The Meaning of Majority: Is the Majority Right? (Column 69)

With God’s help

In column 66 I dealt with the question whether the majority determines. The conclusion was that in democracy it does, and I explained this by saying that the majority is a reasonable measure for determining the opinion of the public as a whole. Against the Platonic proposal of rule by philosophers, that is, giving extra weight to wise people, I argued that in democracy we are not seeking a correct decision but a decision that reflects the will of the public. The question whether the majority is a good criterion for reaching a correct decision remained open, since following the majority in democracy does not depend on that (we follow the majority whether it is right or not). As I promised there, I now come to discuss the second question: is the majority right?

As stated, Plato’s assumption was that a state’s decision-making ought to strive for truth, and within that framework he confronts the intuition that the majority is a measure of truth. His claim was that a minority of wise people comes closer to the truth than a large number of fools, and therefore he recommends rule by philosophers. Is the democratic intuition indeed mistaken? Are there really better algorithms than following the majority for arriving at the truth? We are speaking about contexts in which the goal of the decision is truth, as in some professional decision (judges, judges on a religious court, some scientific or strategic decision, and the like). In addition, at least for the sake of this discussion, I will assume that there exists some sort of wisdom or specific skill that is relevant to the decision at hand.

What information do we have about the discussants?

The answer to this question is of course not simple, and it depends on the information we have about those involved in the discussion. I assume we will agree that on a question requiring knowledge and skill (such as a scientific question), a few experts will arrive at a better answer than a broad public of laymen.[1] And what if there is a dispute among experts? In such a case, is the majority a measure of truth? The author of Sefer HaChinukh claims that even in this case the answer is yes. In commandment 78 (the commandment of to incline after the majority, in a religious court) he writes:

And the choice to follow the majority, in my view, applies when the two disputing sides are equally versed in the wisdom of the Torah; for it cannot be said that a small group of sages should outweigh a large group of ignoramuses, even if they were as numerous as those who left Egypt. But when the wisdom is equal, or nearly so, the Torah informs us that a greater number of opinions will always agree with the truth more than a smaller number. And whether they agree with the truth or do not agree with it, in the opinion of the one listening, reason dictates that we should not depart from the path of the majority. And what I say—that following the majority always applies when the two disputing sides are equal in their grasp of the true wisdom—is so everywhere except with the Sanhedrin. With them, we do not scrutinize, when they disagree, which side knows more; rather, we always act in accordance with the view of their majority. The reason is that their number was prescribed by the Torah, and it is as though the Torah explicitly commanded: after the majority of these, you shall conduct all your affairs. Another reason is that they were all great sages.

First, it is important to understand that in a religious court the goal is to get as close as possible to halakhic truth (this is not a democratic majority based on rights, but a majority within a debate among experts trying to reach the correct answer). At the beginning of his remarks, Sefer HaChinukh argues that even among experts it is not always correct to follow the majority. Even among professionals, a wise minority outweighs a less wise majority. But when the level of wisdom is similar, then his claim is that the majority is a good measure of truth. This of course does not mean that the majority is always right or necessarily right, but rather that if we are looking for a uniform and a priori measure of truth, the majority is a better candidate than the minority.

And what about a situation in which we have no information about the discussants? Or if we have no clear measures of their wisdom and intelligence in the field under discussion? For example, in disagreements within a religious court, when we have no clear measure by which to determine the judges’ level of wisdom (we are all ignoramuses, or postmodernists who do not believe in more or less wisdom), is it correct in such cases to follow the majority? Another example: when there are disagreements among army officers regarding the correct military course of action. In most cases it is difficult to determine the level of military wisdom of those involved in the debate (one of them may have brilliant ideas that the others did not think of. But in a case where no one is persuaded by another’s proposals and sees in them a brilliant solution to the problem, it is hard to determine who is wiser and whose proposal ought to be followed). In such a case, is it correct to follow the majority? True, even in cases like these there is room to decide that, in the absence of a clear way to get closer to the truth, we will adopt the majority as an alternative criterion. Not because it will necessarily bring us closer to the truth, but because in the absence of another criterion it is only natural to adopt it (why? Just because! What other criterion do we have?!). But if we nevertheless want to maximize the chance of arriving at the truth, is the majority the right algorithm?

