Since it is probably easiest to understand how DMP works by doing an example, this post will show how to use DMP to determine the outcome of the 2015 provincial election in PEI.

This example will make use of a reserve factor of 15%, a district threshold of 5%, and the redistricting done by Anna Keenan (shown below).

DISTRICT	GREEN	LIBERAL	NDP	PC
1+2	145	2121	784	2627
3+4	258	2155	801	1988
6	330	1453	350	2155
5+7	411	2287	469	2674
8+9	584	3292	991	2164
10+11	806	2591	652	1730
12+13	808	2009	613	1698
14+15	706	2429	1447	2151
16+17	2454	2490	301	1665
18+19	836	2306	420	3182
20+21	659	2168	617	3089
22+23	561	2393	831	1908
24+25	125	2729	449	1649
26+27	167	3055	269	1984

The first step is to allocate a proportional share of the seats to each party. This is done by multiplying each party’s share of the popular vote by the total number of available seats and distributing the remaining seats to the parties with the largest remainders. For example, the Green Party received 10.8% of the vote, so its deserved seat share is (10.8%)X(28 seats) = (3.02 seats).

Since this example uses a reserve factor of 15%, each party’s seat allocation is split into two groups: core seats and reserve seats. To calculate a party’s contribution to the reserve, its total number of deserved seats is multiplied by the reserve factor and the result is rounded down. Once a party’s contribution to the reserve is known, its core seat allocation can be found by subtracting its reserve seats from its total number of deserved seats. The Liberal Party, for instance, contributes 1 seat to the reserve, because (15%)X(11 seats) = (1.65 seats) rounds down to 1 seat.  This results in a core seat allocation of (11 seats)-(1 seat) = (10 seats) for the Liberal Party.

The results of Step 1 are summarized in the following table.

	GREEN	LIBERAL	NDP	PC
SHARE OF THE POPULAR VOTE	10.80%	40.80%	11.00%	37.40%
DESERVED SEAT SHARE	3.02	11.42	3.08	10.47
TOTAL SEATS DESERVED	3	11	3	11
CORE SEATS	3	10	3	10
RESERVE SEATS	0	1	0	1

The second step is also straightforward. The first seat in each district is assigned to the first ranked candidate of the party that received the largest fraction of the local vote. The party with the highest district vote is highlighted in orange below.

	GREEN	LIBERAL	NDP	PC
1+2	2.6%	37.4%	13.8%	46.3%
3+4	5.0%	41.4%	15.4%	38.2%
6	7.7%	33.9%	8.2%	50.3%
5+7	7.0%	39.2%	8.0%	45.8%
8+9	8.3%	46.8%	14.1%	30.8%
10+11	13.9%	44.8%	11.3%	29.9%
12+13	15.8%	39.2%	12.0%	33.1%
14+15	10.5%	36.1%	21.5%	31.9%
16+17	35.5%	36.0%	4.4%	24.1%
18+19	12.4%	34.2%	6.2%	47.2%
20+21	10.1%	33.2%	9.4%	47.3%
22+23	9.9%	42.0%	14.6%	33.5%
24+25	2.5%	55.1%	9.1%	33.3%
26+27	3.1%	55.8%	4.9%	36.2%

Now we can start the last step. The first part of Step 3 is to transfer half of the votes received by the parties that won first seats to their second listed candidates.  The second listed candidates and their share of the local vote are shown in orange.

	GREEN	LIBERAL	NDP	PC
1+2	2.6%	37.4%	13.8%	23.1%
3+4	5.0%	20.7%	15.4%	38.2%
6	7.7%	33.9%	8.2%	25.1%
5+7	7.0%	39.2%	8.0%	22.9%
8+9	8.3%	23.4%	14.1%	30.8%
10+11	13.9%	22.4%	11.3%	29.9%
12+13	15.8%	19.6%	12.0%	33.1%
14+15	10.5%	18.0%	21.5%	31.9%
16+17	35.5%	18.0%	4.4%	24.1%
18+19	12.4%	34.2%	6.2%	23.6%
20+21	10.1%	33.2%	9.4%	23.6%
22+23	9.9%	21.0%	14.6%	33.5%
24+25	2.5%	27.6%	9.1%	33.3%
26+27	3.1%	27.9%	4.9%	36.2%

Step 3.b is to make a ranked list of each party’s candidates.  Let’s look at the Green Party to see how this is done.  In the above table, the Green candidate with the strongest performance is in district 16+17. As a result, this candidate is the first listed candidate for the Green Party.  The Green candidate with the next best performance is in district 12+13. Therefore, it is the second listed candidate for the Green Party.  This process is repeated until all of the candidates for each party have been ranked. The result is given in the table below.

