by Brian Tomasik
Draft began: 2 Feb 2013; major additions: 22 June 2013; last update: 16 Jan. 2015

Summary

Note: I probably now disagree with Nick Bostrom's self-sampling assumption, which forms the basis of the current essay. For alternate approaches, see "Anthropics without Reference Classes".

It naively seems as though our influence over the far future could be extremely significant, because it's possible we'll be able to affect a vast number of later minds through our actions today. However, if you apply anthropic reasoning using the assumption that you are a random observer from all observers ever (the universal reference class), then the doomsday argument suggests that you're extremely unlikely to have a big influence after all. A typical way to restore a common-sense perspective on the future is to modify your reference class so as to no longer be universal. The Self-Indication Assumption or other novel proposals may also yield a solution. Given the high model uncertainty at play, it makes sense to continue acting as though the future might be extremely important, to be on the safe side.

Update (July 2013): The section Multiple civilizations may change the argument explains why doomsday reasoning might in fact not have the full force that it seems to. For example, in the Appendix, I discuss how, under a reasonable setting of parameters, futures n times as long as our civilization so far may have something like n1/4 as much expected importance relative to short-lived futures even after adjusting for doomsday effects. If correct, this argument invalidates some of the other reasoning in this piece. I conclude that until uncertainty on this topic is resolved, it seems like a good idea in general to consider both the short-term and long-term effects of what we advocate rather than assuming the future always dominates. That said, it's plausible that the long-term future still matters more in expectation, especially after we incorporate model uncertainty.

Update (Sept. 2013): I'm told that Bostrom's SSSA is no longer the state of the art in some communities. It has been replaced by a different approach to anthropics that's something like "SIA for exact copies of you," which removes the doomsday argument entirely. I haven't yet explored this approach in more detail.

Introduction

The future ain't what it used to be. --misattributed to Yogi Berra

If you ask most scientists about our place in the history of the universe, they'll tell you that we're among about the first 100 billion humans to have been born. We may have a long future ahead of us, or we may destroy ourselves early. It doesn't appear inconceivable that humans and their possibly post-human descendants could survive a very long time. Thus, it seems that what we do now to affect the future could be extraordinarily important, given how many future lives may be affected by our actions.

Some altruists take this line of reasoning and conclude what I'll call future fanaticism, the idea that shaping the far future (e.g., by promoting better values) is overwhelmingly more important in expected value than reducing suffering in the short term in more tangible ways (e.g., reducing wild-animal suffering on Earth).

Anthropics

This naive picture of our place in the history of Earth-originating life is conceived before considering anthropic bias. A standard principle in anthropics is the Self-Sampling Assumption (SSA): All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class. Philosopher Nick Bostrom correctly modifies this to the Strong Self-Sampling Assumption, which replaces observers by observer-moments, since personal identity is fuzzy, and you're twice as likely to find yourself as someone who lives twice as long or experiences subjective time twice as fast.

It's not clear what exactly counts as an observer. Is it all conscious minds, or just minds that are smart enough to think about anthropics? Is it all smart-enough minds, or is it only those smart-enough minds when they're actually thinking about anthropics? These questions are crucial, but I don't know their answers now.

Universal reference class

It seems intuitive to me to take your reference class to be all observer-moments ever in the whole multiverse -- what Bostrom calls the universal reference class. Bostrom himself rejects this, because it leads to seemingly insane conclusions. He recommends modifying your reference classes in reasonable ways, in order to avoid these paradoxes. The doomsday argument, discussed below, may be avoidable by this approach, as I explain at the end of this piece. In this essay, I assume the universal reference class and see where that gets us as far as altruism regarding the future, keeping in mind that I may be misguided to use this reference class.

Why do I think the reference class should be universal? It seems any other reference class is cheating by adding information without updating on it. Let me explain.

