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.
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.