I wrote this short piece earlier this year:
A few years ago I read a slate of reviews of James Surowiecki's The Wisdom of Crowds, which cites empirical evidence to argue that the free interplay of diverse and independent opinions results, on the whole, in correct, accurate, beneficial, or, in short, ideal judgments and outcomes. I haven't read the book, which is a pity, but I distinctly remember one particular example that most of the reviewers mentioned. I managed to rediscover it for myself on Wikipedia (http://en.wikipedia.org/wiki/The_Wisdom_of_Crowds), which reads:
'The opening anecdote relates Francis Galton's surprise that the crowd at a county fair accurately guessed the butchered or the "slaughtered and dressed" weight of an ox when their individual guesses were averaged (the average was closer to the ox's true butchered weight than the estimates of most crowd members, and also closer than any of the separate estimates made by cattle experts).'
Also according to the Wikipedia article,
'Will Hutton has argued that Surowiecki's analysis applies to value judgments as well as factual issues, with crowd decisions that "emerge of our own aggregated free will [being] astonishingly... decent". He concludes that "There's no better case for pluralism, diversity and democracy, along with a genuinely independent press."'
Assuming that this is true - and of course we'd have to test its veracity thoroughly - we are left wanting an explanation of why this is the case. What follows are my attempts at just such an explanation.
Let us suppose a giant gum ball machine whose spherical glass container is six feet in diameter. Within the machine are mountains of gum balls of different colors and sizes, and it is up to a crowd of diverse observers - gum ball guessing idiots and savants alike - to privately guess at the correct number. Let us say that
N = the actual number of gum balls;
n = one person's estimate of the actual number of gum balls.
Let us suppose further that each individual estimate n is the result of a complex web of mental processes and algorithms, which are themselves the result of complex neurochemical processes. Therefore, let us say that
f(n) = the aggregate physical-mental function responsible for calculating estimate n,
such that
f(n) => n
and
f(n1) => n1
where f(n1) is the pattern of conscious and unconscious reasoning of the person to guess n1 first.
Let us go on again to suppose that the sum of two people's estimates, n1 + n2, is, in some shadowy way, the collective result of the sum of their mental processes, f(n1) + f(n2). In other words,
f(n1 + n2) => n1 + n2
We could then represent the case of the gumball machine as
f(n1 + n2 + ... + np) => n1 + n2 + ... + np
where
p = the number of people guessing.
The average guess, the sum of all estimates divided by the number of guessers, would accordingly come into existence by means of the mean of the crowd's gum ball guessing algorithms, such that
f(n1 + n2 + ... + np)/p => (n1 + n2 + ... + np)/p
If, as seems likely, the average of the guesses gets closer to the mark, N, as more guessers weigh in with their opinions, then we must also say, continuing in this vein, that the mean mental algorithm approaches a definite and particular expression, f(N), in the same way . In the notation we have
(n1 + n2 + ... + np)/p --> N
and
f(n1 + n2 + ... + np)/p --> f(N)
as
p --> infinity
Therefore, by method of substitution,
f(N) => N
That is, there exists somewhere a process of thought, f(N), which unerringly and unavoidably arrives at the correct number of gum balls. Moreover, all modes of gum ball guessing thought revolve about this inerrant function. It is possible that what we are seeing is a 'master' code from which we inherit copies in various states of corruption. If this is so, and if there are in fact multiple f(N)'s - for problems more weighty and pressing than guessing at gumballs - then it appears we have vastly underestimated both the degree of perfection to which the design of the human species has attained, and also our innate capacity for achieving, together, what can only be called a discernment befitting of the gods.
---
I showed my argument to a few folks, and accordingly there were a few points of contention. One, I guess, is that in proper math one is not supposed to divide by infinity. But I do not pretend that this is a watertight mathematical argument - I am merely using the notation to illustrate an idea. Two, my concluding paragraph is, shall we say, a bit ambitious: if I am to be taken seriously I must speak in hushed and moderate tones. I guess so. The third issue brought to my attention trafficked more heavily in the 'propositional content' of the piece. This is the reply I got from a cognitive science professor I used to know:
I read your argument. I found it interesting but I have to say I didn't find it completely convincing. Imagine that we built a machine to shoot arrows from a distance of 20 feet at a small target on a wall. We just load it up with 1000 arrows and it shoots them automatically. Imagine further that it rarely hits the target, and in fact that the average absolute distance that arrows land from the target is 5 feet. This would be a pretty inaccurate machine, missing on average by 5 feet from a distance of 20 feet.
However, imagine also that the machine is equally likely to miss in any direction -- up, down, up left, down right, etc. If we average across all the locations where the arrows hit the wall, the average location will be exactly at the target location. However, this fact doesn't change the point that the machine is quite inaccurate; it only shows that the machine has no systematic bias to miss in a particular direction.
To make the example more comparable to the average opinion across many people, imagine that we built 1000 identical machines (created with perfect precision) from the same specifications, and shot one arrow from each. Very few arrows would hit the target, but the average across the locations where the arrows hit would still be the target location. But again this fact would not show that the individuals were imperfect copies of a perfect arrow-shooting machine. They would instead be perfect copies of an imperfect machine.
Maybe I've failed to appreciate some aspect of your argument, but those are my thoughts based on what I understand of it.
