In statistics it is relatively easy to reason about expected values. And the statistics-based literature on the rationality of disagreement that I have worked on has interpreted typical human opinions (about matters of fact as opposed to value) as expected values (which include probabilities). But I wonder: is this a big mistake?

Imagine an Easter egg hunt where there was only one egg. Even if we all had the same abilities and all agreed on a probability distribution over where the egg might be, we should still search in different parts of the yard. We should spread ourselves out, with the number of people in each region being roughly proportional to the chance of finding the egg in that region. Could our opinion game be similar?

If we were penalized in proportion to something like the square of the "distance" of our opinion from the truth, we should choose an opinion which was an expected value of the truth (in terms of that distance). And in this case we should not knowingly disagree. But if the opinion game we play instead rewards opinions closer to the truth than other opinions, it can make much more sense to knowingly disagree.

So what are the rewards in typical opinion games we play? What opinion games should we want people to play?

I like the Easter egg hunt example. A good strategy to get someone in the group to find the egg is for each person to randomly sample the probability distrubtion and search at that spot. However, a selfish individual who only wants to find the egg for herself would search at the peak of the probability distribution. If everyone follows that argmax rule then the group outcome is suboptimal.

Say that people can't hold an entire probability distribution as a belief, but rather can only hold a point estimate. If people have to use these argmax estimates then we're all in trouble -- people will gravitate towards local maxima instead of taking into account all the possibilities.

You can look for an easter egg in three typical ways (assuming you can't hang back and wait for the others to do so):1) You individually figure out an algorithm to determine where to look for2) You choose at random, using the probability distribution3) You split the domain of research quasi equaly, and each go over your own bit

Of the three, 3) is the most equitable (everyone has roughly equal chances of finding the egg once the search starts). 2) is quite unequal (choosing zones at randon means a lot of overlap - those few out on their own have much more chances of finding the egg). 1) is the most interesting; at least initially, it will probably be even less equal than 2).

However, if we iterate the search multiple times, and each searcher changes their algorithm to improve it each time, then we get an interesting situation. We'd probably something in between 2) and 3), born of the twin urges to differentiate yourself from the pack and try and search the most likely areas.

These algorithms then become biases, usefull biases. They're usefull because they are different from each other, so cover the field more effectively than randomness. They would be usefull even if most of the searchers were searching randomly, as long as a small group isn't.

They would look just like true biases: use the algorithm to compute a probability distribution for where to search. If one took this new probability distribution to be "true" one would search better than if one took the genuine one to be "true".

This may be an argument in favour of "diversity" - ensuring that different areas are searched outweighs the need to have everyone as accurate as possible.

When does this become harmfull? When the global probability estimate (summing up all the individual ones) no longer resembles the true estimate. This goes back to the whole issue of the wisdom of crowds.

And thinking about it, there may be such a factor afoot in the football example. In everyday life, if you are betting on the games, then you have the incentive to be good, but different (as the odds are set by the betting levels). Maybe this is one factor explaining the overall wisdom of the crowd?