This question will become clearer through a somewhat different discussion.

Arguments from the distribution of opinions: religious/secular and right/left

In Dawkins’s books, as well as on various atheist websites, you will find studies (in my view it is more accurate to call them surveys) showing that, on average, educated people tend toward atheism, and believers are less educated.[2] The same applies to the Right and the Left. Although several explanations have been suggested for this that are unrelated to the question of which side is more correct, it is very common to present these data as an argument in favor of the Left and secularism. If the wise and the educated (I will not enter here into the question of the relation between education and wisdom) support one position, while the less educated and less wise support another, then the first position is probably correct. What is assumed here is that educated people are more likely to hit the truth. At first glance this seems very plausible, does it not?

A side note: the relevance of a majority without discussion

In several halakhic sources a principle is cited, based on reason, that there is no obligation to follow the majority except where the majority and minority have argued the matter among themselves. Thus, for example, Rashba writes in a responsum (vol. 5, sec. 126) regarding a religious court, and also in another responsum (vol. 3, sec. 304) in which he deals with voting by public representatives (the point is repeated as well in Maharit, responsa, vol. 1, sec. 95, and elsewhere). True, one might distinguish between a vote on a question of Jewish law and a vote on a democratic question, since the reasoning deals mainly with the minority’s right to have its opinion heard. But even in a vote on a question of Jewish law, where one is seeking the truth and not an expression of the right of the individual voter (see column 66 and here at the beginning of my remarks), the same logic exists. If the minority and the majority did not debate one another and did not hear each other’s arguments, then the majority’s decision carries no weight. Nor is it any indication of truth. It may be that if they had heard the minority’s arguments they would have been persuaded. Indeed, on that basis several halakhic decisors wrote[3] that in a dispute among different decisors on a question of Jewish law we do not follow the majority, because they did not debate with one another (unlike judges on a religious court, concerning whom it was said to incline after the majority (“incline after the majority”)). Here this is a clear example of a “vote” whose purpose is truth, and nevertheless one does not follow the majority if it did not sit and deliberate with the minority.

The same should be taken into account in surveys of the sort discussed here: the sides are not really sitting and discussing the matter with one another (and certainly there is no real listening to the different arguments. On the contrary, there is usually absolute mutual contempt). In such a case it is difficult to decide in favor of the majority merely because it is the majority. Only when all the information, arguments, and reasons are available to everyone, and nevertheless one side has a majority, does that majority have any weight.

But for the sake of discussion, let us nevertheless assume that everyone arrived at their conclusions after hearing all the arguments and weighing them. What is the meaning of the majority in such a case? Does it constitute any indication of truth?

Usually the minority is right

My initial anarchistic intuition is that in every dispute both sides are wrong unless proven otherwise. But even if the two views being pitted against one another are the only possible ones (so that it cannot be that both sides are wrong), then usually the minority is right. I even had a nice little argument for this. Surely there is a pyramid of intelligence, meaning that in an ordinary group of people there are few wise people and many more fools. So if there is a public dispute about some issue that requires wisdom, it is reasonable to assume that the minority is the view of the wise, while the majority is probably just most people (which of course consists of the more foolish). Therefore, in every dispute the starting assumption should be that the minority is right.

The conclusion is that in disputes about faith or about Right and Left, if the minority among the wise advocates faith and religiosity, it is probably right. Beni Moshe suggested a more complex model. He claims that, according to his impression, the distribution is that the lowest four deciles are right-wing/religious, the next five are left-wing/secular, and the top decile is again right-wing/religious. In the following sections I will suggest a model that explains this proposal.

