Marginal Revolution linked to A Fine Theorem, which has summaries of papers in decision theory and other relevant econ, including the classic “agreeing to disagree” results. A paper linked there claims that the probability settled on by Aumann-agreers isn’t necessarily the same one as the one they’d reach if they shared their information, which is something I’d been wondering about. In retrospect this seems obvious: if Mars and Venus only both appear in the sky when the apocalypse is near, and one agent sees Mars and the other sees Venus, then they conclude the apocalypse is near if they exchange info, but if the probabilities for Mars and Venus are symmetrical, then no matter how long they exchange probabilities they’ll both conclude the other one probably saw the same planet they did. The same thing should happen in practice when two agents figure out different halves of a chain of reasoning. Do I have that right?
ETA: it seems, then, that if you’re actually presented with a situation where you can communicate only by repeatedly sharing probabilities, you’re better off just conveying all your info by using probabilities of 0 and 1 as Morse code or whatever.
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.
But in reality, what happens when people try to aumann involves a different set of problems, such as status-signalling, especially the idea that updating toward someone else’s probability is instinctively seen as giving them status.
Marginal Revolution linked to A Fine Theorem, which has summaries of papers in decision theory and other relevant econ, including the classic “agreeing to disagree” results. A paper linked there claims that the probability settled on by Aumann-agreers isn’t necessarily the same one as the one they’d reach if they shared their information, which is something I’d been wondering about. In retrospect this seems obvious: if Mars and Venus only both appear in the sky when the apocalypse is near, and one agent sees Mars and the other sees Venus, then they conclude the apocalypse is near if they exchange info, but if the probabilities for Mars and Venus are symmetrical, then no matter how long they exchange probabilities they’ll both conclude the other one probably saw the same planet they did. The same thing should happen in practice when two agents figure out different halves of a chain of reasoning. Do I have that right?
ETA: it seems, then, that if you’re actually presented with a situation where you can communicate only by repeatedly sharing probabilities, you’re better off just conveying all your info by using probabilities of 0 and 1 as Morse code or whatever.
ETA: the paper works out an example in section 4.
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.
But in reality, what happens when people try to aumann involves a different set of problems, such as status-signalling, especially the idea that updating toward someone else’s probability is instinctively seen as giving them status.
Thanks a lot for both links. I already understood common knowledge, but the paper is a very pleasing and thorough treatment of the topic.