In a couple of days France will be taking part in the 2017 Presidential elections, after which a new president will be chosen for 5 years. The electoral process in these elections is unlike how elections work in the United Kingdom or the United States. The elections in France are done in two polls. In the first poll, voters can cast a vote for any candidate from a number of candidates. In the second poll, two weeks after the first polls, voters must choose between the two most successful candidates from the first round.

For anyone remembering the UK 2015 General Election the French election system may be appealing. In the UK, in 2015, the Conservative Party won the majority of seats in the House of Commons, but only got 36.1 percent of the popular vote. This came as a shock as many would have preferred Labour over The Conservatives even though Labour wasn’t their first choice. For instance, most Green voters (about 4% of the voters) would have backed Labour (30%).

The reason why the Conservatives got the majority of the votes is something called “First-Past-the-Post“. Loosely speaking in a First-Past-The-Post voting system whoever gets the majority of the votes wins. So for instance if the Conservatives get 45% of the votes, Liberal Democrats 20% and Labour 35%, then the Conservatives won. Plus in the UK, elections are held in every constituency (one seat for each constituency) and the results vary between constituencies. So in practice First-Past-The-Post does not mean gaining all the seats but rather it tends to amplify the success of the party which came first in the polls.

The French electoral system with two rounds promises to improve on First-Past-The-Post by avoiding the least desirable option. In principle if there is a candidate who is the least wanted choice by a majority of the people, even if he/she passes the first round, he/she will lose in the second. Anyone who has been following the French election closely will have heard this claim.

My question is: “Does the French electoral system prevent undesirable candidates in general?” The answer is no, but it does improve.

## What is the limitation with First-Past-the-Post?

Let’s imagine three hypothetical candidates: L, C and R. Now have a look at the following table:

1st choice | 2nd choice | 3rd choice | |
---|---|---|---|

L voter | L | C | R |

C voter | C | L | R |

R voter | R | C | L |

In reality not all supporters of a candidate or party will have the same 2nd, 3rd, … choices but we will suppose so for simplicity sake.

And suppose the following polls: 45% for R, 35% for L and 20% for C. Using First-Past-the-Post R wins with only 45 percent of the vote. R did not need an absolute majority to win.

Let’s analyse what this result means in terms of voter preferences:

- R is the 1st choice for 45% of voters
- R is the 2nd choice for 0% of voters
- R is the 3rd (least preferred) choice for 55% of voters.

The electoral system ended up selecting the least-preferred candidate

Now imagine there is a second round like in France. In the second round, R will get 45% of the vote from R-voters, and L will get 35% from L-voters and 20% from C-voters. In other words, L would win the second round of the election with 55%. The winner:

- L is the 1st choice of 35% of voters.
- L is the 2nd choice of 20% of voters.
- L is the 3rd choice of 45% of voters.

So a two round electoral system would not elect the least-preferred candidate. The new winner is less least preferred, but it also less the most desired. If this is the desired behaviour than we should say that “*an electoral system should select the candidate that is least least preferred of all candidates*” (yes I put twice least on purpose). So does the French election system do a good job at preventing undesirable candidates? So far it seems so.

## The French electoral system still has the same limitation

I will illustrate why the French electoral does not necessarily solve the problem.

My new example needs 5 candidates so bear with me. Here are the new 5 candidates: C, L, M, R and N. And here is the preference table:

1st choice | 2nd choice | 3rd choice | 4th choice | 5th choice | |
---|---|---|---|---|---|

C voters | C | L | M | N | R |

L voters | L | M | C | R | N |

M voters | M | L | R | C | N |

R voters | R | M | L | N | C |

N voters | N | R | M | L | C |

Now the polls are: N and R with 25%, M with 20% and C and L with 15%. In a two round system we would get:

- In the first round M, C and L are eliminated and N with R goes to the second round.
- N gets in the second round 25% + 15% = 40% and R gets in the second round 25% + 15% + 20% = 60%

So R wins. Let analyse that result:

- R is the 1st choice for 25% of voters
- R is the 2nd choice for 25% of voters
- R is the 3rd choice for 20% of voters
- R is the 4th choice for 15% of voters
- R is the 5th choice for 15% of voters

Is there anyone less least desirable? What about M?

- M is the first choice for 20% of voters.
- M is the second choice for 15 + 25 = 40% of voters.
- M is the third choice for 15 + 25 = 40% of voters.
- M is the fourth choice for 0% of voters.
- M is the fifth choice for 0% of voters.

So the French electoral system does not produce the best choices (if we are seeking the least least desirable candidates, of course).