Influence of Social Networks and the majority opinion of the Jury Theorem / Majority Opinion and the Jury Theorem 
The great power of diversity is completely realized by using the opinion of the majority to make decisions. Michael Mauboussin
produces a very good demonstration of this with his students at Columbia Business School. Each year just before the Academy Awards are announced, a vote is made for those who believe they will be winners in each of the 12 categories in which prizes are awarded. Not only popular categories like "Best Actor" category, but more hidden as "Best Film Editing" or "best artistic direction."
In 2007, the average single correct answer was 5, 12. However, the average number of correct answers
for the entire group was 11 December !
Why is the most accurate in their answers so frequently? One reason may be illustrated by the history of the development of U.S. Consititución, and two of his most famous craftsmen, Benjamin Franklin and Thomas Jefferson.
Franklin and Jefferson both spent time in Paris before participating in the creation of the constitution, which were experienced in 1787. Both were involved in doscusiones with French intellectuals pirmera primarily responsible for the French constitution, which was completed in 1789. One such intelectales was Marquis de Condorcet.
Condorecet had begun his career as a mathematician, and in that time worked as an inspector of coins at the Mint of Paris. He was fascinated by the idea that mathematics can be used to support arguments for human rights and moral principles.
Condercet Franklin met with many times after I arrived in Paris, and was very impressed by the progress that Condorcet had reached its "social mathematics," indicating that "should be discussed." There was nothing even Scytho about it, but that all changed after the publication of Condorcet's essay on the application of probability analysis to the decision of majorities, published in 1785.
Franklin was clearly influenced by the ideas of Condorcet, in particular mathematical proof by now known as the "Condorcet Jury Theorem," theorem that is now considered one of the foundations for our understanding of the democratic process.
Condorcet wanted to find a mathematical reason for a rational citizen accept the authority of the state as expressed through democratic election.
He argued that the best reason would be if its individual probability of making the correct decision was less than the collective probability of choosing the correct alternative. His theorem appears to prove that this is almost always the case.
The theorem in its simplest form says that if each group member has a 50% chance correct answer to a problem with only 2 possible answers, then the possibility of a majority verdict
is fast approaching 100% as the group size increases .
Even if the individual can get the correct answer is 60%, the possibility that get the most correct answer increases to 80% for a group of 17 people and 90% for a group of 45 people.The theorem Condorec jury appears as an impressive mathematical justification of the power that has the intelligence group in the democratic process. 5 however, depends on basic assumptions:
1 .- Individuals in the group must be independent, ie no deben influenciar las opiniones entre sí,
2.- No deben tener opinionbes tendenciadas (preconcevidas),
3.- Todos deben estar intentando responder la misma pregunta,
4.- Deben estar suficientemente bien informados: La probailidad de cad aindiviuo de obtener respuesta correcta debe ser mayor al 50%,
5.- Debe haber una respuesta correcta. Estos cinco requerimientos implican que el teorema del jurado es util solo en muy restringido grupo de cisrcumstancias - aunque fue, y continua siendo, el punto de partida para discusiones sobre como se puede hacer funcionar a la democracia.
Caso práctico: Si se analyze practical cases, when you apply this logic to the television show "Who Wants to be a millionaire", is that the answers to "consult the public" (90%) are consistently more accurate than those made by "ask the expert" (66 %)
also the assumption that each member of the audience needs to have a probability of more than 50% of obtaining a correct answer is also not necessary. A closer examination reveals that
group intelligence works even if only a few know the answer and the rest of the individuals only chose to various probabilities of hitting .
To see how this works, try the following question with friends, a situation originally made by Scott Page: Of the following people, who was not a member of musical group the Monkees "in the 60's?: Peter Tork, Davy Jones, Roger Noll, and Michael Nesmith?
If this question is asked of 100 people, one possible scenario is that over two thirds (ie 68%) say they have no idea, 15 would know the miembos one of the group, 10 people could identify 2 members , and only 7 would know the correct answer. The correct answer is Roger Noll, an economist at Stanford. How many votes would it?