Anthropics works as follows. Begin with hypotheses for how the universe as a whole might be. These hypotheses describe everything about the universe, including where each mind is. In this specification, the only thing you don't know yet is which of those minds you are. Who you are is a further piece of evidence that updates your probability assessment for the hypothesis. The posterior probability of the hypothesis given who you are, P(h|o), is proportional to the prior for the hypothesis, P(h), times the likelihood for predicting who you find yourself to be, P(o|h).

For example, say you're uncertain between two hypotheses. With hypothesis 1 (h1), the universe consists of 10 minds: 9 humans and one super-intelligent post-human AI. In hypothesis 2 (h2), the universe consists of 100 minds: 50 humans and 50 super-intelligent post-human AIs. Based purely on the models of physics, biology, social evolution, and so on that you expect to see, you assign P(h1) = P(h2) = 0.5. Now you learn a further piece of information: You're a human, not a post-human AI. This additional evidence (o) updates the probability of each hypothesis. In particular, P(o|h1) = 0.9 and P(o|h2) = 0.5, so the posterior ratio P(h1|o)/P(h2|o) = 9/5.

We might have further information as well. For instance, suppose your name is George -- call this additional evidence g. Suppose h1 says that 3 of the humans are named George, while h2 says that 25 of the humans are named George. Then P(g|h1,o)=3/9 and P(g|h2,o)=25/50. This causes us to further update our probabilities, multiplying the original 9/5 ratio by (3/9)/(25/50), giving a final ratio of 6/5. We could have also derived this by considering the joint evidence all at once: P(o,g|h1)=3/10 and P(o,g|h2)=25/100, so P(h1|o,g)/P(h2|o,g) = (3/10)/(25/100)=6/5.

But what would have happened if we had reasoned in this way: Post-human AIs are very different from humans, so we can't put ourselves in the same reference class with them. We should only consider ourselves a random sample from the humans. Then we would have omitted the o evidence in the above calculations. All we would have had would have been the following: P(h1)=P(h2)=0.5, and we assume we're a human because that's our reference class. Then P(g|h1)=3/9 and P(g|h2)=25/50, so the posterior ratio P(h1|g)/P(h2|g)=(3/9)/(25/50)=2/3. This is a different answer from the 6/5 we saw above. In fact, from this we would conclude that h2 was more likely in view of the evidence, when from the above analysis, we saw that h1 was more likely in view of the evidence. We can't simply omit part of the update because it's inconvenient. This illustrates why I think the reference class must be universal.

Doomsday argument

Assuming a universal reference class, I'll describe a standard version of the doomsday argument. Consider two scenarios. In ShortLived, humanity contains only roughly as many observer-moments as we've had so far -- call this some amount X. In LongLived, humanity expands into the galaxy and has vastly more observer-moments than we've had so far. For concreteness, suppose it's 1000X. We can imagine more possibilities, but these two suffice to illustrate the mechanics of what's going on, so it's better not to complicate the analysis. Also assume that these scenarios seem about equally plausible on their face, with 50%-50% probabilities for each. The argument still works if you don't assume this, but I'm trying to minimize the number of variables to improve intuition at the expense of formality.

We seem to find ourselves among the first X observer-moments. If ShortLived is true, this isn't surprising -- we are necessarily among the first X observer-moments. In contrast, if LongLived is true, then this is rather strange, because the probability of finding ourselves so early is only 1/1000. Using Bayes' theorem to update our 50%-50% prior, the posterior probabilities of ShortLived vs. LongLived are 1000/1001 and 1/1001. It's often easier to think of this in terms of the Bayes factor: LongLived is now 1000 times less likely than ShortLived.

Multiple civilizations may change the argument

Note: This section contains a new realization that seems to overturn much of the rest of this piece. I haven't updated the rest of this essay to account for it.

You might try to escape the doomsday conclusion with the following thought: Okay, the above argument applies if we're considering just one civilization's prospects. But throughout the universe, there are many possible civilizations. Say one of them is short-lived and another is long-lived. Because the long-lived one has lots more observer-moments, we're a priori most likely to be in that one. Then even when we discount by the improbability of being so early, it's still as likely we're in the long-lived as the short-lived universe. Another way to see this is that the short-lived civilization has, say, X observer-moments, and the long-lived one has X observer moments at the beginning and then another 999X beyond that. The probability of being in the short-lived civilization is X/(X + 1000X) = 1/1001. The probability of being in the first X of the long-lived civilization is also X/(X + 1000X) = 1/1001.

