The Mathematics of Dissent

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October 2011
Ivan Obolensky

Is it better to be led by a group or an individual? I have heard it said that a group survives better if run by an individual leader rather than by a committee. I have also heard that humanity is subject to “Groupthink” and the “Herd Mentality”. So which is it?

On the individual side, using history as a guide, I can find some very bright, resourceful leaders (Churchill, Washington) and I can sure find some that would have been better left untried (Nero, Caligula). So the individual has a track record of some success, but it has not been 100%, by any measure.

On the group side, there have not been many countries that have been ruled by a group although ancient Sparta and Athens come to mind. My personal favorite has always been the Persian councils to Cyrus the Great of around 600 BC. According to Strabo, they never undertook serious discussions of policy when sober, although they had the foresight to review and revise their decisions the next morning. This was not some puny, little country they helped run, but an empire of some 40 million souls that stretched from Egypt all the way to India. It, too, passed away only to be eclipsed by Rome in terms of size and complexity hundreds of years later. In review, leadership by groups has not been so brilliant either.1

Surely if the individual acting alone is smarter, then would we not have more of a propensity to be “loners” rather than being the social people that we are? Since humanity has made it this far, perhaps the idea of a group being wise is worth looking into.

In the early 1900s, many county fairs in England had contests whereby the individual would try to guess the weight of a slaughtered and fully dressed ox for the cost of a few pence, and the individual who guessed the closest answer would be the winner of the entire ox. There were often several hundred entries and although there was only one winner, if one took the average of all the answers submitted, the group as a whole would generally be much closer to the correct weight than the majority of the individuals. In the particular case noted by Sir Francis Galton, the Victorian polymath and statistician, the average was within one pound of the correct weight, or within a fraction of 1%.2

I have seen a similar experiment of guessing the number of jelly beans in a jar many times. I was always surprised at how close the group came to the correct answer and that the accuracy increased as the number of participants grew. I wondered why this was so and came across an interesting explanation.

Let’s take the case of the weight of the ox.

Firstly, each participant bought a ticket so that each had a vested interest in guessing as accurately as possible the correct weight.

Secondly, each guess could be considered to be a combination of two elements: the unknown true value and an error term that is either plus, or minus.

As an example, if the correct value is 1047 pounds for the ox and the participant guesses it to be 1100, then the ticket could be represented as the correct weight (1047), plus an error term of +53 since the guesser is 53 pounds too high. In a similar way, if the participant guessed 1000 pounds. the number could be thought to be the correct weight of 1047 with an error term of -47, since this guesser is 47 pounds too low. The idea is that the greater the number of participants and guesses, the more error terms there are and the greater likelihood that the error terms will cancel each other out. Of course none of the participants know the true value beforehand, but it is quite likely that the guesses will be somewhere around the correct value with about half guessing too high and half guessing too low.

In the modern age, this same mathematical explanation is used to explain why stock markets are storehouses of information. If one considers each trade in the stock market to be similar to the weight contest only happening every second, one can see that prices record the consensus on a near instantaneous basis as information is looked at and considered.

Which leads to the question: How come a stock trades at a particular price?

Firstly, price is a real-world property. It may change moment to moment, but each consecutive price is a record of actual shares trading at that price at that exact time. The reported price is only incorrect if the actual price reported is different from the price it traded at. Value on the other hand is a subjective measure. Price and value can appear far apart depending on who you are and what you think.

Unlike the seven hundred participants in the contest who buy a ticket for the ox contest, stock markets have thousands of participants, each with a vested interest in being right. It is similar to a tug-of-war: one side is convinced the price is too high, and the other side is certain that it is too low. But like the weight of the ox contest observed by Galton, not everyone is right but given a large number of participants, the error terms (the pluses and minuses) generally cancel each other out such that the price that is recorded with each sale is thought to be a correct and true value of the underlying security at that moment in time.

This works well during times when there is a balance between buyers and sellers, but what happens when everyone wants to buy or sell and there is a large imbalance?

