Opinion de la mayoria y Teorema del Jurado / Majority Opinion and the Jury Theorem

El gran poder de la diversidad se muestra completamente al usar la opinion de la mayoria para realizar decisiones.

Michael Mauboussin produce una muy buena demostracion de esto con sus estudiantes en la escuela de negocios de Columbia. Cada año justo antes de que los premios de la academia sean anunciados, se efectúa una votacion respecto de quienes creen que serán ganadores en cada una de las 12 categorías en las que se entregan premios. No solo categorías populares como "mejor actor" sino que categorias mas ocultas como "mejor edición de pelicula" o "mejor direccion artistica".
El 2007, el promedio de respuestas correctas individuales fue 5 de 12. Sin embargo, el promedio de respuestas correctas para el grupo completo fué 11 de 12!

Porque esta la mayoria acertada en sus respuestas de forma tan frecuente? Una razón puede ser ilustrada por la historia del desarrollo de la Consititución de USA, y dos de sus mas famosos artifices, Bejamin Franklin y Tomas Jefferson.

Franklin y Jefferson ambos pasaron un tiempo en Paris antes de participar en la creación de la constitución, la que fue adptada el 1787. Ambos se involucraron en doscusiones con intelectuales franceses responsables principalmente por la pirmera constitucion francesa, la que fue completada en 1789. Uno de esos intelectales fue Marquis de Condorcet.

Condorecet había comenzado su carrera como un matematico y que en ese tiempo trabajaba como inspector de monedas en la Casa de moneda de Paris. A él le fascinaba la idea de que la matematica puediese ser utilizada para obtener argumentos que apoyaran los derechos humanos y los principios morales.

Franklin se reunión con Condercet muchas veces despues de llegar a Paris, y quedó muy impresionado de los avances que Condorcet había alcanzado con su "matematica social", indicando que "debía ser discutida". No había nada aun escito al respecto, pero todo eso cambió luego de la publicación del ensayo de Condorcet sobre la aplicacion de Analisis a la probabilidad de la decision de las mayorias, publicado en 1785.

Franklin claramente fue influenciado por las ideas de Condorcet, en especial por su prueba matemática hoy conocida como el "Teorema del Jurado de Condorcet", teorema que es considerado hoy una de los fundamentos para nuestro entendimiento del proceso democrático.

Condorcet quería encontrar una razón matemática para que un ciudadano racional aceptara la autoridad del estado como se expresa a través de la eleccion democrática. Él argumentaba que la mejor razón sería si su probabilidad individual de efectuar la decision correcta era menor que la probabilidad colectiva de escoger la alternativa correcta. Su teorema aparece probar que este es casi siempre el caso.

El teorema en su forma mas simple dice que si cada miembro del grupo tiene un 50% de posibilidad de obtener la respuesta correcta a un problema con solo 2 posibles respuestas, entonces la posibilidad de un veredicto por mayoría se acerca rapidamente al 100% a medida que el tamaño de grupo aumenta.

Incluso si la posibilidad individual de obtener la respuesta correcta es de 60%, la posibilidad de que la mayoría obtenga la respuesta correcta aumenta al 80% para un grupo de 17 personas y a 90% para un grupo de 45 personas.El teorema de jurado de Condorec aparece como una impresionante justificación matemática de la potencia que tiene la inteligencia grupal en el proceso democratico. Depende sin embargo de 5 supuestos basicos:

1.- Los individuos del grupo deben ser independientes, esto es, 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 analizan casos prácticos, cuando se aplica esta logica al show de televisión "quien quiere ser millonario", se encuentra que las respuestas de "consultar al publico" (90%) consistentemente son mas acertadas que las hechas por "pregunte al experto" (66%)

Además el supuesto que cada miembro de la audiencia necesita tener una probabilidad de mas del 50% de obtener una respuesta correcta, tampoco es necesaria. Un examen mas cercano revela que la inteligencia de grupo aun funciona si solo unos pocos conocen la respuesta y el resto de los individuos solo escogen con variadas probabilidades de acertar.

Para ver como funciona esto, intenten la siguiente pregunta con amigos, situación originalmente formulada por Scott Page: De las siguientes personas, ¿Quién no fué miembro del grupo musical "the Monkees" en los años 60?: Peter Tork, Davy Jones, Roger Noll, o Michael Nesmith?

Si esta pregunta se formula a 100 personas, un escenario posible es que mas de dos tercios (por ejemplo 68%) diga que no tiene idea, 15 conocerían el de uno de los miembos del grupo, 10 personas podrían identificar a 2 miembros, y solo 7 podrían saber la respuesta correcta. La respuesta correcta es Roger Noll, economista de Stanford. Cuantos votos obtendría él?

- 68 personas sin idea: implica que van a escoger al azar cualquiera de las 4 opciones: 25% de ellos escogerán la respuesta correcta: 17
- 15 identifican solo 1 miembro del grupo, escogerán al azar de cualquiera de las 3 opciones restantes: 33% de respuestas correctas: 5
- 10 personas identifican a 2 miembros del grupo: 50% de posibilidades de respuesta correcta: 5
- 7 personas que tienen 100% de obtener respuesta correcta: 7

Esto da un total de 34 respuestas correctas, superior al 22% de respuestas para cada una de las otras opciones, una mayoria clara.

Por tanto la inteligencia de grupo funciona en estos casos con solo unas pocas personas tengan conocimiento de las respuestas. En el problema anterior, la probabilidad de escoger la respuesta correcta sería mayor incluso si 68 personas no tuviesen idea de la respuesta correcta, y las 34 restrantes solo supieran el nombre de 1 integrante del grupo. Esto daría 28 votos a la respuesta correcta y solo 24 a cada una de las restantes posibilidades.

La Distribución estadística de conocimiento puede tornar este pronostico algo menos efectivo, pero si el tamaño de grupo aumenta, la diferencia se hace mas significativa hacia la respuesta correcta.

Ahora, cuando la población alcanza los millones, el voto de la mayoría puede proveer de una guia muy acertada, razón por la cual motores de busqueda como Google, Yahoo o Digg.com, lo usan como una guia muy importante en sus algoritmos de ranking.

The remarkable power of diversity reveals itself fully when 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.