That's all correct, but it relied on a crucial assumption: That there is one short-lived civilization and one long-lived one. But in fact, we don't know how many short-lived vs. long-lived civilizations there are. We have some prior probability distribution over this. In particular, say there were two options: ManyLong, in which there are 50% short-lived and 50% long-lived civilizations, and FewLong, in which there are 1000 times fewer long-lived than short-lived civilizations. Denote our observation of being in the first X observer-moments as "obs." If ManyLong is true, then the probability we'd be in the first X observer-moments is only 2X/(X + 1000X) = 2/1001, because there's one X for the 50% that are short-lived and another X for the first X of the 50% that are long-lived. That is, P(obs | ManyLong)=2/1001. In contrast, if FewLong is true, then the probability we'd be in the first X observer-moments is (1000X + X)/(1000X + 1000X) = 1001/2000, because for every long-lived civilization (which has X observer-moments in the first X and 999X not in the first X), there are 1000 short-lived civilizations that all have observer-moments in the first X. So P(obs | FewLong) = 1001/2000. Now, the probability that we ourselves are in a long-lived civilization given our observations is

P(long-lived | obs) = P(long-lived | ManyLong, obs)P(ManyLong | obs) + P(long-lived | FewLong, obs)P(FewLong | obs)
= P(long-lived | ManyLong, obs)P(obs | ManyLong)P(ManyLong)/P(obs) + P(long-lived | FewLong, obs)P(obs | FewLong)P(FewLong)/P(obs)
= (1/2)(2/1001)(1/2)/P(obs) + (1/1001)(1001/2000)(1/2)/P(obs).      (eqn. *)

To compute P(obs):

P(obs) = P(obs | ManyLong)P(ManyLong) + P(obs | FewLong)P(FewLong)
= (2/1001)(1/2) + (1001/2000)(1/2)
= 0.2512.

Substituting back into (eqn. *) gives P(long-lived | obs) as approximately 3/1000. So unlike if we knew for sure that ManyLong was true, in this case we still observe the doomsday argument taking effect and downweighting the likelihood that we're in a long-lived civilization. By the way, this reasoning suggests the doomsday argument's answer to the Fermi paradox: Most ETs go extinct early (i.e., FewLong is true), because if they didn't, we would expect to be in a long-lived ET civilization.

In Philosophical Implications of Inflationary Cosmology (section 6), Knobe, Olum, and Vilenkin make a similar distinction. For any given fraction of long-lived civilizations f, our own probability of being in one of them is just f, because the increased number of observers exactly cancels the reduced probability of being so early. But observing ourselves early implies a downward shift in our beliefs about what f is likely to be, which the authors call universal doomsday because it suggests that across the universe, fewer civilizations than we thought are long-lived.

But notice something interesting: Whereas in the standard doomsday argument, given equal prior probabilities of a short- or long-lived civilization, the posterior probability for being in the long-lived one was ~1/1000, now with multiple civilizations, it's ~3/1000 (given the particular choice of hypotheses above -- not in general). Intuitively, the reason it's more likely we're in a long-lived civilization is because when they exist, they have lots of observers, so it's more likely a priori that we'd be one of them.

In the Appendix, I develop this reasoning further, extending it to the case of a continuous probability distribution over what fraction of civilizations is long-lived. There I show that if we have a uniform probability distribution over what fraction of civilizations is long-lived, then after observing ourselves to be in the first X observer-moments, the updated odds ratio for being in a short-lived vs. long-lived civilization is only about ln(1000)-1 ~= 6, not 1000 as was the case for the standard doomsday argument. For prior probabilities that assume most civilizations are short-lived, the posterior odds ratio is bigger than ln(1000)-1 but still less than 1000, showing that this sub-linearity in the doomsday-update Bayes factor is robust. If my reasoning is accurate, this finding invalidates the assumption made throughout this essay that the doomsday penalty for a civilization n times as long is always 1/n. If I verify that this is accurate, I'll have to update my conclusions.

