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 effectively represented by MP; my “Representation Effectiveness 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 former MP – and that the others have more than their fair share of representation.

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

Figure 1: Distribution of representation effectiveness 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 almost 20% of the population have Representation Effectiveness 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 less than 1% of voters are not represented at all and less than 2% have about double the representation that they should.

Most voters have an REScore near the ideal of 1.0.

Figure 2: Distribution of representation effectiveness scores for Single Transferable Vote, 2019

Representation Equity 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 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 Representation Effectiveness Score instead of income. We call the result the Representation Equity Index.

The 2019 Canadian federal election had a Gini Coefficient or Representation Equity Index of 0.55. When measuring income inequality, a Gini Coefficient of 0.55 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 Representation Effectiveness.

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

But the 2019 FPTP election was not that fair. The blue (lower) line shows the cumulative representation for the first 50% of the voters is zero. The first 80% of the voters accumulated about 52% of the representation leaving the remaining 20% of the population with 48% of the representation.

Figure 3: Representation Equity Index for SMP (FPTP), 2019

As expected from the graph showing the distribution of Representation Effectiveness 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 0.22.