A democratic paradox

Democracy depends on people voting for their leaders, but there are many different voting systems in use, all of which aim for a fair result: first past the post; single transferable vote; alternative vote plus are three examples. All have advantages, all have disadvantages.

Many popular voting methods use a system which allows voters to select candidates in order of preference. Suppose there are three candidates, A, B and C. You would expect that if more voters preferred A to B, and more voters preferred B to C, then it must be true that more prefer A to C – and yet an 18th century French philosopher, the Marquis de Condorcet, showed that this isn’t necessarily true.

The Condorcet Paradox

Let’s take our three candidates, and let’s take 220 voters (this makes the numbers simple). Each voter places the candidates in order of preference: if a voter chooses the order A then B then C, let’s write this as A > B > C. There are six possible orderings available to the voter, and in the table below I’ve shown how many voters went for each possibility.

A > B > C B > C > A C > A > B A > C > B B > A > C C > B > A
40 60 40 40 20 20

If we take the first column, we can see that there are 40 people who prefer A to B, and then B to C, which means they also prefer A to C. Now, how many people in total prefer A to B? There are the 40 in column 1, another 40 in column 3, and 40 more in column 4: 120 in all. Let’s draw up another table showing the totals for all the possible pairings.

A > B B > C C > A B > A C > B A > C
120 120 120 100 100 100

So 120 prefer A to B, but only 100 prefer B to A – so A wins over B. 120 prefer B to C, only 100 prefer C to B – so B wins over C. And then the paradox: 120 prefer C to A, only 100 prefer A to C, so C wins over A. A beats B beats C beats A!

So who should win this election? We can solve this problem using the ….

…Single Transferable Vote

From the top table, we can see that as their first preference 80 voters chose A; 80 also chose B as their first preference, but only 60 chose C. So we eliminate C, and their votes are then transferred like this: the 40 voters who chose C > A > B have their vote transferred to their second choice, A, giving A 120 votes in all; the 20 voters who chose C > B > A have their vote transferred to their second choice, B, giving B 100 votes in all.

A is our new leader!

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