Modal realism and doomsday.

In the original version of this piece, I made the claim that in a modal-realism multiverse, the more intuitive [S]SSA implies SIA. SIA is the Self-Indication Assumption, which roughly speaking says that a world with twice as many observer-moments is twice as likely, relative to whatever prior probabilities you assigned it. While Carl Shulman later objected to the semantics of my use of the SIA terminology, the intuition behind my statement was exactly what I articulated in the first paragraph of the section Multiple civilizations may change the argument: Namely, that if you knew there was one short-lived and one long-lived civilization, the posterior probability of being in either given that you find yourself in the first X observer-moments is the same because the prior probability of being in the longer-lived one was higher. The reason the prior probability was higher for the longer one was that the observer-moments it contains actually exist somewhere in the multiverse, so weighting by number of observer-moments as SIA would have prescribed for the case of a single existing universe is what SSSA itself tells us to do when both universes actually exist.

If you exactly knew the true measure of all the possible worlds in a modal-realism multiverse, then the prior probability of being in any of those worlds would be proportional to the physical measure of the world times the number of observer-moments it contains, just by the regular SSSA. But as we saw in the case of uncertainty between the ManyLong and FewLong hypotheses (i.e., uncertainty about what the true modal-realism measure is), our anthropic updates no longer looked quite like SIA; instead, the SIA-like effect of huge numbers of observers in big universes is tempered though not completely eliminated. So uncertainty about the true modal-realism measure means modal realism alone can't prevent doomsday effects, though as we've seen, it can reduce them relative to what would be the case in a single-civilization universe.

Implications for altruism

If LongLived is very unlikely, this seems to dampen the fanaticism that we might have had about trying to influence the future. It's very unlikely there will be a future on which that influence would operate. So, for instance, ensuring that the future cares about wild-animal suffering and avoids spreading it is less important than it might seem relative to reducing wild-animal suffering on Earth in the near term.

What about actions that aim not to influence a given future population but that aim to change the size of the future population? If you take an action that aims to reduce extinction risk, that action seems very likely to fail, because if it was likely to succeed, this would imply a higher probability of a big future, in which case it's then unlikely you'd be so early. The bigger the increase in observer-moments that an action aims to effect, the less likely it is to work. Inversely, actions that aim to reduce future observer-moments have decent prospects, though of course any given one of them is by no means guaranteed to go through; what is likely is that one factor or another at some point will cut off future observer-moments.

Pascalian response

Altruists maximizing expected value can still respond to the doomsday argument as follows. Even if LongLived is 1000 times less likely than ShortLived, if LongLived is true, then my actions affect 1000 times as many observer-moments. Therefore, choosing to focus on the long term has the same expected value as choosing to focus on the short term, because the 1000 times greater impact exactly cancels the 1/1000 probability penalty.

This response is reminiscent of, though not equivalent to, SIA. In the case of the doomsday argument, SIA says that whatever else you thought, the LongLived scenario is 1000 times more likely than that, and this balances out the 1/1000 penalty due to being so early. In the Pascalian version, we don't actually think the LongLived civilization is 1000 times more likely, but we think it's 1000 times more important. The cancellation of the doomsday dampening happens in expected values rather than in probabilities.

The gaping problem with the Pascalian response is that, even if the future contains 1000 more observer-moments under LongLived, our ability to reduce suffering is not literally 1000 times as much. There may be entropy effects that diminish our influence over events further and further into the future. If humans create a goal-preserving AI, then this entropic reduction in influence needn't approach zero for very far-out times, but it should still be lower than our influence now, and in any event, it's not at all clear that a goal-preserving AI will be created. There are a whole lot more uncertainties at play in trying to influence the future, such that we should by default apply a large non-anthropic probability penalty to our ability to make an impact there. Say there's a 1% probability that what we do now can reliably affect the far future in predictable ways (this seems unrealistically high, but just take it for illustration). Then if we work on the present, we affect X observer-moments. If we work on the future, we might affect 1000X * 1%, but there's only a 1/1000 chance there will be a far future like this, so the total value is 0.01X. It seems we should work on the present due to greater certainty of immediate impacts.

