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Math. proof of the superiority of independent guesses?

by Milton1 min read19th Jul 20207 comments

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Alice, Bob, Chris, Dana and Erica have set out to guess the number of candies in a glas jar just by looking at it. How do I mathematically show that the average guess will come closer to the real number if they guess independently, i.e without Bob hearing Alice's guess, Chris hearing Alice's and Bob's guess and so on?

And finally, I'm pretty confident I've read about this on LessWrong before, but I can't seem to find the specific post. Can anyone help me recollect?

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If the guesses are unbiased, the law of large numbers can be used to show this:

https://en.wikipedia.org/wiki/Law_of_large_numbers

(If you look at the proof, you can see where the independence assumption comes in.)

5 comments, sorted by Highlighting new comments since Today at 10:17 PM

How do I mathematically show that the average guess will come closer to the real number if they guess independently, i.e without Bob hearing Alice's guess, Chris hearing Alice's and Bob's guess and so on?

Things like that can't be illustrated about the real world directly, i.e., just with math.

But if people's guesses are clustered around the number, then the average will be close to the number on account of the errors being in both directions (not biased). Maybe there's risk associated with "anchoring", but I haven't heard about it being a risk in that particular setup before.*

There was a setup involving a wheel with numbers written on it being spun, and then people being asked how many countries there are in the world.

*Usually the effectiveness of averaging guesses that way is demonstrated by having a lot of people write their guess down on a piece of paper.

 

the specific post.

Sounds like the Sequences, stuff about entropy, probability distributions, or biases.

Thanks! And sorry for being such a noob.

Concerning "anchoring", do you know where I can go to read more?

(I don't know the state of the research with regards to replication.)

 

Asking this question in the right subreddit might turn up better results from people up to date on where the field is today.

 

If you're interested in the research at the time the sequences was done, here's the bibliography, which might include what you're looking for (I've heard it's not complete overall, but there's 5 things that came up searching that page for "anchoring"):

https://www.readthesequences.com/Bibliography

That is old research (I think it's all from well before 2000*), and as for where things are at now, I'm not sure. I turned up this literature review (2011):

http://www.communicationcache.com/uploads/1/0/8/8/10887248/a_literature_review_of_the_anchoring_effect.pdf

 

*Some of the same stuff is covered on Wikipedia: Anchoring (Cognitive Bias):

In another study by Tversky and Kahneman, participants observed a roulette wheel that was predetermined to stop on either 10 or 65. Participants were then asked to guess the percentage of the United Nations that were African nations. Participants whose wheel stopped on 10 guessed lower values (25% on average) than participants whose wheel stopped at 65 (45% on average).[3] The pattern has held in other experiments for a wide variety of different subjects of estimation.

[Emphasis added - looking up those researchers might be a good source, if you're interested in the start of the field.]

There's more on Wikipedia, but that link in the quote above seems to be from a study published in 1974. https://arxiv.org/ doesn't have a paywall, but doesn't seem to have a category for psychology or biases. 

Thanks again!

I guess what I'm pointing to is that it might be individually rational to update on previous guesses (perhaps this could be called "deliberate anchoring"?), whilst it might not be group rational.

The relevant post sounds like "Hold off on proposing solutions".