An Example: Using DMP to Determine the Outcome of the 2015 PEI Election

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

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