Anthropic approaches for saving future fanaticism

1. Pick a non-universal reference class

At the beginning I assumed a universal reference class, saying that we should consider ourselves a random sample from all observer-moments, not just some subset of them that share given properties. The only relevant feature of whether something has anthropic weight is whether it's an observer; it shouldn't matter if it's human or not.

However, if you allow your reference classes to be narrower than this, you can cheat away doomsday reasoning. For instance, if your reference class is only the set of humans ever, then you have an easy solution to the doomsday argument: Humans soon go extinct by replacing themselves with post-humans that aren't in their reference class.

Bostrom seems to advance a view of this kind:

Take the Doomsday argument [DA]. In order for it to work, one has to assume that the beings who will exist in the distant future if humankind avoids going extinct soon will contain lots of observer-moments that are in the same reference class as one's current observer-moment. If one thinks that far-future humans or human descendants will have quite different beliefs than we have, that they will be concerned with very different questions, and that their minds might even be implemented on some rather different (perhaps technologically enhanced) neural or computational structures, then requiring that the observer-moments existing under such widely differing conditions are all in the same reference class is to make a very strong assumption. [...] These arguments will fail to persuade anybody who doesn't use the particular kind of very inclusive reference class they rely on -- indeed, reflecting on these arguments may well lead a reasonable person to adopt a more narrow reference class. [...]

As regards DA, we can distinguish versions of it that have a greater degree of persuasiveness than others. For example, DA provides stronger grounds for rejecting the hypothesis that humans will exist in very great numbers in the future in states that are very similar to our current ones (since for this, only relatively weak assumptions are needed: that the reference class definition be at least somewhat inclusive) than for rejecting the hypothesis that humans will continue to exist in any form in large numbers (which would require that a highly diverse set of possible observer-moments be included in our current reference class).

2. Adopt SIA

I feel uncomfortable about SIA but wouldn't rule it out. According to the Wikipedia article, Many people [...] believe the leading candidate for Doomsday argument refutation is a Self-Indication Assumption of some kind. One classic intuition pump against SIA is Bostrom's presumptuous philosopher thought experiment.

3. Model uncertainty

Many smart people disagree about anthropic reasoning, and even if there were a consensus, it might easily be overturned at some point in the future, as have consensuses in other fields (physics, biology, religion, etc.). We shouldn't place too much faith in doomsday dampening of our influence on the future, and a natural fallback position is to give some credence to the common-sense perspective. Reference classes and SIA are two examples of the general category of ways in which doomsday reasoning could be defused.

Other approaches for saving future fanaticism

Sentient non-observers?

Note (Nov. 2014): I now disagree with the binary notion of consciousness discussed here. For an example of my updated views, see "Consciousness Is a Process, Not a Moment".

It's not clear which non-human animals would count as anthropic observers within the universal reference class. If there are animals that we consider sentient that aren't intelligent enough to fall into the observer category for anthropic purposes, then the doomsday argument would have less force in updating away from thinking there could be astronomical numbers of such animals in the future. The same applies for artificial sentient minds that might be able to suffer but wouldn't have the capacity to observe themselves as being minds.

If the mere fact of being conscious implies that you are an observer, then there may not be any minds I care about that aren't observers. If you have to be not just conscious but able to understand being an observer, then many minds I care about wouldn't be observers.

