tl;dr: there is no real Doomsday argument in ADT. Average utilitarians over-discount the future compared with total utilitarians, but ADT can either increase or decrease this effect. The SIA Doomsaday argument can also be constructed, but this is simply a consequence of total utilitarian preferences, not of increased probability of doom.

I've been having a lot of trouble formulating a proper version of the doomsday argument for Anthropic Decision Theory (ADT). ADT mimics SIA-like decisions (for total utilitarians, those with a population independent utility function, and certain types of selfish agents), and SSA-like decisions (for average utilitarians, and a different type of selfish agent). So all paradoxes of SIA and SSA should be formulatable in it. And that is indeed the case for the presumptuous philosopher and the Adam and Eve paradox. But I haven't found a good formulation of the Doomsday argument.

And I think I know why now. It's because the Doomsday argument-like effects come from the preferences of those average utilitarian agents. Adding anthropic effects does not make the Doomsday argument stronger! It's a non-anthropic effect of those preferences. ADT may allow certain selfish agents to make acausal contracts that make them behave like average utilitarian agents, but it doesn't add any additional effect.

Doomsday decisions

Since ADT is based on decisions, rather than probabilities, we need to formulate the Doomsday argument in decision form. The most obvious method is a decision that affects the chances of survival of future generations.

But those decisions are dominated by whether the agent desires future generations or not! Future generations of high average happiness are desired, those of lower average happiness are undesirable. This effect dominates the decisions of average utilitarians, making it hard to formulate a decision that addresses 'risk of doom' in isolation. There is one way of doing this, though: looking at how agents discount the future.

Discounting the future

Consider the following simple model. If humanity survives for n generations, there will have been a total of Gqⁿ humans who ever lived, for some G (obviously q>1). At each generation, there is an independent probability p of extinction, and pq < 1 (so the expected population is finite). At each generation, there is an (independent) choice of consuming a resource to get X utilities, or investing it for the next generation, who will automatically consume it for rX utilities.

Assume we are now at generation n. From the total utilitarian perspective, consuming the resource gives X with certainty, and rX with probability p. So the total utilitarian will delay consumption iff pr>1.

The average utilitarian must divide by total population. Let C be the current expected reciprocal of the population. Current consumption gives an expected XC utilities. By symmetry arguments, we can see that, if humanity survives to the next generation (an event of probability p), the expected reciprocal of population is C/q. If humanity doesn't survive, there is no delayed consumption; so the expected utility of delaying consumption is prXC/q. Therefore the average utilitarian will delay consumption iff pr/q > 1.

So the average utilitarian acts as if they discounted the future by p/q, while the total utilitarian discounts it by p. In a sense, the average utilitarian seems to fear the future more.

But where's the ADT in this? I've derived this result just by considering what an average utilitarian would do for any given n. Ah, but that's because of the particular choice I've made for population growth and risk rate. A proper ADT average utilitarian would compute the different p_i and q_i for all generation steps and consider the overall value of "consume now" decisions. In general, this could result in discounting that is either higher or lower than the myoptic, one-generation only, average utilitarian. The easy way to see this is to imagine that p is as above (and p is small), as are almost all the q - except for q_n. Then the ADT average utilitarian discount rate is still roughly p/q, while the myoptic average utilitarian discount rate at generation n is p/q_n, which could be anything.

So the "Doomsday argument" effect - the higher discounting of the future - is an artefact of average utilitarianism, while the anthropic effects of ADT can either increase or decrease this effect.

SIA Doomsday

LessWronger turchin reminded me of Katja Grace's SIA doomsday argument. To simplify this greatly, it's the argument that since SIA prefers worlds with many people in them (most especially many people "like us"), this increases the probability that there are/were/will be many civilizations at our level of development. Hence the Great Filter - the process that stops the universe from being filled with life - is most likely in the future for our kind of civilizations. Hence the probability of doom is higher.

