After Monday's election results were in, the distortions of the first-past-the-post electoral system were again on display. The Liberal party won the most seats but with less votes than the Conservatives, because of how votes were distributed regionally (see my blog Canada's growing urban-rural political divide). How would the 43rd parliament have looked like if the election had used provincial proportional representation? Below you see the results of this calculation, based on the following assumptions:

1. Each province retains the number of seats it currently has in the federal parliament—from the smallest province (Prince Edward Island's 4 seats) to the largest province (Ontario's 121 seats).
2. Seats are allocated proportionally within each province using the Webster/Sainte-Laguë method.
3. There is a five percent hurdle in each province. Any party that fails to reach this threshold will be excluded from the calculation. (In the small provinces, the threshold is effectively higher.)
4. The three northern territories retain their first-past-the-post voting.

The rationale for provincial proportional representation (rather than nationwide proportional representation) is Canada's federalism and unique regional differences. Quebec's political landscape and existence of a regional party makes it almost inevitable to provide for regional representation. Even for the small provinces it makes sense to have them represented by politicians elected only in that province, even if it dilutes proportionally somewhat. Other researchers often propose Atlantic Canada as a single regional district in order to improve proportionality, but arguably that comes at the cost of diluting regional representation. More importantly, the small provinces have been purposefully over-represented in parliament, and grouping all Atlantic provinces into a single regional district would dilute this purposeful over-representation of the small provinces.

There is an important caveat that applies to translating FPTP votes into outcomes under proportional representation. Under FPTP, many voters cast their ballots strategically. They would have voted differently under proportional representation. This may matter most for the parties on the progressive side of the political spectrum where voter have reciprocal second preferences. Someone may have voted NDP where the Liberal candidate did not have a chance of beating the Conservative candidate, and someone may have voted Liberal instead of NDP for the same reason.

With these assumption, the table below shows the seats for each province and how they were allocated under the current first-past-the-post (FPTP) electoral system, and how they would have been allocated under provincial proportional representation (P-PR).

The big picture: Conservatives and Liberals would have received 117 and 116 seats, respectively, rather than 121 and and 157. Both large parties would lose seats under P-PR as they have been the major beneficiary of the current FPTP. Conversely, the smaller NDP and the Green Party would be the main beneficiaries of P-PR, with the NDP more than doubling its seats from 24 to 57, and the Green Party going from 3 to 23 seats. Furthermore, the Bloc Québécois is over-represented in Quebec with 32 rather than 25 seats.

Let us look at some of the provinces in detail. Alberta's 34 seats were almost all taken by the Conservatives bar one seat in Edmonton that went to the NDP. Under P-PR, Liberals and NDP would have received 5 and 4 seats, respectively, while the Conservatives would have gained only 24. Even the Green Party would have received one seat in Alberta!

British Columbia is actually not too far off the mark. Under FPTP, Conservatives gained two more seats than under P-PR, and the Independent seat plus the two Conservative seats would have gone to the Green Party instead.

While BC would not see too dramatic changes, Ontario would look rather different. Liberals held on to 79 seats compared to 36 by the Conservatives. Under P-PR the gap would shrink from 43 to just 10 seats. However, Conservatives would only gain a few more seats, with most of the seats moving from the Liberals to the NDP. The Liberal's win in Ontario came mostly at the expense of the NDP (6 instead of 21 seats) and the Green party (0 instead of 8 seats).

Quebec's election results would look rather different as well. Liberals and the Bloc Québécois are over-represented and would lose 8 and 7 seats, respectively. These 15 seats would shift to the NDP (8 more seats), the Green Party (4 seats), and Conservatives (3 more seats).

Atlantic Canada would also look more diverse. Instead of the 6-1 split between Liberals and NDP in Newfoundland and Labrador, under P-PR the province would have elected two Conservatives and NDP members each, and only three Liberals. Nova Scotia's 11 seats, instead of being split 10-1 between Liberals and Conservatives, would have gone to a broad array of parties: 5 to Liberals, 3 to Conservatives, 2 to NDP, and one to the Greens.

What about forming a government? The diagram below shows the feasible (formal or informal) coalitions based on the above results. Other than a "grand coalition" of Liberals and Conservatives, the NDP could form coalitions with either Liberals or Conservatives.

The Bloc Québécois and Green Party would not be able to form a coalition with either the Conservatives and Liberals; they would fall short several votes.

As I was writing these lines I saw Prof. Hayes's article in The Conversation, who calculated both national proportional representation and regional proportional representation. The latter differs from provincial proportional representation in important ways. Rather than treating the North as FPTP districts and provinces separately, Atlantic Canada, Prairie provinces, and the Northern territories would be grouped together. This method leads to more proportionality, but dilutes provincial representation into larger regions. This results in fewer seats for Liberals and the Green Party. My argument is that provincial PR is more in line with the constitutional history and configuration than creating artificial new electoral regions. To get provinces to agree to PR (or at least feel content with it), their numerical strength of representation in parliament cannot be diminished. With provincial proportional representation, each province would send their own representatives to the federal parliament. This is a slight nod to federalism at the expense of very little disproportionality.

In my previous blog on Electoral reform with strong local representation I had pointed to my related research paper, Electoral economics: Maximizing local representation under proportionality. Local representation can be enhanced both by recognizing the importance of provincial boundaries and by assigning local representation in a way that maximizes the representation preferences of electoral districts. PR does not need to forego effective and strong local representation.

Technical note about PR methods. The above calculations employ the Webster/Sainte-Laguë method, which uses a slightly different formula then the d'Hondt method that is also widely used. Specifically, under P-PR, the d'Hondt method would allocate three less seats to the Green Party, one less to the NDP, one more each to Liberals and Bloc, and two more to Conservatives. This does not change t