- 68 people with no idea, means that will randomly pick one of 4 options: 25% of them choose the correct answer:
17 - 15 identify only 1 member of the group, chosen at random from all 3 options remaining: 33% of correct answers: 5
- 10 people identify 2 members of the group: 50% chance of correct answer: 5
- 7 persons who have obtained 100% correct answer: 7
This makes a total of 34 correct answers, more than 22% of responses for each of the other options, a clear majority. Therefore
group intelligence works in these cases with only a few people are aware of the answers. In the above problem, the probability of choosing the correct answer would be greater even if 68 people had no idea of \u200b\u200bthe correct answer and 34 restrant only knew the name of 1 of the group. This would give 28 votes to the right answer and only 24 to each of the remaining possibilities.
The statistical distribution of knowledge may make this forecast a little less effective, but if the group size increases, the difference becomes more significant towards the correct answer. Now, when the population reaches the millions, the majority vote can provide a very accurate guide, which is why search engines like Google, Yahoo or Digg.com, use it as an important guide in their ranking algorithms.
The Remarkable Power of diversity Reveals Itself When Fully it comes to using majority opinion to make decisions. Michael Mauboussin produces a neat demonstration in another experiment with his Columbia Business School students. Each year, just before the Academy Awards are announced, he gets the students to vote on who they think will win in each of twelve categories—not just popular categories like best actor but relatively obscure ones, like best film editing or best art direction.
In 2007, the average score for individuals within the group was 5 out of 12. The group as a whole, though, got 11 out of 12 right!
Why is the majority so often right? One reason can be illustrated by the story of the Constitution, and of two of its principle framers, Benjamin Franklin and Thomas Jefferson.
Franklin and Jefferson both spent time in Paris before working on framing the Constitution, which was adopted in 1787. Both of them became involved in discussions with French intellectuals who were primarily responsible for the first French constitution, which was completed in 1789. One of those intellectuals was the Marquis de Condorcet, a corresponding member of the American Philosophical Society, founded by Franklin in 1743 (and still going strong).
Condorcet had begun his career as a mathematician, but when Franklin met him he had been appointed as inspector-general of the Paris Mint at the instigation of the reforming economist Anne-Robert-Jacques Turgot. Turgot didn’t last long in the atmosphere of intrigue and double-dealing that characterized Louis XVI’s court, but Condorcet prospered. He also became fascinated by the idea that mathematics could be used to support arguments for human rights and moral principles.
Franklin met up with Condorcet many times after he arrived in Paris and was impressed by the progress that Condorcet had made with his “social mathematics,” saying at dinners he attended that it “had to be discussed.” Nothing was yet on paper, but that soon changed with the publication of Condorcet’s remarkable work Essay on the Application of Analysis to the Probability of Majority Decisions, published in 1785.
There is a copy of the book, signed by Condorcet himself, in Jefferson’s library.
Franklin was clearly influenced by Condorcet’s ideas, in particular by his mathematical proof of what is now known as “Condorcet’s jury theorem.” John Adams told Jefferson that Condorcet was a “mathematical charlatan,” but this was far from being the case, and Condorcet’s theorem is now regarded as a cornerstone for our understanding of democratic decision-making processes.
Condorcet wanted to find a mathematical reason for a rational citizen to accept the authority of the state as expressed through democratic choice. He argued that the best reason would be if his or her individual probability of making a correct choice was less than the collective probability of making a correct choice. His theorem appears to prove that this is nearly always the case.
The theorem in its simplest form says that if each member of a group has a better than 50:50 chance of getting the right answer to a question that has just two possible answers, then the chance of a majority verdict being correct rapidly becomes closer to 100 percent as the size of the group increases. Even if each individual has only a 60 percent chance of being right, the chance of the majority being right goes up to 80 percent for a group of seventeen and to 90 percent for a group of forty-five.
Condorcet’s jury theorem looks like a stunning mathematical justification of the power of group intelligence in the democratic process, but it relies on five crucial assumptions, some of which are similar, though not identical, to the elements of cognitive diversity:
1.- the individuals in the group must be independent, which means that that they mustn’t influence each other’s opinions
2.- they must be unbiased
3.- they must all be trying to answer the same question
4.- they must be well-informed enough to have a better than 50:50 chance of getting the right answer to the question
5.- there must be a right answer
These requirements mean that the jury theorem is useful only in a very restricted range of circumstances—although it was (and continues to be) a concrete starting point for discussions on how democracy can best be made to work, and on the way that consensus decisions are arrived at in nature. Condorcet even used it after the French Revolution to suggest the best method of jury trial for the king, but his ideas were not taken up in an atmosphere that was more concerned with retribution than with fairness.
