## The Mythology of the Ranked Ballot

It is widely believed that the ranked or 'preferential' ballot would produce a more accurate representation of the wishes of the voters in an election. Keep in mind that this form of voting is another form of 'First Past The Post' since the candidate with the most votes wins as happens in the single choice ballot form of election.

To assess the accuracy of this belief we need to carefully examine the mathematics involved in the ranked ballot scoring methods. I say 'methods' because there are many ways of determining the winner in this system. All of them appear simple but can, in fact, be extremely complicated and can produce wildly different results. All are also open to manipulation by those with vested interest in the outcome.

## The Reality

Looking first at what seems to be the simplest form of 'counting' the votes, namely choosing the candidate with the highest point score in the numerical ranking is not as straightforward as it seems. You can, in fact, get very different results depending on the distribution of votes between the candidates and the method used to calculate the winner.**Condorcet's Paradox**

A simplified illustration of this, which has become known as Condorcet's Paradox, is as follows: with 3 candidates (A, B and C) and 3 voters (1, 2 and 3) if voter 1 ranks candidates A B C, voter 2 ranks candidates C A B and voter 3 ranks candidates B C A, you end up with a tie at 3 votes each. It could easily be 300 voters or 300,000 voters ending up with the same tie. Of course, you could end up with a 2 way tie as easily.

Just counting the totals, however, is not the only, and likely not the most used way of calculating the winner. There are many other methods, all of them open to the possibility of the results not being a true reflection of voters preferences or even to the possibility of intentional manipulation.

**Plurality Rule**

Each voter ranks candidates in order, and the candidate(s) with the higest total wins.

Plurality rule is a very simple method that is widely used despite its many problems. The most pervasive problem is the fact that plurality rule can elect a Condorcet loser. Borda (1784) observed this phenomenon in the 18th century.

Example: Assuming 21 voters:

- RANK 1 2 3
- 1 voter ranks candidates: A B C
- 7 voters rank candidates: A C B
- 7 voters rank candidates: B C A
- 6 voters rank candidates: C B A

However, Candidate A is the Condorcet loser (both B and C beat candidate A, 13 – 8). The Condorcet method of counting the votes considers head to head matches. So A is ranked above B 8 times and B is ranked above A 13 times. Likewise A is ranked above C 8 times while C is ranked above A 13 times. C is the Condorcet winner since it is ranked above B 20 - 13.

So we can see that using just 2 different methods of counting votes in ranked ballots we can get two very different results from the same set of votes. There are in fact many different ways of calculating the winner with ranked ballots, raising the possibility of determining most, if not all of the candidates a winner depending on the method of tabulating the votes. There is the Borda Count, The Hare Rule, Plurality with Runoff, Coombs Rule as well as numerous others that determine the way in which votes are counted, even within a methodology such as Condorcet's (eg. Dodgson's Method and Black's Procedure). There are also Negative Voting, Approval Voting, Cumulative Voting and possibly many other variables that can be applied and will influence the results.

The above is a very (and perhaps overly) simplistic overview of the complexities of ranked ballot voting. It is intended only to illustrate that the possibility of a ranked ballot system of voting NOT reflecting the true wishes of the electorate is very real. Indeed the results may, in fact, be completely contrary to those wishes.

And how do we determine what the true wishes of the electorate are? If we can choose a different winner by using different counting methods, would not the same apply to deciding what the wishes of the electorate are? Are they the candidate chosen first the most times overall or the candidate that outscores each of the other candidates in 'head to head' battles? Or is it some other determiner decided by one of the other scoring methods? The first order of business should be to agree on how we determine what represents a true picture of voter preferences.

In what is perhaps the most likely system to be promoted for Canada, and one of the 4 currently proposed for Prince Edward Island, each voter selects multiple candidates and ranks them in order of preference. The first choice votes are counted. If no one party (or individual) has a 50 per cent majority, the last place candidate is eliminated and those who voted for that candidate have their second choice votes counted and tacked onto the first vote count. If, after redistributing, there’s still no one with 50 per cent of the vote, then the new last place candidate is eliminate, and whoever voted for that candidate has their vote for next preferred candidate redistributed. And so on until one candidate has the majority of the vote.

Example: Assuming 33 voters.

- RANK 1 2 3 4
- 1 voter ranks candidates: A B C D
- 7 voters rank candidates: A C D B
- 2 voters rank candidates: B C A D
- 6 voters rank candidates: D B C A
- 3 voters rank candidates: C B D A
- 9 voters rank candidates: C D B A
- 5 voters rank candidates: D C A B

Candidate C is the WINNER (Due to choices of the first round third place Candidate A)

Round 1 2 3 Candidate A 8 11 X Candidate B 2 X X Candidate C 12 12 19 Candidate D 11 13 12

- RANK 1 2 3 4
- 1 voter ranks candidates: A B D C
- 7 voters rank candidates: A D B C
- 2 voters rank candidates: B C A D
- 6 voters rank candidates: D B C A
- 3 voters rank candidates: C B D A
- 9 voters rank candidates: C D B A
- 5 voters rank candidates: D A B C
**Compare other systems:**- Mixed Member Proportional
- Dual Member Proportional
- First Past The Post

Candidate D is the WINNER (Due to strategic voting by supporters of third place Candidate A).

Round 1 2 3 Candidate A 8 11 X Candidate B 2 X X Candidate C 12 14 14 Candidate D 11 13 20

For more detailed information see Stanford Encyclopedia of Philosophy "Voting Methods" at http://plato.stanford.edu/entries/voting-methods/#3.1 and Voting Procedures, Stevens J. Brams and Peters C. Fishburn at http://www.eecs.harvard.edu/cs286r/courses/fall11/papers/Brams-Fishburn02.pdf.

All of this does not take into account the possibility of other forms of manipulation through strategic voting or the "Arrow" effect of a candidate (or party) that syphons off a portion of the vote from a candidate. Those will have to be the topic of another time.