How does this work, translated into ADT format? Well, imagine there were two options: either the great filter is in the distant evolutionary past, or is in the future. The objective uncertainty is 50-50 on either possibility. If the great filter is in the future, your civilization has a probability p of getting through it (thus there is a total probability of p/2 of your civilization succumbing to a future great filter). You have the option of paying a cost C to avoid the great filter entirely for your civilization. You derive a benefit B from your civilization surviving.

Then you will pay C iff C<Bp/2. But now imagine that you are a total utilitarian, you also care about the costs and benefits from other civilizations, and you consider your decision is linked with theirs via ADT. If the great filter is early, let's assume that your civilization is the only one still in existence. If the great filter is late, then there are Ω civilizations still around.

Therefore if the great filter is early, the total cost is C (your civilization, the only one around, pays C, but gets no benefit as there is no late great filter). However, if the great filter is late, the total cost is ΩC and the total benefit is ΩBp (all of Ω civilizations pay C and get benefit B with probability p). So the expected utility gain is ΩBp-(Ω+1)C. So you will pay the cost iff C < BpΩ/(Ω+1).

To an outsider this looks like you believe the probability of a late great filter is Ω/(Ω+1), rather than 0.5. However, this is simply a consequence of your total utilitarian preferences, and don't reflect an objectively larger chance of death.

[EDIT: I think the SIA doomsday argument works after all, and my objection to it was based on framing the problem in a misguided way. Feel free to ignore this post or skip to the resolution at the end.]

ORIGINAL POST:

Katja Grace has developed a kind of doomsday argument from SIA combined with the Great Filter. It has been discussed by Robin Hanson, Carl Shulman, and Nick Bostrom. The basic idea is that if the filter comes late, there are more civilizations with organisms like us than if the filter comes early, and more organisms in positions like ours means a higher expected number of (non-fake) experiences that match ours. (I'll ignore simulation-argument possibilities in this post.)

I used to agree with this reasoning. But now I'm not sure, and here's why. Your subjective experience, broadly construed, includes knowledge of a lot of Earth's history and current state, including when life evolved, which creatures evolved, the Earth's mass and distance from the sun, the chemical composition of the soil and atmosphere, and so on. The information that you know about your planet is sufficient to uniquely locate you within the observable universe. Sure, there might be exact copies of you in vastly distant Hubble volumes, and there might be many approximate copies of Earth in somewhat nearer Hubble volumes. But within any reasonable radius, probably what you know about Earth requires that your subjective experiences (if veridical) could only take place on Earth, not on any other planet in our Hubble volume.

If so, then whether there are lots of human-level extraterrestrials (ETs) or none doesn't matter anthropically, because none of those ETs within any reasonable radius could contain your exact experiences. No matter how hard or easy the emergence of human-like life is in general, it can happen on Earth, and your subjective experiences can only exist on Earth (or some planet almost identical to Earth).

A better way to think about SIA is that it favors hypotheses containing more copies of our Hubble volume within the larger universe. Within a given Hubble volume, there can be at most one location where organisms veridically perceive what we perceive.

Katja's blog post on the SIA doomsday draws orange boxes with humans waving their hands. She has us update on knowing we're in the human-level stage, i.e., that we're one of those orange boxes. But we know much more: We know that we're a particular one of those boxes, which is easily distinguished from the others based on what we observe about the world. So any hypothesis that contains us at all will have the same number of boxes containing us (namely, just one box). Hence, no anthropic update.

Am I missing something? :)

RESOLUTION:

The problem with my argument was that I compared the hypothesis "filter is early and you exist on Earth" against "filter is late and you exist on Earth". If the hypotheses already say that you exist on Earth, then there's no more anthropic work to be done. But the heart of the anthropic question is whether an early or late filter predicts that you exist on Earth at all.

Here's an oversimplified example. Suppose that the hypothesis of "early filter" tells us that there are four planets, exactly one of which contains life. "Late filter" says there are four planets, all of which contain life. Suppose for convenience that if life exists on Earth at all, you will exist on Earth. Then P(you exist | early filter) = 1/4 while P(you exist | late filter) = 1. This is where the doomsday update comes from.