.
• For CHPV, ballot, block, candidate and party sticks represent tallies displayed on a binary logarithmic scale for one voter, one block of voters, one candidate and one party respectively.
• For elections with two competing parties, their respective tally shares can be displayed using a one-dimensional scale; this line being the two-party map.
• For elections with three competing parties, their respective tally shares can be displayed using a two-dimensional area; this equilateral triangle being the three-party map.
• For elections with four competing parties, their respective tally shares can be displayed using a three-dimensional solid; this regular tetrahedron being the four-party map.
• For more than four parties, maps cannot be drawn as more than three dimensions are required.
• Each point on a map represents a unique combination of the (per-unit) party tally shares.
• All possible combinations of (per-unit) party tally shares are contained within a map.
• For a given number of winners (W), a map can be partitioned into several individual domains where within each domain the same share of seats for each party results.
• There are as many domains as there are combinations of the numbers of seats won by the various parties.
• The boundary between two (or more) adjacent domains equates to a critical tie between two (or more) parties.
• For each seat share domain, there will be one fixed point on the map (called a dot) where the share of the tally for each party perfectly matches its share of the seats.
• For an optimally proportional voting (OPV) system, each dot is at the centre of its own domain; except when domains are truncated by a map edge.
• For CHPV, the domain boundary positions are deduced using party stick diagrams to identify all possible critical ties.
• CHPV domain boundaries do not generally coincide with OPV domain boundaries.
• In CHPV, an optimum outcome only occurs when a tally share point is located in equivalent OPV and CHPV domains.
• For two parties competing in a three-winner CHPV election, 93% of all possible outcomes are optimally proportional.
• For three parties competing in a four-winner CHPV election, 84% of all possible outcomes are optimally proportional.
• For two or three parties competing for up to five seats in a party-list CHPV election, over 70% of all possible outcomes are optimally proportional.
• Optimality peaks and a systemic bias towards any size of party is minimal where there are 'few' winners (W ≤ 6).
• With more or less than the optimum number of seats, there is some bias towards a small or large party respectively.
• If desired, a small party is prevented from gaining its first seat too easily by marginally lowering the number of seats from its value for maximum optimality.
• An approximate model based on extreme seat share domains can be used to estimate the extent of optimal proportionality for CHPV elections with more than three competing parties.
• Using this model, it is recommended that five seats should be allocated to a party-list CHPV election in which more than five parties are likely to compete.
• OPV is however susceptible to strategic nominations where a party clones itself to gain more seats.
• Cloning is helpful to the cloning party when it consequently lowers its own tally share threshold to gain a given number of seats but cloning is harmful when this same threshold is raised.
• For two-party OPV elections, one in six outcomes are altered by cloning and at least half of them are harmful.
• For two-party CHPV elections with two or three winners, cloning is never helpful to the cloning party.
• For two-party CHPV elections with up to five or up to six winners, cloning is never helpful to the cloning party provided it has a vote share of less than two thirds (t < 2/3) or one half (t < 1/2) respectively.
• The Least Squares Index (LSI) can be used to assess the disproportionality of an election outcome.
• For OPV, only rounding due to seat resolution gives rise to any disproportionality.
• For OPV, the worst case LSI equates to the difference between seat shares as indicated by the dot at a domain centre and tally shares as indicated by a point furthest from the dot but within that same domain.
• Naturally, this seat resolution LSI decreases as the number of winners is increased.
• For CHPV, the LSI due to seat resolution is likely to be very similar on average to that for OPV (as the number of domains is fixed for a given number of seats).
• To assess the extent of disproportionality introduced by choosing CHPV over OPV, a separate LSI is adopted.
• A centre offset LSI measures the disproportionality resulting from the displacement of the dot in the centre of an OPV domain to the centre of the corresponding CHPV domain.
• The centre of a CHPV domain is where the multiple intervening (non-critical) two-way ties intersect.
• The individual centre offset LSIs are then combined into one overall voting system LSI.
• The voting system LSI for CHPV can then be compared directly to those for other electoral methods.
• For CHPV, the voting system LSI indicates minimal disproportionality is achieved with around five seats where between two to six parties compete.
• Due to its uniquely 'peaked' performance regarding both optimality and disproportionality, party-list CHPV is ideally suited to few-winner elections (W ≤ 6); where typically five seats are recommended.
• CHPV is inappropriate for a single-constituency election where a large number of seats are to be filled.
• Where in total a large number of winners is required, simultaneous party-list CHPV contests across numerous few-winner constituencies should be held instead.

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