Two hidden assumptions

On second thought, there are two assumptions underlying my intuition, and at least one of them is unreasonable: a. We are assuming here that the distribution is homogeneous (that is, that the minority is made up of the minority of wise people, and the majority is made up of the majority of fools). Could it not be that both camps are each composed of both wise people and fools? b. The distribution of wisdom has the shape of a pyramid, with a broad base and a narrow top.

The first assumption actually sounds fairly reasonable to me. It is of course not necessary, but at least in the absence of other information I would be willing to adopt it. If we are dealing with a decision that requires wisdom, and we have a distribution of people along the axis of that particular wisdom, then it is reasonable to think that people with the same capacity and the same intelligence will arrive at the same answer. It is therefore only natural to conclude that the minority and the majority are segmented by intelligence. You may ask: what about the distribution between Right and Left, religious and secular, or socialism and capitalism? First, it may be that even in these disputes this is indeed the case. Second, these are disputes that depend on values as well and not only on facts and wisdom, and therefore it is less reasonable there to assume that the segmentation is specifically by intelligence.

But the second assumption is plainly problematic. For just as there is a minority that is very wise, there is also a small minority that is very foolish. The majority lies somewhere in between those two extremes. In other words, the distribution of wisdom is not a pyramid but something like a symmetric Gaussian. Here is an example of a Gaussian distribution of grades on some test:

The vertical axis describes the number of people who received the grade marked on the horizontal axis.[4] The average grade is 70 (400 people received it. In the Gaussian case, the average grade is also the grade that occurs in the highest number of cases), and around it there is a symmetric distribution of grades in the group. Most examinees receive grades between 60 and 80 (one standard deviation to the right and left of the average), and few receive higher and lower grades. The more extreme the grade, the fewer people received it. If we treat this graph as the distribution of wisdom among our discussants (we gave them an intelligence test, and this is the distribution of the results), you can immediately see that there are few wise people and few fools, and even fewer very wise people and very foolish people. The majority are found, at better or worse levels, somewhere in the middle.

If so, even if we adopt assumption a, that the distribution is homogeneous and segmented by intelligence, we will have to assume that the minority belongs to one of the extremes, while the majority is everyone else. If we assume that in the military dispute 30% of the officers propose course of action X and the remaining 70% propose Y, we still have two interpretive possibilities: a. The minority consists of those who would receive a score above 80, and the majority is all those who would score below that. b. The minority consists of those who would score below 60, and the majority is all those who would score above. Thus, even if we adopt assumption a, the question whether the minority is right or the majority is right (that is, whether to do X or Y) remains open, and this is because assumption b is not correct. The conclusion is that although there is no reason to assume that the majority is right, on the other hand there is also no reason to assume that it is wrong. So half our anarchistic desire is in our hands. But do not worry—we are only halfway there.

Truncated Gaussian

Now think of a situation in which the discussion is taking place among students in a university biology department, for example. A reasonable assumption is that there are no complete fools there, since they passed a reasonable screening process when admitted to study (a psychometric exam and matriculation grades). The meaning of this is that our Gaussian is truncated from below. A reasonable model will assume that the composition of the class is described by the right-hand part of the Gaussian (from some score upward). In such a situation we have the few wise people, but not the few fools. True, the greatest sages in the universe are not necessarily in that classroom, but it is clear that the distribution of intelligence there does not look like that of the population at large. The screening is directional (upward), and it breaks the symmetry. The result resembles the pyramid distribution from which I started in my anarchistic phase, and therefore for the purposes of the discussion we may assume that this is a Gaussian truncated from below. Even if this is a simplistic model, it is still clear that in such a situation the distribution is no longer symmetric, and therefore the conclusion, in the case of a dispute among students in a biology department, or in a dispute among academics (assuming that there too there is screening that truncates the distribution from below), is that the minority is right. Within a population that has been screened upward, there is much logic to the claim that the minority is right. By contrast, in a population screened downward, the majority is right.[5]

