Comparisons: D'Hondt ~ Optimality 1

Proportion of Optimal Outcomes in Two- and Three-Party D'Hondt Party-List Elections

On the two-party and three-party maps in the previous section, the D'Hondt domain boundaries are shown as full lines while those of an optimally proportional voting (OPV) system are shown as dotted lines. An OPV system is one where the sum of the magnitudes of the differences between the tally share and seat share for each party is always minimal for any election outcome. In other words, any tally share point within an OPV domain is closest to the seat share dot at the centre of that domain than to any other domain centre.

When the tally shares for any election generate identical seat shares whether the D'Hondt or OPV system is used, then the D'Hondt method here yields optimally proportional results. On the maps, these results occur when the D'Hondt seat share domains overlap with their equivalent OPV ones. On the remaining parts of the maps where the D'Hondt and OPV domains generate dissimilar seat share outcomes, then the D'Hondt method does not produce optimally proportional results.

Therefore, the proportion of optimal outcomes for the D'Hondt method can be expressed as the ratio of those parts of a map that do generate an optimal outcome relative to the map as a whole. As both the D'Hondt and OPV domain boundaries are precisely positioned on the maps, an accurate ratio for any map can be derived by calculating and then summing the relevant areas on a three-party map or the relevant lengths on a two-party map.

The table above provides the exact fractional proportion of optimal D'Hondt method outcomes for two- and three-party elections with up to six winners (W).

The bar chart opposite illustrates these same proportions as percentages rounded to the nearest integer.

[Note: For the three-party map, both the whole area and any component part can be expressed as an exact fraction times the square root of three (√3). When the ratio is calculated, the √3 surd in the numerator and denominator cancel each other out leaving an exact fraction as the answer.]

For two-party contests, the proportion of D'Hondt method outcomes with optimum proportionality is at a maximum with two winners and declines very slowly as the number of seats rises. For three-party elections, the proportion of optimal D'Hondt method outcomes is again at a maximum with two winners and also declines slowly as the number of seats increases. However, optimality is higher with two parties than with three and it declines more slowly as the number of seats rises. For up to six winners, both two- and three-party D'Hondt Party-List elections generate optimally proportional outcomes in over 60% of all possible cases.

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