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HalFinney comments on Open Thread: June 2010 - Less Wrong
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I thought of a simple example that illustrates the point. Suppose two people each roll a die privately. Then they are asked, what is the probability that the sum of the dice is 9?
Now if one sees a 1 or 2, he knows the probability is zero. But let's suppose both see 3-6. Then there is exactly one value for the other die that will sum to 9, so the probability is 1/6. Both players exchange this first estimate. Now curiously although they agree, it is not common knowledge that this value of 1/6 is their shared estimate. After hearing 1/6, they know that the other die is one of the four values 3-6. So actually the probability is calculated by each as 1/4, and this is now common knowledge (why?).
And of course this estimate of 1/4 is not what they would come up with if they shared their die values; they would get either 0 or 1.
Here is a remarkable variation on that puzzle. A tiny change makes it work out completely differently.
Same setup as before, two private dice rolls. This time the question is, what is the probability that the sum is either 7 or 8? Again they will simultaneously exchange probability estimates until their shared estimate is common knowledge.
I will leave it as a puzzle for now in case someone wants to work it out, but it appears to me that in this case, they will eventually agree on an accurate probability of 0 or 1. And they may go through several rounds of agreement where they nevertheless change their estimates - perhaps related to the phenomenon of "violent agreement" we often see.
Strange how this small change to the conditions gives such different results. But it's a good example of how agreement is inevitable.