Even if some animals or artificial sentients aren't observers, this doesn't wipe away the impact of doomsday reasoning. By default you'd expect that in order to have vast numbers of non-observer sentients, you'd need large numbers of more intelligent minds that were observers to build and safeguard them. If the ratio of non-observer sentients to observers was roughly constant throughout history, then the doomsday argument would work against non-observer sentients just as strongly as against observers. If, in the future, the ratio of non-observer sentients to observers increased significantly, the future could remain very important even after a doomsday update. And of course, if the ratio of non-observer sentients to observers decreased, the future would be even less important than the doomsday argument made it seem.

If non-human animals are observers, this leaves us with a puzzle about why I'm not a non-human animal. Neurologically, it certainly seems like at least some non-human animals should be conscious, although the primates and cetaceans are few enough that the anthropic update against their being conscious is smaller than against, say, insects. Anyway, this discussion is all conditional on the assumption that animals should be observers, which is not at all clear. The possibility that animals are not observers is a reason why concern for potential animal suffering remains extremely important in expectation. It would also remain important if my general anthropic framework were wrong -- e.g., if we used a reference class that excluded non-human animals or if we adopted SIA. In general, doomsday reasoning about the future is directly parallel with these anthropic questions about animal consciousness.

Update: My recent realization about the multiple-civilizations deflation of the doomsday argument also has something to say about the animal question. Consider two planets: One where only human-level life is conscious, and one where insects and pre-human animals are as well. Even after finding yourself as a human, you're equally likely to be on either of these planets, because the second one has a lot more observers offsetting the low probability that any particular one is human. There is a universal update away from animal sentience across all planets, but that update changes the probability distribution slower than a single-planet update would. (Of course, if the animals are identical on both planets, then knowing whether they're conscious on one tells you whether they're conscious on the other, but in general, the animals on different planets should be different, allowing for some to be conscious while others are not.)

If there's a nontrivial chance of suffering in fundamental physics, this would be a more extreme version of the argument from sentient non-observers as to why the far future might be extremely important in expectation. Of course, this assumes that protons and the like aren't observers. Such an assumption is granted by most reference classes that people would ordinarily use, though it's not true for my physics-sampling assumption, which yields a doomsday argument even from very long periods of just fundamental physical processes in the late universe.

Non-conscious observers?

Above I noted that it's not clear if being conscious implies you're an anthropic observer. Conversely, does being an anthropic observer imply that you're conscious? Here by conscious I mean a computation that reflects on itself in a similar way as humans do and that I would consider to have moral importance.

We can imagine AIs of the future who are not conscious in the sense used above but who still decide the fate of our future light cone. If they weren't observers, then doomsday reasoning wouldn't militate against them (except insofar as they might create lots of conscious observers).

Would they be anthropic observers?

  • Yes: They would probably do anthropic reasoning. They would have leftover writings from humans about how to do it properly, and at the very least, they could just follow that guidance. Probably they could devise an even better anthropic theory.
  • No: They wouldn't have to do anthropic reasoning. They could be dumb machines that follow the anthropic rules that humans programmed in. Even if they were smarter than humans, maybe humans could deliberately prevent them from doing their own anthropic reasoning (e.g., by an unchangeable internal prohibition against that) in order to prevent them from being considered observers. Even absent that, I can't imagine myself being a non-conscious anthropic-reasoning computer. It's not clear that doing reasoning about anthropics makes you any more an observer than doing reasoning about quantum mechanics does.

I think the possibility of non-observer AIs shouldn't be ruled out as a reason why future fanaticism might survive doomsday reasoning, although it does seem odd that in these scenarios, the AIs aren't creating many conscious observers (since if they did so, we'd doomsday-update against those scenarios).

Caring about non-conscious things

If, unlike me, you care about things that aren't conscious minds, then the doomsday argument may be less dramatic for you. For instance, if you want to create paperclips, you could aim to build just a few AIs to do this for you. Even if the AIs are observers, you might not need huge numbers of them, and since paperclips aren't observers, you could create lots of paperclips without affecting anthropics too heavily. The same goes if you care about things like knowledge, complexity, beauty, art, etc., as long as you keep the number of observers required to produce them to a minimum. Keep in mind, though, that AIs would run at high clock speed, so even a few individuals could imply lots of observer-moments.