Condorcet also invoked the jury theorem in a discussion about the structure of government under the new U.S. Constitution. A point on which all the Framers were firm was that the new government should consist of two houses—a House of Representatives, representing the people, and a Senate, representing the states. When copies of the U.S. Constitution arrived in Paris in November 1787, Condorcet wrote to Franklin, complaining that such a bicameral legislature was a waste of time and money because, according to his mathematical approach to decision making, “increasing the number of legislative bodies could never increase the probability of obtaining true decisions.”
The point that Condorcet missed was that the two houses were put in place to answer slightly different questions. The U.S. Supreme Court made this clear in a 1983 judgment about the functions of the two houses when it said, “the Great Compromise [of Article I], under which one House was viewed as representing the people and the other the states, allayed the fears of both the large and the small states.” In other words, the House of Representatives is there to ask, “Is X good for the people?” while the Senate’s job is to ask, “Is X best implemented by the federal government or by the states?” The fact that the two houses are answering slightly different questions negates Condorcet’s argument that one of the houses is redundant.
It might appear that the jury theorem is more relevant to the functioning of juries themselves, but here again it is a matter of how juries are set up. To take maximum advantage of group intelligence, jurors need to be truly independent, which means that each would need to listen to the arguments of both sides and then make a decision without discussing it with the other jurors. The decisions would then be pooled, and the majority decision accepted.
Condorcet suggested that Louis XVI’s jury be set up in this way, but his ideas were rejected, and as far as I can find there have been no tests of his proposal since, in France or elsewhere. It does seem a pity, because discussions between jury members before coming to a decision mean that one of the main foundations of group intelligence (that of independence) is lost. Discussions certainly have their value—allowing people to change their minds under the influence of reasoned argument—but other forces can also be at work. One of these is the social pressure to conform with other members of the group that goes under the name of “groupthink,” and which I discuss in the next chapter. So long as members of juries continue to thrash out the merits of a case between themselves before coming to a conclusion in the manner depicted in the film Twelve Angry Men, the jury theorem will largely be irrelevant to their deliberations.
It comes into its own, however, when applied to the game show Who Wants to Be a Millionaire? although it turns out that our collective judgment is even more reliable than the theorem suggests. James Surowiecki points out that the “Ask the Audience” option consistently outperforms the “Call an Expert” option. This group of “folks with nothing better to do on a weekday afternoon” produces the correct answer 90 percent of the time, while preselected experts can only manage 66 percent.
It seems like an ideal case for the jury theorem. The selections are independent. The audience is presumably unbiased. Its members are all trying to answer the same question, and the question has a definite right answer.
The assumption that all members of the audience need to have a better than 50 percent chance of getting the answer right, however, is not necessary. Close examination reveals that their group intelligence still works even if only a few people know the answer and the rest are guessing to various degrees.
To see how this works, try the following question, originated by Scott Page, on your friends. Out of Peter Tork, Davy Jones, Roger Noll, and Michael Nesmith, which one was not a member of the Monkees in the 1960s?
If you ask this question of 100 people, one possible scenario is that more than two-thirds (68, say) of them will have no clue, 15 will know the name of one of the Monkees, 10 will be able to pick two of them, and only 7 know all three. The non-Monkee is Roger Noll, a Stanford economist. How many votes will he get?
Seventeen of the 68 will choose Noll as a random choice. Five of the 15 will select him as one choice out of three. Five of the 10 will select him as one choice out of two. And all of the 7 will choose him. This gives a total of 34 votes for Noll, compared to 22 for each of the others—a very clear majority.
So group intelligence can work in this case with only a few moderately knowledgeable people in the group. It would even have a fair chance of working if 68 people had no clue and the remaining 32 only knew the name of one Monkee. One-third of these (11 to the nearest whole figure) would choose Noll as the exception, giving an average total of 28 votes for Noll and 24 for each of the others.
Statistical scatter makes this prognostication less sure, but with increasing group size the difference becomes more meaningful.
When it reaches the millions, the majority vote can provide a very sure guide, which is why search engines such as Google, Yahoo, and Digg.com use it as an important guide in their ranking algorithms.
