Beyond Condorcet winners: Majority Judgement

631 words

There have been rumours on Wikipedia that Michael Balinksi passed away this Februari 4th. If this is indeed the case, then my heartfelt sympathies go out to those around him. I never knew him personally, but he was an amazing scientist and I’ve used results of his more than once. He was quite a phenomenal speaker.

Today I want to talk about a single-winner voting system that Balinksi introduced with Rida Laraki. It is called majority judgement and it is so brilliant that it almost makes me wonder what voting theorists could have been doing both before and since then.

One big concept in social choice theory is majority rule: if most people think that A is better than B, then A should be the winner. Most multi-candidate voting systems generalize this in various ways, always preserving that if candidate A would beat every other candidate B, C, etc, in a pairwise competition, then A should win the election. If candidate A satisfies this criterium, we call A a Condorcet winner. The leading wisdom in social choice theory was that any good voting rule should let the Condorcet winner win (if it exists).

According to my informal sample from the Effective Altruism community, EA’s favourite voting system seems to be approval voting, which is one of these voting systems that generalizes majority rule to multiple candidates.

The genius of majority judgement is that it moves past the Condorcet winner paradigm and considers a perspective beyond that.

To illustrate, let’s assume we’re running an election among two candidates, the red candidate and the blue candidate, and every voter gets a ballot with for each candidate the options “Amazing”, “Good”, “Mediocre” and “Terrible” to express how good of a president they think the candidate would be. Let us for simplicity assume that our voting population consists of 5 groups, the A’s, B’s, C’s, D’s and E’s, and everyone in a group votes the exact same way. The outcome of the election is in the table below.

ABC
% of population402040
Red candidateAmazingMediocreTerrible
Blue candidateGoodTerribleAmazing

The red candidate wins the Condorcet-style election: 60% of the population prefers the red candidate over the blue candidate. But the blue candidate is clearly better: 80% of the population considers the blue candidate to be “Good” or better, while 60% of the population considers the red candidate to be “Mediocre” or worse.

Majority judgement is a voting rule designed to have the blue candidate win in the above election. The actual algorithm is a bit involved, but the first step is to compare the median vote: if at least 50% of the voters think the blue candidate is “Good” or better and at least 50% of the voters think that the red candidate is “Mediocre” or worse, than the blue candidate will win. If the median opinion is a tie, a more complicated tie-breaking rule is entered. ((The exact rule satisfies a number of optimality criteria and is the only rule to do so. For this post I want to skip the details.))

I think the concept is very elegant, and I believe that the outcomes really would be better than with a system that elects Condorcet winners. In a talk that Balinksi gave, which I was lucky enough to attend, he pointed out another advantage of the majority judgement rule: it allows voters to express what they think of the candidates. You wouldn’t be asking anyone to “vote for the lesser evil”, everyone can keep their conscience clear. Majority judgement admits a clear way of expressing frustration with both candidates: rank both of them very badly. It also helps that the different options are given in words instead of ranking by numbers, for the latter turns out to entice voters to rate their favourite candidate 10/10 and all others 0/10.

Post-scripts

  1. This excerpt from Walter Lippmann’s 1925 book The Phantom Public is so good.

    “But what in fact is an election? We call it an expression of the popular will. But is it? We go into a polling both and mark a cross on a piece of paper for one of two, or perhaps three or four names. Have e expressed our thoughts on the public policy of the United States? Presumably we have a number of thoughts on this and that with many buts and ifs and ors. Surely the cross on a piece of paper does not express them. It would take us hours to express our thoughts, and calling a vote the expression of our mind is an empty fiction.”

  2. I received a message from a reader who recommended this rebuttal of MJ. Most of it is pretty good, definitely worth checking out. As with every voting system, it is not hard to find bad inputs for it, which the linked article rightly does. There are some good comments about ways in which MJ does not behave as well under exaggeration of preferences as MJ’s proponents advertise.
    My main issue with the article is an aggregious misinterpretation right at the beginning about how scoring using words (bad, good, superb) would be the same as scoring using numbers (1, 2, 3).

    It seems that the authors of the piece missed one of the essential ideas about why MJ is worth considering. Namely, that averaging scores is a political act whose importance we should not overlook. The unique average of 1 and 3 is 2, but why would the average of “bad” and “superb” be “good”?

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