Political representation matters. In xxxx I became concerned about a political issue and went to see my Minister of Parliament (MP) to discuss it. I hoped that my MP would discuss the issue with his colleagues and that Parliament might ultimately do something about it. My MP listened politely, but it became clear that our fundamental world views were very different. The issue that I cared passionately about would never be represented by him in Parliament.

As I left his office, I was not particularly surprised. As a candidate in the most recent election, this person was my least favourite choice of those running. It was obvious that his world view differed significantly from mine and that the policies he would advocate in Parliament would usually be different from what I would want.

That day I did not feel represented by MP; my “Legislative Power Score” was low. I wondered how it would feel if the candidate I had actually voted for was my representative in Parliament.

Canada’s 2019 Election

In Canada’s 2019 election with the Single Member Plurality electoral system, many people are unrepresented – like I felt when I went to see my MP – and others have more than their fair share of legislative power.

Figure 1 shows the distribution of Legislative Power Scores for the 2019 federal election. Just over 50% of voters had an LPScore of 0. In many ways they are unrepresented.

Figure 1: Distribution of legislative power scores for Single Member Plurality (FPTP), 2019

Representation is a zero-sum game. If 50% of the voters are not effectively represented, where did that representation go? It went to other voters. The graph shows that more than 10% of the population have Legislative Power Scores of 2 – twice what they should have. Others have somewhat less or somewhat more, depending on the outcome of the vote in their particular riding as well as the size of their riding.

The shape of this graph is very characteristic of Single Member Plurality elections. The graphs for the 2006, 2008, 2011, and 2015 federal elections are nearly indistinguishable from the above graph.

In each of these elections, just over half of the voters were not represented (voted for someone who was not elected). Their representation was effectively transferred to voters who voted for someone who was elected.

In contrast to Canada’s current election rules, simulating the 2019 federal election with Single Transferable Vote gives a much more equitable result, as shown in Figure 2. The graph shows that more than 50% of voters have about their fair share of legislative power and only about 11% have no legislative power.

Figure 2: Distribution of legislative power scores for Single Transferable Vote, 2019

Legislative Power Index

The study of income inequality often refers to the Gini Coefficient. The Gini Coefficient can be presented two ways. The first is a simple number between 0 and 1 (or sometimes 0 and 100) where 0 means perfect equality (everyone has the same income) and 1 means perfect inequality (one person has all the income while everyone else has none).

We can apply the same idea to an election where the Gini Coefficient uses each person’s Legislative Power Score instead of income. We call the result the Legislative Power Index.

The 2019 Canadian federal election had a Gini Coefficient or Legislative Power Index of 0.57. When measuring income inequality, a Gini Coefficient of 0.57 is considered very inequitable. See, for example, this world map from Wikipedia.

The second way of presenting the Gini is as a graph showing the cumulative percentage of the income vs. the cumulative percentage of people. We can adapt that to Legislative Power.

In the following graph, if representation were evenly (fairly) distributed, it would follow the green line straight from the bottom left to the top right. The first 20% of voters would have 20% of the cumulative legislative power. The first 80% of the voters would have 80% of the cumulative power.

But the 2019 FPTP election was not that fair. The red (lower) line shows the cumulative legislative power for the first 50% of the voters is zero. The first 80% of the voters accumulated about 51% of the representation leaving the remaining 20% of the population with 48% of the representation.

Figure 3: Legislative Power Index for SMP (FPTP), 2019

As expected from the graph showing the distribution of Legislative Power Scores, a Single Transferable Vote simulation shows a much fairer result.

Figure 3: Representation Equity Index for Single Transferable Vote, 2019

The Representation Equity Index of this simulation is 0.31.