GREEN	LIBERAL	NDP	PC
16+17	5+7	14+15	3+4
12+13	1+2	3+4	26+27
10+11	18+19	22+23	22+23
18+19	6	8+9	24+25
14+15	20+21	1+2	12+13
20+21	26+27	12+13	14+15
22+23	24+25	10+11	8+9
8+9	8+9	20+21	10+11
6	10+11	24+25	6
5+7	22+23	6	16+17
3+4	3+4	5+7	20+21
26+27	12+13	18+19	18+19
1+2	14+15	26+27	1+2
24+25	16+17	16+17	5+7

Step 3.c requires that the first seats won by each party be subtracted from their core seat allocation.

	GREEN	LIBERAL	NDP	PC
CORE SEATS	3	10	3	10
FIRST SEATS WON	0	9	0	5
CORE SECOND SEATS	3	1	3	5

Next, we attempt to assign each party’s deserved number of core second seats to their top ranked candidates. These are shown in orange below.

GREEN (Needs 3)	LIBERAL (Needs 1)	NDP (Needs 3)	PC (Needs 5)
16+17	5+7	14+15	3+4
12+13	1+2	3+4	26+27
10+11	18+19	22+23	22+23
18+19	6	8+9	24+25
14+15	20+21	1+2	12+13
20+21	26+27	12+13	14+15
22+23	24+25	10+11	8+9
8+9	8+9	20+21	10+11
6	10+11	24+25	6
5+7	22+23	6	16+17
3+4	3+4	5+7	20+21
26+27	12+13	18+19	18+19
1+2	14+15	26+27	1+2
24+25	16+17	16+17	5+7

Step 3.e is to resolve conflicting seat assignments by electing the better performing candidate. For example, in the above table, both the Green and PC parties have been assigned the second seat in district 12+13. Since the Green Party received 15.8% of the vote in that district and the PC Party received 33.1%, the district is assigned to the PC Party. To make up for this loss, we attempt to elect the Green Party’s highest ranked candidate of those remaining on its list.

Step 3.f is to repeat step 3.e until no conflicts remain and all of the core second seats have been assigned.  The result is given in the following table.

GREEN (Needs 3)	LIBERAL (Needs 1)	NDP (Needs 3)	PC (Needs 5)
16+17	5+7	14+15	3+4
12+13	1+2	3+4	26+27
10+11	18+19	22+23	22+23
18+19	6	8+9	24+25
14+15	20+21	1+2	12+13
20+21	26+27	12+13	14+15
22+23	24+25	10+11	8+9
8+9	8+9	20+21	10+11
6	10+11	24+25	6
5+7	22+23	6	16+17
3+4	3+4	5+7	20+21
26+27	12+13	18+19	18+19
1+2	14+15	26+27	1+2
24+25	16+17	16+17	5+7

Now that all of the core seats have been assigned, we can assign the reserve seats. These seats are assigned using steps 3.d to 3.f, with the additional rule that a conflict between a reserve and core seat is always won by the core seat. The assigned reserve seats are shown in green below.

GREEN (Needs 0)	LIBERAL (Needs 1)	NDP (Needs 0)	PC (Needs 1)
16+17	5+7	14+15	3+4
12+13	1+2	3+4	26+27
10+11	18+19	22+23	22+23
18+19	6	8+9	24+25
14+15	20+21	1+2	12+13
20+21	26+27	12+13	14+15
22+23	24+25	10+11	8+9
8+9	8+9	20+21	10+11
6	10+11	24+25	6
5+7	22+23	6	16+17
3+4	3+4	5+7	20+21
26+27	12+13	18+19	18+19
1+2	14+15	26+27	1+2
24+25	16+17	16+17	5+7

The table below summarizes the outcome of the election.

DISTRICT	Winner of First Seat	Winner of Second Seat
1+2	PC	NDP
3+4	LIBERAL	PC
6	PC	LIBERAL
5+7	PC	LIBERAL
8+9	LIBERAL	NDP
10+11	LIBERAL	GREEN
12+13	LIBERAL	PC
14+15	LIBERAL	NDP
16+17	LIBERAL	GREEN
18+19	PC	GREEN
20+21	PC	PC
22+23	LIBERAL	PC
24+25	LIBERAL	PC
26+27	LIBERAL	PC