From the mathematics above, groups appear to be more accurate in their assessments when there is a balance of opinion spread across both sides of an issue. This would seem to make sense. If the error terms have to cancel each other out, there must be positive and negative views. If there are only minus error terms because everyone believes prices are too low, then the consensus may be in error. Once this error comes to light after an extended period, prices can adjust very quickly creating panics and stunning advances as these imbalances are corrected. A group can get it wrong as badly as any individual. It would seem that when everyone has the same thinking about an issue it might be worth a look at the opposite side.

In the case of the toxic mortgage-backed securities debacle not only was the consensus wrong, but there was no perceived risk because participants had no vested interest in the outcome.

Many of the original issuers of mortgages sold them as soon as possible to firms that packaged them. Those firms were convinced that credit quality did not particularly matter, provided it was a mortgage on real property because a complex mathematical algorithm that linked credit scores and defaults said so. The algorithm (called a Gaussian copula function) turned out to be based on information that extended over too short a time to be meaningful, with alarming consequences.

So when a group contains many diverse points of view, decisions made by that group after a consensus is reached seem to be more accurate than when the entire group believes in the same thing. There is a market adage that crowds are wrong at extremes of either fear or greed. This would seem to be borne out when diversity of opinion is lost.3

Part of the difficulty of government is found in exactly this discrepancy.

It would seem that when there are extremes of public sentiment, with the majority galvanized by fear and panic or by greed and lust for personal gain, irrational decisions are often made. The Peloponnesian War between Athens and Sparta from 431 BC to 404 BC, as described by Thucydides, illustrates how the Athenian democracy became so one-sided in its desire to be supreme over Sparta, that those who counseled caution were deemed unpatriotic and those that believed in circumspection, cowards. Athens went over a cliff and never recovered.4

Rome of 200 BC was also aware of the problem of governmental bodies such as the Senate becoming so locked in disagreement and divisiveness that nothing was done. Their solution was the election of a dictator. If a particularly troubling situation arose, a dictator was elected for the length of the emergency, usually six months, after which power was handed back to the Senate. The dictator’s power was absolute. One of the symbols of his power was the fasces or a bundle of wooden rods with an axe in the middle. We derive the modern word fascist from that symbol. During the later Republic of 50 BC, crisis grew upon crisis and the dictator became the preferred governmental solution until it became permanent under the title of “Emperor”. Dissent under the Emperor was not a healthy option.

Since then, the idea of the Roman Emperor was repeated in the monarchies that sprang up throughout European history until the eighteenth century. Many of them were replaced by some sort of representative government where dissent was more possible but even then it has not been encouraged, particularly by those in power. There is still a tendency to look to individual leaders in times of crisis, in the Roman tradition.

It is not that individuals and groups cannot make correct decisions, but when it comes to leadership, neither is infallible and both must be watched carefully by everyone. Groups are not inherently infallible. They contain many viewpoints that are often divisive, but without such points of view taken into account and consensus achieved it is not likely that correct decisions will be made.

From a mathematical perspective, if there are no plus and minus points of view, if there is insufficient participation, and if there is no vested interest in the outcome, it is not possible for the errors terms to cancel out and a better decision worked out. Even single leaders are wise to surround themselves with those who can and will offer contrary opinions. Dissent matters.


1 Durant, W. (1963). Our Oriental Heritage. New York, NY: Simon & Schuster.

2 Surowiecki, J. (2004). The Wisdom of Crowds: Why the Many are Smarter than the Few and How Collective Wisdom Shapes Business, Economies, Societies, and Nations. New York, NY: Doubleday.

3 Harris, L. (2003). Trading and Exchanges: Market Microstructure for Practitioners. New York, NY: Oxford University Press, USA.

4 Thucydides, Strassler, R. B., Crawley, R., & Hanson, V. D. (1996). The Landmark Thucydides: A Comprehensive Guide to the Peloponnesian War. New York, NY: Free Press.


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© 2011 Ivan Obolensky. All rights reserved. No part of this publication can be reproduced without the written permission from the author.

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