Just to make sure I understood: you can value the existence of a nonexistent person of whom you have memories that you know are delusional more than you value your own continued existence, as long as those memories contain certain properties. Yes?

Yes.

So, same question: can you say more about what those properties are? (I gather from your example that being your daughter can be one of them, for example.)

I discovered the daughter example purely empirically when doing thought experiments. It seems plausible there are other examples.

Also... is it important that they be memories? That is, if instead of delusional memories of your time with Anna, you had been given daydreams about Anna's imagined future life, or been given a book about a fictional daughter named Anna, might you make the same choice?

Both of these would have significantly increased the probability that I would choose Anna over myself, but I think the more likley course of action is that I would choose myself.

If I have memories of Anna and my life with her, I find basically find myself in the "wrong universe" so to speak. The universe where Anna and my life for the past few years didn't happen. I have the possibility to save either Anna or myself by putting one of us back in the right universe (turning this one into the one in my memory).

In any case I'm pretty sure that Omega can write sufficiently good books to want you to value Anna or Alice or Bob above your life. He could even probably make a good enough picture of a "Anna" or "Alice" or "Bob" for you to want her/him to live even at the expense of your own life.

Suppose one day you are playing around with some math and you discover a description of ... I hope you can see where I'm going with this. Not knowing the relevant data set about the theoretical object, Anna, Bob or Cthulhu, you may not want to learn of them if you think it will make you want to prefer their existence to your own. But once you know them by definition you value their existence above your own.

This brings up some interesting associations not just with Basilisks but also with CEV in my mind.