Application to the education-faith surveys

I will now apply this to the claims about the superiority of the Left and the secular among the educated, and you will see what surprising result emerges. The basic claim is not that all educated people are left-wing and secular and all uneducated people are right-wing and religious, but rather that among the educated the majority is left-wing/secular, and among the uneducated the majority is right-wing/religious. In order to analyze these results and draw conclusions from them, let us first look at the group of the educated. It is described by a Gaussian truncated from below, and therefore in this case it is specifically the minority that is right (because in such a Gaussian the minority is the wiser group). Let me just remind you that among the educated the minority supports the Right and faith. The conclusion is that from an analysis of the results among the educated, it would be more correct to infer in favor of the right-wing religious outlook. Now let us turn to the second group, that of the less educated. Here the Gaussian is truncated from above, and therefore in such a group it is specifically the majority that is right (because the majority are on the wiser side). But here the majority is right-wing/religious, and therefore the conclusion again is that the right-wing religious outlook is the correct one. That is, from both sides of the equation it emerges that these data actually lead to the religious-right conclusion, and not as people tend to think at first glance (it may be that the first glance is left-wing/secular, that is, less intelligent).

One can of course dispute quite a few of the foundations of this analysis, and there is no doubt that it contains a great deal of oversimplification (to the point of absurdity). I will gladly join those who laugh at such analyses, and in fact that is precisely my purpose in writing this. Still, one thing is clear: drawing conclusions from the survey results cited above (about the relation between education and faith) is far more simplistic than the analysis I have offered here. Therefore this simplistic analysis is enough to show that the left-wing/secular conclusion certainly does not follow from such data, and is not even supported by them. In other words, if anyone is at all trying to infer some conclusion from these data (as stated, this is not recommended), then at most one can derive from them the right-wing/religious conclusion. So we have more or less returned to Beni Moshe’s claim cited above.

 

Note: Long after the writing and publication of this column, in a lesson I gave on the topic of majority (see the video lessons, P.T., Majority, dated 18.10.2018), I received from Arik Peshdatski an objection to the reasoning I had offered regarding the claim that usually the minority is right. The objection is correct, and I therefore amend the point here.

He argued that even if we accept the model of a pyramid of talents, or a state of a truncated Gaussian (that is, where there are few wise people and many fools), the majority is still right. We are dealing with a binary question, one that has a yes-or-no answer. Contrary to my assumption here, the fools do not give the wrong answer but a random one. Therefore, on average, half of them will answer correctly and half will not. By contrast, among the wise, most will answer correctly. Therefore, all in all, there will be a majority for the correct answer. Of course this too can be challenged, but it is clear that the basic model I proposed here is not correct. He is right!

 

[1] In column 62 I touched a bit on the wisdom of crowds, and explained why it is not relevant to questions of our kind.

[2] See here and here (around note 34) for a more balanced presentation. Additional sources regarding evolution and creationism are brought in the introduction to my book God Plays Dice and in note 2 there.

In a discussion like this it is important to distinguish between surveys that show that believers are less educated (or that atheists are more educated) and surveys that show that the less educated are believers (or that the more educated are atheists). The direction of the correlation has a different significance for our purposes. In the discussion that follows, I will assume results that express the percentage of believers/atheists according to level of education, and not the other way around.

[3] See Shakh on Choshen Mishpat, sec. 25, s.k. 19, and Netivot HaMishpat (Novellae) there, s.k. 18, and Urim there, sec. 20, and many others.

[4] The description is simplistic because I did not provide the resolution. Try to think about how many examinees there were in total and you will see that there is no way to know that from the graph. But for our purposes here, it is enough.

[5] A question for thought: even in classes of biology students, should we expect a Gaussian distribution of grades (with two tails in both directions)? If so, does that contradict the description I gave here?