Warning: this post is somewhat technical – looking at this summary should help.

1,000,000 people are in a giant urn. Each person is labeled with a number (number 1 through number 1,000,000).
A coin will be flipped. If heads, Large World wins and 999,999 people will be randomly selected from the urn. If tails, Small World wins and 1 person will be drawn from the urn.
After the coin flip, and after the sample is selected, we are told that person #X was selected (where X is an integer between 1 and 1,000,000).
Prior probability of Large World: P(heads)=0.5
Posterior probability of Large World: P(heads|person #X selected)=P(heads)=0.5
Regardless of whether the coin landed heads or tails, we knew we would be told about some person being selected. So, the fact that we were told that someone was selected tells us nothing about which world we are in.

Jason Roy argues that the self indication assumption (SIA) is equivalent to such reasoning, and thus wrong. For the self indication assumption to be legitimate it would have to be analogous to a selection procedure where you can only ever hear about person number 693465 for instance – if they don’t come up you hear nothing.

In both cases you can only hear about one person in some sense, the question is whether which person you could hear about was chosen before the experiment, or afterwards from those which came up. The self indication assumption looks at first like a case of the latter; nothing that can be called you existed before the experiment to have dibs on a particular physical arrangement if it came up, and you certainly didn’t start thinking about the self indication assumption until you were well chosen. These things are not really important though.

Which selection procedure is analogous to using SIA seems to depend on what real life thing corresponds to ‘you’ in the thought experiment when ‘you’ are told about people being pulled out of the urn. If ‘you’ are a unique entity with exactly your physical characteristics, then if you didn’t exist, you wouldn’t have heard of someone else – someone else would have heard of someone else. Here SIA stands; my number was chosen before the experiment as far as I’m concerned, even if I wasn’t there to choose it.

On the other hand ‘you’ can be thought of as an abstract observer who has the same identity regardless of characteristics. Then if a person with different characteristics existed instead of the person with your current ones, it’s just you observing a different first-person experience. Then it looks like you are taking a sample from those who exist, as in the second case, so it seems SIA fails.

This isn’t a question of which of those things exists. They are both coherent enough concepts that could refer to real things. Should they both be participating in their own style of selection procedure then, and reasoning accordingly? Your physical self discovering with utmost shock that it exists while the abstract observer looks on non-plussedly? No – they are the same person with the same knowledge now, so they should really come to the same conclusion.

Look more closely at the lot of the abstract observer. Which abstract observers get to exist if there are different numbers of people? If they can only be one person at once, then in a smaller world some observers who would have been around in the bigger world must miss out. Which means finding that you have the person with any number X should still make you update in favor of the big world, exactly as much as the entity defined by those physical characteristics should; abstract observers weren’t guaranteed to have existed exist either.

What if the abstract observer experiencing the selection procedure is defined to encompass all observerhood? There is just one observer, who always exists, and either observes lots of creatures or few, but in a disjointed manner such that it never knows if it observes more than the present one at a given time. If it finds itself observing anyone now it isn’t surprised to exist, nor to see the particular arbitrary collection of characteristics it sees – it was bound to see one or another. Now can we write off SIA?

Here the creature is in a different situation to any of Roy’s original ones. It is going to be told about all the people who come up, not just one. It is also in the strange situation of forgetting all but one of them at a time. How should it reason in this new scenario? In ball urn terms, this is like pulling all of the balls out of whatever urn comes up, one by one, but destroying your memories after each one. Since the particular characteristics don’t tell you anything here, this is basically a version of the sleeping beauty problem. Debate has continued on that for a decade, so I shan’t try to answer Roy by solving it now. SIA gives the popular ‘thirder’ position though, so looking at the selection procedure in this perspective does not undermine SIA further.

Whether you think of the selection procedure experienced by an exact set of physical characteristics, an abstract observer, or all observerhood as one, using SIA does not amount to being surprised after the fact by the unlikelihood of whatever number comes up.