Mathematics of Voting: 4. Condorcet and Borda

Suppose we have the following situation:

    \[ \begin{array}{ccc|c}A&B&C&\mbox{Votes}\\  \hline 1&2&3&30,000\\  1&3&2&10,000\\  2&1&3&15,000\\  3&1&2&10,000\\  2&3&1&5,000\\  3&2&1&30,000 \end{array} \]

If we were to distribute preferences, we would first distribute 40,000 to A, 25,000 to B and 35,000 to C. Then B would be eliminated, and B‘s papers would be distributed to A and C: 15,000 to A and 10,000 to C, thus giving A the win with 55,000 votes to C‘s 45,000.

But this, it can be argued, is not really fair. If we examine the above ballot distribution, we see that a total of 55,000 voters have given B a higher preference than A, and a total of 55,000 voters have given B a higher preference than C. Since B is preferred to either A or C by a majority of voters, the win should really go to B.

This can be more easily seen by a slight rewriting of the above ballots:

    \[ \begin{array}{c@{\,>\,}c@{\,>\,}c|c} \multicolumn{3}{c|}{\mbox{Preferences}}&\mbox{Votes}\\ \hline  A&B&C&30,000\\  A&C&B&10,000\\  B&A&C&15,000\\  B&C&A&10,000\\  C&A&B&5,000\\  C&B&A&30,000 \end{array} \]

In such a case, where one candidate is preferred to all others individually by a majority of voters, that candidate is called a Condorcet winner, after the Marquis de Condorcet, Marie Jean Antoine Nicolas de Caritat, (1743 – 1794) who first described this type of situation in 1785. A Condorcet winner does not exist for all voting situations, but if there is a Condorcet winner, it seems reasonable to elect that particular candidate. Any voting system which chooses a Condorcet winner, if there is one, is called a Condorcet system.

One problem with the ranking “X is preferred to Y by a majority of voters” is that it is not transitive. For example:

    \[ \begin{array}{c@{\,>\,}c@{\,>\,}c|c} \multicolumn{3}{c|}{\mbox{Preferences}}&\mbox{Votes}\\  \hline A&B&C&33\%\\   C&A&B&34\%\\  B&C&A&33\% \end{array} \]

Here A is preferred to B by 67\% of B is preferred to C by 66\% of voters, and C is preferred to A by 66\% of voters. This is an example of a voting cycle. In such a case there is no Condorcet winner, and so, if a Condorcet method is being used to select the winner, some method must be used to “break” such cycles. We’ll look at a few of these later.

We also see that it is not necessarily fair to eliminate a candidate on the grounds of first preferences alone, as there may be good support for that candidate in the second and lower preferences. Indeed, the automatic elimination of a candidate on the grounds of first preferences alone is acknowledged to be a weakness in the IRV method.

Enter another Frenchman and early voting theorist, Jean-Charles de Borda (1733 – 1799), who devised a counting method, known as the Borda count, to take into account all preferences. Each candidate is assigned a numerical value based on the number of preferences received. If there are n candidates, each first preference counts n-1 and each second preference counts n-2 down to preference n which counts 0. In general, a preference i will count n-i. For example, in the situation given above, candidate A has received 40,000 first preferences, 20,000 second preferences and 40,000 third preferences, so the Borda value is

(40,000\times 2)+(20,000\times 1)=100,000.

The values for the other candidates can be similarly calculated:

B:\quad (25,000\times 2)+(60,000\times 1)=110,000

C:\quad (35,000\times 2)+(20,000\times 1)=90,000

The highest value is that of B, who wins.

Even if a Condorcet winner exists, that winner is not necessarily chosen by the Borda count. A simple example is provided by the following voting situation:

    \[ \begin{array}{c@{\,>\,}c@{\,>\,}c|c} \multicolumn{3}{c|}{\mbox{Preferences}}&\mbox{Votes}\\  \hline A&B&C&30\\   B&C&A&20\end{array} \]

The Borda counts values are

A:\;30\times 2=60

B:\;30+(20\times 2)=70

C:\;20

giving the win to B. However, on inspection there is a Condorcet winner in this case, and it is candidate A.

Nanson’s Method

The Borda count is by no means perfect as a general method for establishing a majority choice. If we consider the situation:

    \[ \begin{array}{ccc|c} A&B&C&\mbox{Votes}\\  \hline  1&2&3&60,000\\  2&1&3&2,000\\  3&1&2&33,000\\  3&2&1&2,000\\  2&3&1&3,000 \end{array} \]

we see that candidate A has a clear majority on first preferences alone, and so should win. If, however, we calculate the rank-order values, we obtain:

A:\;(60,000\times 2)+(5,000\times 1)=125,000

B:\;(35,000\times 2)+(62,000\times 1)=132,000

C:\;(5,000\times 2)+(33,000\times 1)=43,000

which would give the win to B.

Nanson’s method (named after E. J. Nanson, a professor of mathematics at the University of Melbourne, who described this method in 1882) is designed to be an all purpose method that will deal with situations like this.

To begin, the Borda counts of the candidates are calculated, and the average (the arithmetic mean) found. Any candidate whose value is less than the average is eliminated. In the situation described above, the average is

(125,000+132,000+43,000)/3=100,000

and so C would be eliminated.

The remaining candidates are then ranked as before, and the new Borda values calculated. We continue eliminating candidates whose values fall below the average, and ranking the remaining candidates, until one candidate is left.

Consider the situation as described above. After C has been eliminated, we rank A and B, writing 1 for a high preference and 2 for a lower preference:

    \[ \begin{array}{cc|c} A&B&\mbox{Votes}\\  \hline  1&2&60,000\\  2&1&2,000\\  2&1&33,000\\  2&1&2,000\\  1&2&3,000 \end{array} \]

Note that A and B have retained their orders with respect to each other – only the numbers used to indicate preferences have changed. The new rank-orders are 63,000 for A and 37,000 for B. Thus B is eliminated, and the winner is A.

There is another version of Nanson’s method, where instead of eliminating all candidates whose rank-order values are less than the average, only the candidate with the lowest rank-order value is eliminated at any stage – this is called Baldwin’s method. Both methods are Condorcet methods, in that they will always choose a Condorcet winner if one exists.

Nanson’s method has not been used for parliamentary elections, but it has been used for elections to the council of the University of Melbourne.

A problem with Condorcet

A major flaw with all Condorcet methods is that they suffer from vulnerability to irrelevant alternatives. Suppose A is preferred to B in the final rankings. It may be possible to include a third candidate X who does not change the rankings of A and B on the individual ballots, but which has the effect of changing the final rankings of A and B.

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