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What does Brexit mean? ‘Majority judgment’ can solve the puzzle

Something is going badly wrong in elections and referenda across the world. The culprit is an age-old voting system that prevents voters from fully expressing themselves.

Iain McLean Rida Laraki
27 February 2019

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Screenshot: Demonstrator at a “gilets jaunes” demonstration near Narbonne in December 2018."We vote strategically in the first round, we vote against on the second round, blank votes are thrown in the trash. What if we finally vote by choice with Majority Judgment?”

On roundabouts across France, the “gilets jaunes” (yellow vests) have galvanised French politics. What started as a protest against fuel prices has morphed into a challenge to the presidency of Emmanuel Macron – and the voting method by which he was elected.

Indeed, one of the rallying cries of the protesters has been that Macron should resign from the presidency because he represents only 24.01% of the electorate (his share of the vote in the first round of the 2017 presidential election). It’s true that he won the second by a decisive margin with 66.10% of the vote – against Marine Le Pen’s 33.90%. But, 4,085,724 million of the 31,381,603 votes cast were either blank or invalid. And overall, 1.5 million fewer people took part in the second round than did in the first.

Electoral reform is now a key demand of the yellow vests. They want blank votes to be officially recognized. And they want to introduce what they are calling “Citizens’ Initiative Referendum” (or RIC) or the ability to put any policy proposal that garners 700,000 signatures to a popular vote. The RIC – and the intrinsic tension between representative democracy and direct democracy – is the subject of passionate debate across the country. According to the latest public opinion polls, nearly 80% of French voters support introducing RIC. But many commentators – including president Macron – warn of how easy it is to manipulate public opinion through referenda.

Their number one case in point is what happened in the UK in 2016 with the Brexit vote.

If one thing is clear from the last two years and a half in British political life, it is that the politicians campaigning for “leave” did not agree as to what Brexit would actually mean.

This in turn made the binary yes/no voting system of the 2016 referendum particularly inappropriate because it counted votes of different meanings as if they were the same. “Remain” may have had a more or less a clear meaning but different voters understood “leave” in different ways.

Similarly in the parliamentary vote of January 15 2019, the 432 MPs who voted against Theresa May’s deal meant different things by their “no.” Some hope for a second referendum, others think the UK can obtain a better deal, and some MPs want a “no deal”.

How best, then, to represent what people really want with Brexit and come to a consensual solution?

The first thing to do is to let voters express their opinions about all feasible options of which there are at least nine:

- Revoke Art 50 notice and remain

- 2nd referendum

- Extend Art 50 and call a general election to change government

- Extend Art 50 for current government to renegotiate with EU

- May's current deal

- May's deal with a time limited backstop

- May's deal with a unilateral backstop exit mechanism

- New deal with permanent customs union with EU

- New deal with single market membership

If people have to use traditional plurality voting, they choose only one option among the above nine (and so implicitly reject all other options). There is then, no guarantee that the winning option will be the one closest to the general consensus. It may be, for example, that one option wins 20% of the vote, but 80% of the voters are against it and feel terrible about it.

Majority Judgement

There is another way. We propose a new method: majority judgment (MJ). It makes the establishment of consensus more likely because voters are offered the opportunity to give their opinion on each of the options on offer, by evaluating them on a scale of five grades such as Very Good, Fairly Good, Acceptable, Fairly Bad, and Very Bad. Each voter is free to assign the grades she wishes, and the option best evaluated by a majority wins!

In the case of Brexit and the nine options listed above, a voter can evaluate one of the options (for example “New deal with single market membership”) as very good, two fairly good, two acceptable, three fairly bad and one (for example “no deal”) very bad. Another voter will have a different opinion and believe that “no deal” is very good, but think that all other options are very bad.

To understand how majority judgement works and how using it could radically change the outcome of an election, let us see how it would have worked in the US 2016 presidential election where we are lucky to have opinion survey data that correspond to MJ grades. Next we will use a GB recent survey to illustrate what it teaches us about Brexit.

Who were the consensus candidates in the US 2016 presidential election?

During the 2016 presidential primaries in the US, the respected pollster Pew Research Center asked 1,787 randomly chosen voters to answer the following question[1]:

Regardless of who you currently support, I’d like to know what kind of president you think each of the following would be if elected in November 2016… [D]o you think (he/she) would be a Great, Good, Average, Poor, or Terrible president. The results were as follows:

Pew Political Survey

Table 1: Pew Research Center, Poll of 1787 voters, 17-27 March 2016.

When we apply majority judgment we see a huge difference with plurality voting. The result for a candidate with majority judgement is not the percentage of people who voted for that candidate, but a precise picture of how many voters think she is excellent, how many think she is good, etc.

To calculate the MJ rank order from the evaluations in Table 1, start from each end of the spectrum and add up the percentages until you have a majority of voters’ opinions. Taking John Kasich as an example, 5% believe he is Great, 5+28=33% that he is Good or better, and 33+39=72% (a majority) that he is Average or better. So Kasich’s majority-grade is Average.

