Imagine that the only way that civilization could be destroyed was by a large pandemic that occurred at the same time as a large recession, so that governments and other organisations were too weakened to address the pandemic properly.

Then if we looked at the past, as observers in a non-destroyed civilization, what would we expect to see? We could see years with no pandemics or no recessions; we could see mild pandemics, mild recessions, or combinations of the two; we could see large pandemics with no or mild recessions; or we could see large recessions with no or mild pandemics. We wouldn't see large pandemics combined with large recessions, as that would have caused us to never come into existence. These are the only things ruled out by anthropic effects.

Assume that pandemics and recessions are independent (at least, in any given year) in terms of "objective" (non-anthropic) probabilities. Then what would we see? We would see that pandemics and recessions appear to be independent when either of them are of small intensity. But as the intensity rose, they would start to become anti-correlated, with a large version of one completely precluding a large version of the other.

The effect is even clearer if we have a probabilistic relation between pandemics, recessions and extinction (something like: extinction risk proportional to product of recession size times pandemic size). Then we would see an anti-correlation rising smoothly with intensity.

Thus one way of looking for anthropic effects in humanity's past is to look for different classes of incidents that are uncorrelated at small magnitude, and anti-correlated at large magnitudes. More generally, to look for different classes of incidents where the correlation changes at different magnitudes - without any obvious reasons. Than might be the signature of an anthropic disaster we missed - or rather, that missed us.

Ok, I don't like gnomes making current decisions based on their future values. Let's make it simpler: the gnomes have a utility function linear in the money owned by person X. Person X will be the person who appears in their (the gnome's) room, or, if no-one appeared, some other entity irrelevant to the experiment.

So now the gnomes have subjectively indistinguishable utility functions, and know they will reach the same decision upon seeing "their" human. What should this decision be?

If they advise "buy the ticket for price $x", then they expect to lose $x with probability 1/4 (heads world, they see a human), lose/gain nothing with probability 1/4 (heads world, they don't see a human), and gain $1-x with probability 1/2 (tails world). So this gives an expected gain of 1/2-(3/4)x, which is zero for x=$2/3.

So this seems to confirm your point.

"Not so fast!" shouts a voice in the back of my head. That second head-world gnome, the one who never sees a human, is a strange one. If this model is vulnerable, it's there.

So let's do without gnomes for a second. The incubator always creates two people, but in the heads world, the second person can never gain (nor lose) anything, no matter what they agree to: any deal is nullified. This seems a gnome setup without the gnomes. If everyone is an average utilitarian, then they will behave exactly as the total utilitarians would (since population is equal anyway) and buy the ticket for x<$2/3. So this setup has changed the outcome for average utilitarians. If its the same as the gnome setup (and it seems to be) then the gnome setup is interfering with the decisions in cases we know about. The fact that the number of gnomes is fixed is the likely cause.

I'll think more about it, and post tomorrow. Incidentally, one reason for the selfish=average utilitarian is that I sometimes model selfish as the average between total utilitarian incubator and anti-incubator (where the two copies hate each other in the tail world). 50%-50% on total utilitarian vs hatred seems to be a good model of selfishness, and gives the x<$1/2 answer.