The information between the two in the meeting are not exactly the same. From the A person's perspective this is a meeting between a specific A, I, meeting an unspecific B. From the B person's perspective this is a meeting between a specific B, again I, meeting an unspecific A. This importance of specification can be checked by changing the particular individual in the meeting to a different one and see if that affects the reasoning. For example from that A's perspective if this particular A, I, did not meet a B then his reasoning would be entirely different. It doesn't matter if another A have met with a B. Yet if he did not meet this particular B, but some other B instead, his reasoning would still be the same. This difference of specification is entirely due to their different perspectives. It is incommunicable.
Consider this experiment. An alien has abducted you and one of your friends. You are put to sleep. The alien then tosses a fair coin. If it lands on heads it won’t do anything to you. If it lands on tails it will clone you and put the clone into another identical room. The clone process is highly accurate so that the memory is retained. As a result the clone, as well as the original, can not tell if he is old or new. Meanwhile your friend never goes though any cloning process. After waking you up the alien let your friend choose one of the two rooms to enter. Suppose your friend has chosen your room. As a result you guys meet each other inside. How should you reason about the probability of the coin toss? How should your friend reason it?
For my friend the question is non-anthropic thus very simple. If the coin landed heads then 1 out of the 2 rooms would be empty. If the coin landed tails then both rooms would be occupied. Because the room that she randomly chose is occupied she now has new evidence favouring tails. As a result the probability of heads can be calculate by a simply bayesian update to be 1/3.
For halfers the question is not too complicated either. After waking up I have no new evidence about the fair coin toss. So I ought to believe the probability of heads is 1/2. Because my friend is randomly choosing between two rooms, regardless of the coin toss result the probability of my room being chosen is always half. Therefore seeing my friend gives me no new information about the coin toss either. This means I should keep believing that the probability of heads to be 1/2.
Here the disagreement is apparent. Even though the two of us appear to have the same information about the coin toss we assign different probability to the same proposition. To make the matter more interesting nothing I could say would change her mind and vice versa. We can communicate however we like but nobody is going to revise their answer. This may seem strange but it is completely justified. The cause of this disagreement is our different interpretations of who is exactly in this meeting. Remember according to my friends’ reasoning the evidence that causes the probability update is “the chosen room is occupied.” The occupant, in case there are duplicates, could not be specified from her perspective. In other words, as long as there is someone in the room she will reason as such. This is expected since the cloning procedure is highly accurate so there is no objective feature relevant to the coin toss to differentiate each duplicate. However from my first-person perspective I can inherently specify the one whom she is in meeting with. It is me, myself. But this specification is only valid from first-perspective. First-person identity is based on the immediacy to perception, which is primitive. It is incommunicable. I can keep telling her “this is me” and it would not mean anything to her. As a result the two of us would keep our own answers and remain in disagreement
This disagreement is also valid with a frequentist interpretation, which in my opinion is also easier to understand. The experiment can be repeated many times and the relative frequency can be used to show the correct probability. From my perspective repeating the experiment simply involves me going back to sleep, and wake up again after a coin toss and the potential cloning process. Of course after waking up I may not be the same physical human being just as the case of the first experiment. But this does not matter because in first-person perspective I am defined primitively base on subjective identity instead of objective features or qualities. So I would always regard the one falling asleep on the previous night as part of my subjective persistent self since it was the center of my perspective To make the procedure easier suppose I can check the previous coin toss result before going back to sleep again. So each iteration can be summarize as to go to sleep, wake up, and check the coin. Imagine repeating this iteration 1000 times by my count. I would have experienced about 500 heads and tails each. Furthermore if my friend is involved then I would see her about 500 times with about equal number of occurrences after heads or tails. However for these 1000 coin tosses my friend would see an occupied room about 750 times. The extra 250 times would be due to seeing the other duplicate instead of me after tails. It is easy to see our relative frequency of heads with a meeting are indeed different, half for me, a third for her. Of course my friend should be involved in far more repetitions than I do since every duplication of me are indifferent from her perspective so she shall be involved with their repetitions as well. However her relative frequency would remain unchanged by the higher number of iterations.
The disagreement arises because for my friend meeting someone in the room is technically not the same event for me meeting her. This is quite clearly so once the experiment is repeated a large number of times as discussed above. Her seeing someone in the room contains more experiments than me meeting her (750 vs 500). So we are actually assigning probability to different events. In my opinion this means it does not technically violate the theorem. Even though it may seems so superficially.
Aumann’s agreement theorem says that Bayesians with common priors who know one another’s posteriors must agree. There’s no apparent reason this shouldn’t apply to posteriors arrived at using indexical information. This does not mean that you and I should both believe we are as likely to be the author of this blog, but that we should agree on the chances that I am.
The Self-Sampling Assumption (SSA) does not allow for this agreement between people with different reference classes, as I shall demonstrate. Consider the figure below. Suppose A people and B people both begin with an equal prior over the two worlds. Everyone knows their type (A or B), but other than that they do not know their location. For instance an A person may be in any of eight places, as far as they know. A people consider their reference class to be A people only. B people consider their reference class to be B people only. The people who are standing next to each other in the diagram meet and exchange their knowledge. For instance an A person meeting a B person will learn that the B person is a B person, and that they don’t know anything much else.
When A people meet B people, they both come to know what the other person’s posterior is. For instance an A person who meets a B person knows that the B person doesn’t know anything except that they are a B person who met an A person. From this the A person can work out the B person’s posterior over which world they are in.
Suppose everyone uses SSA. When an A person and a B person meet, the A people come to think they are four times as likely to be in World 1. This is because in world two, only a quarter of A people meet a B person, whereas in world 1 they all do. The B people they meet cannot agree – in either world they expected to talk with an A person, and for that A person to be pretty sure they are in world 1. So despite knowing one another’s posteriors and having common priors over which world exists, the A and B people who meet must disagree. Not only on one another’s locations within the world, but over which world they are in*.
An example of this would be a husband and wife celebrating their wedding in a Chinese town with poor census data and an ongoing gender gap. The husband exclaims ‘wow, I am a husband! The disparity between gender populations in this town is probably smaller than I thought’. His wife expected in any case that she would end up with a husband who would make this inference from their marriage, and so cannot update and agree with him. Notice that neither partner need think the other has chosen the ‘wrong’ reference class in any way, it might be the reference class they would have chosen were they in that alternative indexical position.
In both of these cases the Self-Indication Assumption (SIA) allows for perfect agreement. Recall SIA weights the probability of worlds by the number of people in them in your situation. When A and B knowingly communicate, they are in symmetrical positions – either side of a communicating A and B pair. Both parties weight their hypotheses by the number of such pairs, and so they agree. Incidentally, when they first found out that they existed, and later when they learned their type, they did disagree. Communicating resolves this, instead of creating a disagreement as with SSA.
*If this does not seem bad enough, they each agree that the other person reasoned as well as they did.
Another implausible implication of this application of SSA is that you will come to agree with creatures that are more similar to you, even if you are certain that a given creature inside your reference class is identical to one outside your reference class in every aspect of its data collection and inference abilities.