Andrzej Machowski D’Hondt calculator

1.

Let's start with a single circle, e.g. 11-mandate. Let's presume that we have 5 parties participating in the allocation of mandates.

A – 35 percent; B – 32 percent; C – 15 percent; D – 10 percent; E – 8 percent (this is alleged effective support – percentages add up to 100 percent)

A has 0.35 x 11 = 3.85 mandates; so it has 3 mandates and "relays" to the pool of 0.85 mandates (the alleged rest)

B has 0,32 x 11 = 3,52 of the mandate; so it has 3 mandates, and "relays" to the mandate pool 0.52 (the alleged rest)

C has a 0.15 x 11 = 1.65 mandate; it so has 1 mandate and "relays" to the mandate pool 0.65 (this is the rest)

D has a 0,10 x 11 = 1,10 mandate; it so has 1 mandate and "relays" to the 0,10 mandate pool (this is the rest)

E has 0,08 x 11 = 0,88 mandates; so it has 0 mandates, and it ‘gives’ to the pool of 0,88 mandates (this is the rest)

In the case of specified percent results in this district, on average, each batch ‘sold’ to the pool of 0,58 mandates (this is the average of 0,85; 0,52 ....; 0,88).

2.

Let's add the another districts to that. Let's figure out how. d. Here. d = 41. In each of the d the districts are doing the same operation. It is easy to note that the alleged expected value (average) of the remainder for each batch in 41 districts will aim at 0.5. In another words, each organization will average 0.5 seats in each district.

3.

How many tickets will the jackpot yet count? If each organization puts an average of 0.5 seats per territory and the districts are 41; then it can be assumed that each organization puts 41 x 0.5 seats in the pool, or 20.5 seats. If the parties active in the division of mandates are 5, for example, the pool is 5 x 20.5 of the mandate, or 102.5 of the mandate.

4.

What happens to the tickets from the pool? They are shared between parties proportionally at national level (of course we are talking about a mathematical model, not a real division, due to the fact that there is no real division at national level). And so, if, for example, organization A has 40% of the support at the country level, it receives 0.4 x 102.5 mandates, or 41 mandates.

5.

All this can be expressed as:

Number of seats for the lot A = (ods._A × m) – (d × 0,5) + (ods._A × n × d × 0,5) [Model 1]

Number of seats for the lot A = (ods._A × m) – 20,5 + (ods._A × n × 20,5) [Model 1a]

where:

• ods._A is an effective percent of the votes for organization A on a country-wide basis (e.g. organization A gained 32%; but the parties that had reached the threshold voted for a full of 90% of voters; so the effective percent of votes for organization A is 32/90 = 0,3555);
• m is the number of tickets to be obtained on a national scale (with us m = 460; provided that it is actually 459, due to the fact that 1 MN mandate in Opolskie is removed from the national model – but here we presume for simplicity that an anomaly with a mandate for MN does not exist);
• d is the number of constituencies (we 41);
• n is the number of parties participating in the allocation of mandates (presumably n = 5).

Let's base on expression [1], assuming that the effective percent for lot A is 0,3555 :

Number of seats for the lot A = (0,3555 × 460) — (41 × 0,5) + (0,3555 × 5 × 41× 0,5) =

= (0,3555 × 460) – (20,5) + (0,3555 × 102,5) =

= 163,53 – 20.5 + 36,44 = 179,47

Note that it can besides be said that in the D’Hondt strategy each organization receives a precise proportional number of mandates at the beginning and then pays a taxation of 0.5 for each territory (total of 20.5 mandates). It's like a "major" taxation – each organization pays it at the same amount, but for a lot at 0,40 – erstwhile it comes out in proportion of 0,4 x 460 = 184 mandates – this taxation is only 11% of 184 mandates; while for a lot at 0,10 (0.1 x 460 = 46) this taxation is as much as 44,5% of 46.

Model [1a] can be converted:

Number of seats for the lot A = (ods._A × m) – 20,5 + (ods._A × n × 20,5) =

= ods._A × m – (((n × 20,5) / n) + (m ♪ Oh, yeah ♪ n) – (m ♪ Oh, yeah ♪ n) + (ods._A × n × 20,5) =

= (m ♪ Oh, yeah ♪ n) + (ods._A – (1 / n) × (m + n × 20,5) [Model 2]

Where m = 460; n = 5 the expression is:

Number of mandates for lot A = 92 + (ods._A – 0,2) × (460 + 102,5) =

= 92 + (ods._A – 0,2) × (562,5) [Model 3]

If we usage percentages in the expression [3] alternatively of percentages (e.g. 0.352) (e.g. 35.2) the expression takes the form of:

Number of mandates for lot A = 92 + (percent._A – 20) × (4,60 + 1,025) =

= 92 + (percent._A – 20) × (5,625) [Model 4]

That is why specified a precise value (5,625) comes as a reward or punishment for each percent as a consequence of above/below 20% – erstwhile 5 parties participate in the division of mandates. If 4 lots are involved, then the expression [2] n = 4; erstwhile 3 lots n = 3; erstwhile 2 lots n = 2.

From here

Number of mandates for organization A (where 4 lots participate in the allocation of mandates) = 115 + (percent._A – 25) × (5,42)

Number of mandates for organization A (where 3 lots participate in the allocation of mandates) = 153,33 + (percent._A – 33,33) × (5,215)

Number of mandates for organization A (where 2 lots participate in the allocation of mandates) = 230 + (percent._A – 50) × (5,01)

Andrzej Machowski