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 Parado****x**

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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