---
I replied:
I am tempted to apply my argument to your example as well as mine, though I hope I don't start reasoning in circles.
You say that the arrow-shooting machine would have no bias in any particular direction, and that it would be built according to certain specifications. So suppose you wanted to build such a machine. If your design was off in a particular way, if your physics was off in a particular way, and so on, then so too would your aim deviate from the target in specific, measurable, and persistent ways. The machine being imperfect in practice, the result would be a scatter-shot of arrows - yet averaging around some spot a fixed distance away from the target. Unless your specifications were spot-on perfect, the scatter-shot of arrows could average around any possible point (as, statistically speaking, it would be difficult to achieve a cumulative bias of zero). But the scatter-shot does in fact revolve about the target perfectly; therefore the specifications for the machine that produced that scatter-shot must have been 'just right'.
So the question I am trying to raise is, what would it mean for us to have spot-on perfect 'specifications' for machines that compute something as mind-numbingly particular as the number of a gum balls in a gum ball machine? Or for anything else which aggregate opinion applies to? This is the question I'm struggling with. But then, maybe my counter-argument wasn't convincing?
---
Unfortunately our correspondence was terminated at that point, as he never responded.
The fourth issue I was confronted with, is that our opinions are often wildly wrong. A famous example is our intuitive physics, according to which a ball flying through the air must have some acting force 'pushing' on it at all times - which flies in the face of the much more predictively successful Newtonian dynamics (which says simply that bodies in motion tend to stay in motion). With this issue I do not take issue; rather I see in it much potential for honing and applying the theory I have laid out above. On the one hand, we can use the golden mean of aggregate opinion to discover just what problems human beings are naturally and intuitively 'really good at' solving (without the aid of the sciences); and on the other hand, we can apply the golden mean of aggregate opinion to those realms of reasoning we know already are implanted in every member of the species.
Consider, for instance, moral-ethical-political reasoning, about which there is much agreement these days: for the scientists now argue that morality is an evolved trait, and the religionists have long maintained that the law of God is written upon the hearts of all men. If we apply the example of the gum ball machine to the sphere of governance, then we find - not surprisingly - that dictatorships fare the worst when tasked with discerning the solution to any particular social problem: for only one or a few get to guess at the correct 'number of gum balls', and such severely limited data sets are almost always destined to arrive at inaccurate and unfortunate estimates.
We find, likewise, that healthy democracies - i.e. those in which the general public is most participant - result in far more accurate estimates of the 'number of gum balls', and thus in far greater popular satisfaction: for nothing can make more people happier than to arrive at the midpoint of the normal distribution curve for any given problem.
We can say, furthermore, coming from the other end of things, that sick and unhappy societies are sick and unhappy only inasmuch as they suffer from lack of popular political involvement. If, say, the people of the United States (to pick a not irrelevant example) are dissatisfied, it must be because they have little say in the affairs of government. To the extent that decision-making is taken out of the hands of the public, and placed ever more into the hands of experts, technocrats, centralized bureaucracies, and other special and specialized interests - to such an extent will you find a skewed estimate of the correct 'number of gum balls', and thus to greater mismanagement and greater unhappiness with the general drift of things.
To be more specific, and to apply the gum ball example to contemporary American electoral politics, we find that not everyone gets to cast his or her own personal vote on the correct 'number of gum balls'. Rather, the American individual gets to choose within one range of estimates or another, the two ranges largely overlapping and excluding the opinions of roughly three-fifths of the eligible population, who would rather not cast a vote at all than to wager on what they believe to be an inaccurate 'number of gum balls'. Of the remaining two-fifths of the population, who resign themselves to voting for the 'lesser of evils', the best and 'most legitimate' that can be hoped for is an outcome in which a statistical mode can be clearly discerned. It is obvious that the design of such a system is destined to fail spectacularly to arrive at the golden, statistical mean of the aggregate opinion of the whole five-fifths, and thus it is destined to lead to disaster.
The ethical moralist, the political scientist, the democrat, the patriot, and the statistician, then, should endeavor by all means to design a system in which the golden mean of aggregate opinion can be first determined, then implemented. It is the opinion of this author that the above named parties should enlist the help of the software designer as well. For imagine, first, if you will, a website freely available to all, which incorporates the best elements of the codes that determine individual preferences (Amazon.com? Pandora.com?), that facilitate individual expression (Blogger.com? Myspace.com?), that allow for moderated modification (Wikipedia.org?), and that lift the cream to the top (Google.com? Digg.com?), in order to formulate a crystal-clear and coherent message most representative of the greatest number of people. Imagine, next, that a candidate in a given electoral race runs not on any party platform, not on his own personal platform, and not on any ideological platform at all, other than the platform of the will of the people as determined by the best that modern statistical and computer science can offer. Such a candidate would have little need for advertisements, fund-raising, debates, or any else of all that sorry political lot; such a candidate would represent both the necessary technological innovations of the future, and the necessary democratic traditions of the past; such a candidate would be, without effort, among the best representatives there ever were; and such a candidate could spring up all around the country overnight. With such a candidate, and with such tools at his disposal, you might even say that we could flush the existing Washington power structure in a single election cycle.
It sounds silly, I admit, but if you have any better ideas please let me know.
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