Some limits of the median voter theorem

General elections are coming in France, and parties are campaigning.

Unprecedented situation: 5 big candidates. Meaning: 5 candidates whose probability to gather at least 10% of the votes is significant – and it is unlikely any of them exceed 30% as a result. These candidates range as follows: radical left, left wing, center, right wing, extreme right.

According to the median voter theorem, the candidate that has the support of the median elector should win. In real life, this is a bit more complex.

Firstly, there are propositions that transcend traditional borders. Basic income is one such example, it has proponents and opponents in every party in France. It cannot be reduced to a left vs right simple divide.

Secondly, there are more than one dimension to consider. Let’s take for instance security and health. Traditional right wing parties have a tougher stance on security and they want to privatise the healthcare system,  whereas left wing parties have a softer approach to security and want to make/keep healthcare public. Well, what is the median voter actually wants tough security and public healthcare? Who is he going to vote for?

If the utility of public healthcare is more important for the median voter (MV) than the utility of tougher security, then the MV will vote for the left wing candidate. The latter may then be misguided into thinking they were elected for their programme as a whole – which is not the case, since the MV actually favoured not softer but tougher security. The only way to overcome this issue would to have two distinct votes: one on healthcare and one on security.

Thirdly, utility can be complex and have non-linear characteristics, due to heterogeneity of preferences. Some people give extreme importance to issues other people couldn’t care less about, for instance environmental issues. People will vote for the same candidate for different reasons.

Fourthly, people have taboos. The fact that the National Front is in favour of reinstating death penalty or limiting women’s rights to abort will definitely prevent certains people from voting for them, even if they shared many other aspects of their programme more than they did other candidates’.

The problem then becomes a kind of super-paper-scissors-rock conundrum: there is no way to know who is going to win in advance, it is all down to the technicalities of the electoral process.

In France, general elections take place in two rounds. During the first round, people can vote for any of the candidates that were allowed to run – the condition being to gather 500 signatures from town mayors. The two candidates with the best scores make it to the second round, except if the first one exceeds 50% of the vote and is therefore elected straight away. Otherwise, the second round is there to make sure one of the candidate will get more than 50% of the votes (say, X%) and the other one will therefore get (100-X)%.

To predict the winner, you therefore need to consider who can make it to the second round then beat their opponent. It is quite ofter that the candidate who arrived first during the first round gets beaten because of alliances between their opponent and other parties.

If the French general elections used the British or American system, which is “first pass the post” in every district or constituency, the result would likely be different. But still, it couldn’t be explained on the basis of the MV theorem. The reason behind the choice of a candidate would instead be that the candidate had attributes whose utility was greater to many voters, compared to the attributes of other candidates.

And again, these attributes may differ from one voter to another.


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