A similar calculation shows that the Clinton, Sanders, and Cruz majority-grades are also Average, and that the Trump majority-grade is Poor. To determine the MJ ranking among the four who have the same majority-grade (in this case Average), we calculate a number we have called the “gauge.” In Kasich’s case, 5+28=33% evaluated him higher than average and 13+7+9=29% rated him below average. Because the larger share is on the positive side (33>29), Kasich’s gauge is +33%. The Sanders gauge is -39%, for Cruz and Clinton the gauge is respectively, -40% and -47%. Thus, the two best candidates according to MJ are Kasich and Sanders. The rules to determine MJ ranking are the logical consequence of some basic democratic principles[2].

Thus, the US voting process failed because it designated the two “worst” candidates for each party – Clinton and Trump – while the two “best” candidates were Kasich and Sanders. When voters are able to express their evaluations of every candidate – the good and the bad – the results are turned upside-down from those with plurality voting.[3]

Which Brexit option is the best with majority judgment?

On January 30-31 2019, YouGov and The Times commissioned a poll from another respected pollster, NatCen Social Research. They asked 1650 GB adults respondents across the UK to evaluate a range of potential Brexit outcomes. Voters were asked: Do you think it would be a good or bad outcome if:

NatCen Social Research: YouGov and The Times survey 31 January 2019

Table 2: NatCen Social Research: YouGov and The Times survey 31 January 2019

We are lucky that this opinion survey data correspond to MJ grades. From table 2, we can see that B (“remain”) and C (“no deal”) are judged very bad outcomes by 36% of voters. Option A (“government deal”), in contrast, may have only 3% of very good grades but overall it is less negatively evaluated (only 16% of very bad evaluations). According to MJ, option A (“government deal”) is the best of the three options, with majority-grade an acceptable compromise, while the others are judged by the majority: fairly bad.

Interestingly, those results are very different from that of another YouGov survey carried out by NatCen Social Research 13 days earlier, on 18 January 2019, that asked 1754 GB adult respondents to choose between options with the question: If there were a referendum, and the choice were “to stay in the EU” or “leave on the terms recently negotiated by the Government” or “without a deal”, how would you vote? (see Table 3 for the results)

NatCen Social Research: YouGov survey 18 January 2019

Table 3: NatCen Social Research: YouGov survey 18 January 2019

Here, the best option according to plurality voting is “remain in the EU” because leave votes are split between A (deal) and C (no deal).

The fact is that a poll is not a true referendum, and there are more options to consider. Happily, the poll in table 2 asked voters their opinion about a fourth option:

D: Imagine that the final outcome of Brexit was Britain leaving the European Union with an alternative deal that included remaining in the single market and customs union.

NatCen Social Research: YouGov and The Times survey 31 January

Table 4: as Table 2

According to MJ, option D is better than A (government deal). Would this “alternative deal” be accepted by the EU? We don’t know. But the analysis shows, we hope, that letting voters express more precisely their opinions about all feasible options allows them to reach a better decision.

How majority judgment can improve democracy?

Something is going badly wrong in elections and referenda across the world. The culprit, as we see it, is an age-old voting system that prevents voters from fully expressing themselves.

Majority judgment, the outcome of an elaborate mathematical theory of voting, was designed to satisfy some basic democratic principles. In particular:

- With MJ, voters can better express themselves. They are able to give more precisely their opinion about all options/candidates, and are not limited to backing just one to the exclusion of all others in the running.

- With MJ, vote splitting is neutralised. All candidates or options with similar profiles can compete without impinging on each other’s chances because voters can give a good evaluation to all options/candidates they like.

- With MJ, there is no need for voters to protest by abstaining or voting blank because they can negatively judge all of alternatives they dislike.

- Because of its mathematical design, MJ is the most difficult system to manipulate: blocs of voters who exaggerate the grades they give beyond their true opinions have a limited influence on the results.

- Most importantly, MJ elects the candidate (or chooses the referendum option) highest in the esteem of the electorate.

Majority judgment in practice

MJ was designed only recently by Michel Balinski and one of us – our book on the subject was published by MIT Press in 2011[4]. Nevertheless, it is already in use.

For example, the British Academy, which is the UK's academy for humanities and social sciences, uses MJ for electing new fellows. “LaPrimaire.org” (a French political initiative to organize on an online primary election open to all citizens) used MJ in 2016 to nominate its candidate for the 2017 presidential elections with upwards of 32.625 participants. “Generation.s”, a French political movement, uses MJ in its internal decision making and “Yashar”, a Israeli party emphasizing participatory direct democracy, is looking into MJ as a way to improve its processes.

Yellow vest protesters, too, are calling for MJ. In the “Grand National Debate” launched by president Macron this January to solicit citizens’ ideas, 197 proposals – to date – recommend that majority judgment be adopted as the country’s voting system.

What to do now?

As the UK faces the most important decision for its future since the Second World War, is it not time to think about how that decision is going to be made? Voting systems, after all, are human constructs and can be changed for ones that have been shown to work better.

Dedication On February 4 2019, Michel Balinski, co-inventor of majority judgment, passed away at 85 years old. He was a pioneer in designing fair well-founded and practical voting systems. We will miss him.

Thanks: all our gratitude to Maria Balinska and Michael Gordon for useful comments


[1] Pew Research Center. 2016.``March 2016 political survey.''

[2] Balinski M. and R. Laraki (2011) «Majority Judgment: Measuring Ranking and Electing». MIT Press.

[3] For a detailed analysis of this US example, see here.

[4] Balinski M. and R. Laraki (2011) «Majority Judgment: Measuring Ranking and Electing». MIT Press.

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