Representation Equity Index

Traditional measures of an election’s fairness have focused on party proportionality: does the proportion of votes for party X match the proportion of seats party X is awarded? This is measured by the Gallagher Index, the Loosemore-Hanby Index, and others.

This paper proposes a complementary voter-centric measure, the Representation Equity Index (REIndex), which measures the fairness of an election as a whole from the voter’s perspective. It is built on a Representation Equity Score (REScore), which is assigned to each voter in the election. In a perfect world each voter would have a REScore of 1 and the election as a whole would have a REIndex of 0.

Of course, we do not live in a perfect world. We describe the application of Representation Equity to what we believe would be typical of real-world Canadian elections. Unfortunately, our current Single Member Plurality (sometimes called First-Past-The-Post) does not fare well. When other common electoral systems such as Alternative Vote, Single Transferable Vote, and Mixed Member Proportional1 are considered, the Representation Equity Index shows significant differences between them that are not revealed by the traditional party proportionality measures.

To see the simulations, click here. For the theoretical basis, read on.


Political representation matters. In xxxx I became concerned about a political issue and went to see my 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 felt particularly unrepresented by my “representative” and wondered how it would feel different if the candidate I had actually voted for was my representative in Parliament.

(There's a bunch of text that I no longer like that's been deleted. Sorry for the discontinuity.)

Canada's 2019 Election

When we turn from hypotheticals to Canada’s 2019 election with the Single Member Plurality electoral system, we see that 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 – reminiscent of the dictatorship.

2019 FPTP

The Representation Equity Index for the 2019 Canadian Federal election was 1.15.

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. The REIndex for the four elections are also very close.

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.

The Representation Equity model treats representation as a zero-sum game. A voter who does not have representation (or less representation than is equitable) is offset by someone who has more. The bell-shaped distribution in is caused by the variance in riding sizes. If all the ridings had equal voter populations, the 50.95% with a REScore of 0.00 would be offset by 49.05% with a REScore of 2.04.

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 . The graph shows that less than 5% of voters are not represented at all and about 3.5% who have about double the representation that they should.

The REIndex for STV in this election was 0.49.

2019 STV FirstPrefPlusSameParty

But – and this is a big “but” – this is using a representation utility function (see below) that does not apply well to FPTP. It gives a representation of 1 (perfect) to voters whose first preference was elected and a representation of 0.5 if someone from the same party was elected.

If we switch the representation utility function to give 1 to voters whose first preference was elected and 0 otherwise (which better matches what happens with FPTP), we get a graph that is much more similar to FPTP:

2019 STV FirstPrefElected

What is needed (future work!) is a utility function that is more general as discussed below.


The introduction leaves many questions unanswered, primarily about methodology. We shall address them as follows:

  1. What are the basic assumptions behind this methodology?
  2. How is the REScore for each voter calculated?
  3. Are there alternative calculations that should be considered?
  4. How are Representation Equity Scores for individual voters used to generate the Representation Equity Index?
  5. How are voting preferences simulated, given the lack of such data for Canadian elections?


The core assumptions are that each voter should have a representative2 they can turn to with their policy concerns and that equal representation is better than unequal representation.

We assume that voting uses a ranked ballot (not necessarily exhaustive), even in the case of Single Member Plurality. This allows us to more easily compare several different electoral systems, although modelling ranked ballots in the Canadian context is difficult.

We assume that each voter ranks their ballot sincerely and that a higher-ranked candidate will represent the voter better than a lower-ranked candidate.

Calculating the REScore for each voter

This calculation involves two steps. The first step is to apply a utility function, u(b,e) to each voter. The utility function consumes a ranked ballot, b, and the set of elected candidates, e, that could have appeared on the voter’s ballot (i.e. that the voter could have voted for). It produces a non-negative number where 0.0 indicates no substantive representation for that voter and larger numbers indicate better representation. This function is discussed in more depth in a moment.

The second step is to normalize these values so that 1.0 is the average representation. 0.0 still means no representation and larger values still mean better representation. The REScore equation for the ith voter is



  • n is the number of voters
  • b_i is the ballot for the ith voter.

That is, it’s simply the voter’s utility function divided by the average of the function applied to all voters.


  1. How deep in the preferences list to go
  2. FPTP: go to 2nd pref?
  3. Count winning candidates in the same party?
  4. Multiple vote counting (eg: use effective votes for STV?) Probably not. We’re counting the voter’s happiness and if they got their top two choices, they’ll be happier.

There are many possible implementations of the utility function, u(b,e). The ones we have considered sum the contributions from each preference on the ballot that was elected.

Mathematically, the utility function is implemented as follows

Utility function


  • b is the ranked ballot cast by the voter
  • e is the set of elected candidates that could have appeared on b
  • w is a vector of weights in [0, 1] giving the contribution of each preference
  • m is the maximum of the district magnitude + 1 and the number of preferences expressed on ballot b. The district magnitude is considered because as it goes up the voter has a larger selection of candidates of which they approve.
  • f(c) is a function that returns 1 if candidate c appears in the set of elected candidates (e) and 0 otherwise
  • |c| is the number of votes received by candidate c

In plain English, we sum the contribution of each successful candidate on the voter’s ballot up to the limit of the district magnitude plus 1. Each of those contributions is weighted, presumably with the first preference being weighted more heavily than the second preference, which is weighted more heavily than the third. Finally, the contribution is divided by the number of voters that candidate represents. Representing fewer voters means the candidate can give each one more attention and thus represent them better.3

Alternatives for REScore

The equation used for calculating REScore specifies a family of calculations that depend on the weights given for each preference. One could also apply a function to the number of voters before dividing. For example, applying a log function would imply a law of diminishing returns (as the number of voters grows) that is not nearly as steep as simply dividing.

<Other, completely different, approaches?>

Calculating the Representation Equity Index for an election

Simulating voter preferences


  1. What’s the effect of the electoral system compared to riding size differences? (Electoral systems have the greater effect)
  2. Are there real-world electoral systems that do significantly better than Single Member Plurality (FPTP)? (Yes)



Possible REIndex calculations I’m not yet satisfied with the REIndex calculation. The fundamental issue is that (REScore - 1) – the distance from the ideal, is usually 1 or less. Squaring, as inspired by least squares and Gallagher, doesn’t do much in that range to penalize the outliers.

Multiplying by a constant inside the sum is the same as multiplying outside, so doesn’t help spread things out.

Getting rid of the square root and just taking the average difference doesn’t spread things out much either. Using an exponential spreads out the dictator so far that we need to cap it or we get errors writing it to disk. I don’t like the complexity and arbitrariness of some of the terms. I wish Unity was a bit closer to zero.

  1. This hasn't actually been implemented yet. ↩︎

  2. We distinguish between the representation work of an MP from the constituency work of an MP. The former consists of representing a voter’s policy concerns in Parliament. The latter consists of helping the voter navigate government bureaucracy. We are interested in the former and uninterested in the latter as it could be easily replaced by a non-partisan ombudsman. ↩︎

  3. This is taking into account, roughly, the issues that gave rise to the “one person, one vote” Supreme Court case in the United States. ↩︎