With First Past the Post, a voter is said to have a share in a representative’s legislative power if their vote was counted for that representative (MP). To determine the legislative power share of a voter who voted for their MP, we first divide the average number of votes cast in that voter’s riding by the number of votes received by the elected MP. We then apply a weighting factor based on the relative size of that riding to adjust the share of each voter’s legislative power (since voters in ridings with fewer voters have more impact on the election of an MP than voters in ridings with more voters). Because the number of seats per province is set by federal law, we perform these calculations at a provincial level and do not include disparities arising from differences in the average number of voters per riding across provinces.
More specifically, we calculate the LPS score for an individual voter as follows:
Calculate the provincial quota, Qp = Vp / Sp, where Vp = the total number of votes cast in a particular province, and Sp = the total number of seats in that province.^{1}
In a particular riding i, we calculate a local relative influence value as Ii = Vti/Vi, where Vti = the total number of votes cast in riding i and Vi = the number of votes cast for the winning candidate. Ii = 0 for voters who did not vote for the winning candidate. For example, if a candidate receives 20,000 votes out of 40,000 cast, then the local relative influence of the voters who supported that candidate will be I = 40,000/20,000 = 2, or twice the influence voters would have if all their votes had equal influence on the makeup of parliament.
We then apply a weighting factor, Wi, to that riding to reflect any deviation from the average riding size in the province. We define Wi as Wi = Qp/Vti.^{2} For example, if 50,000 votes are cast in an average riding, and 40,000 votes are cast in riding i, then Wi = 50,000/40,000 = 1.25, which reflects the fact that voters in that riding have greater legislative influence due to the fact that a smaller number of voters than average share in the legislative influence of the elected MP there.
We multiply these two factors together to calculate the LPS score in riding i for those voters who voted for the winning candidate^{3}: LPSi = Ii * Wi.^{4} In the example above, we would say that the 20,000 voters who voted for the winning candidate in a riding of 40,000 people would have a Legislative Power Share of 2 x 1.25 = 2.5, or two and a half times as much influence as parity would warrant.
For a single election, we place each voter’s LPS into one of nine bins: a bin for LPS scores of 0, a bin for scores in the range (0.00, 0.50]^{5}, and so on. Graphing the percentage of values in each bin yields the following graph:
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By stacking each of the bars on top of each other, left to right, we can represent all of the elections since Confederation:
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The following summary statistics show that the number of voters no Legislative Power has been increasing over time. This has been at the expense of the voters in the green range (LPS in 0.75 - 1.25) or the green + yellow range (LPS in 0.50 - 1.50).
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Note that, in principle, we would prefer to define the quota as the average population per riding, but these historical details are difficult to acquire and to connect to the voting history records, so we instead make the simplifying assumption that riding populations are approximately proportional to the number of votes cast in each riding. Variations in the ratio of votes cast to riding population will introduce some corresponding variations in our resulting estimates, but we anticipate that the overall comparative patterns will be largely unaffected as we make the same approximation for all voting systems considered. ↩︎
Note that if we were in a position to define the provincial quota based on population instead of number of votes cast, we would similarly define the weighting factor based on riding population rather than votes cast. ↩︎
In a number of early elections (prior to and including 1958), a total of 263 candidates have been acclaimed. This happened primarily in the earliest years, as there were only six acclaimed candidates from 1921 to 1958, but as many as 55 (15.2% - in the 1874 election). In such cases, there was effectively no vote in some ridings, so it does not make sense to calculate an LPS score for the voters in these ridings. We therefore exclude them from our analysis. This has no impact on the analysis back to 1958, and, with the exception of the 1917 election in which 32 candidates were acclaimed, only minimal impact on the analysis back to 1887. ↩︎
Since we are basing the quota and riding weight calculations on votes cast, this equation simplifies to LPSi = (Vti/Vi)*(Vp/Sp)/Vti = (Vp/Sp)/Vi. I.e., the total number of votes cast in the riding cancels out. ↩︎
The notation (x, y] means values strictly larger than x and less than or equal to y. That is, all values z such that x < z <= y. ↩︎