More dramatic hedonic experience

Anthropics weights by observer-moments but not hedonic intensity. Even if we thought we were about halfway through humanity's future, if we thought the second half would be vastly more intense per unit time in hedonic terms, this would push up the value of working on the future.

Closing remarks

When people first hear the doomsday argument, they think it sounds crazy. The other anomalies that anthropics leads to when a universal reference class is used sound even more insane.

Yet at the same time, anthropics with a universal reference class seems to be the great equalizer. Radical Pascalian wagers about far-future influence are humbled at its feet. Mediocritarian thinking defuses Pascalian fanaticism in many other domains. That said, even the universal reference class is not powerful enough to defeat the mighty force of Pascalian model uncertainty, so it seems that fanaticism still rules the day for now.

Acknowledgements

I first discovered the doomsday argument in 2006 but mostly ignored its implications until I more recently read a comment by DanielLC suggesting that he accepts it.

Appendix: If many civilizations exist, the doomsday effect is reduced

Only one civilization

Suppose the entire multiverse contains only one civilization (or multiple approximate copies of this one civilization, such that all the copies are either jointly ShortLived or LongLived). The civilization could be either ShortLived, with X observer-moments, or LongLived, with nX observer-moments. In the text I used n = 1000. You have some probability f that the civilization is LongLived, but you're not quite sure what f should be, so you take a probability distribution P(f) over its possible values. This is a probability distribution over a probability, so we can collapse it to just a single probability. Call P(L) the collapsed probability that the one civilization that exists is LongLived:

P(L) = ∫01 P(f) f df.

Let S represent the event that this one civilization is ShortLived.

Now we observe that we're in the first X observer-moments; call this "obs." We update our probabilities:

P(S|obs)/P(L|obs) = [P(obs|S)/P(obs|L)][P(S)/P(L)]
= [1/(1/n)][P(S)/P(L)]
= n [P(S)/P(L)]
= n [(1-P(L))/P(L)].      (eqn. A)

So the odds ratio for S vs. L has been fully multiplied by n as a result of the update.

We can substitute the integral equation for P(L):

P(S|obs)/P(L|obs) = n [1-∫01 P(f) f df]/[∫01 P(f) f df].      (eqn. B)

If, as an example, we took P(f) = 1 for all f in [0,1], each integral would evaluate to 1/2, and we'd have P(S|obs)/P(L|obs) = n.

If we took P(f) = (1-y) f-y for y in [0,1), then

01 P(f) f df = ∫01 (1-y) f-y f df
= [(1-y)/(2-y)] ∫01 (2-y) f1-y df
= [(1-y)/(2-y)] [f2-y]01
= [(1-y)/(2-y)] [12-y - 02-y]
= (1-y)/(2-y).

Then

P(S|obs)/P(L|obs) = n [1-(1-y)/(2-y)]/[(1-y)/(2-y)]
= n [1/(2-y)]/[(1-y)/(2-y)]
= n/(1-y).      (eqn. C)

If y = 0, then P(f) = 1, and (eqn. C) reduces to P(S|obs)/P(L|obs) = n, just as we saw before. In general, (eqn. C) says that if our prior for f is more weighted toward low values of f (i.e., if y is bigger), then the amount by which we favor S in the posterior odds ratio is higher.

Many civilizations

Now suppose there are many different civilizations in the multiverse. (This seems far more plausible than that the whole multiverse contains just one kind of civilization.) Some fraction f of these are LongLived, but we don't know what f is, so we assign a probability distribution P(f) over the possible values of f. Note that this uncertainty has the same form as in the one-civilization case, but this time f refers to the fraction of actually existing civilizations that are LongLived, rather than our own uncertainty about whether the unique civilization that exists is LongLived or not. In this case, P(L) refers to expected fraction of LongLived civilizations.

Upon learning that we're in the first X observer-moments, we can update our probabilities using exactly the same (eqn. A) as in the one-civilization case. Where this scenario differs from that with one civilization is in P(L). It no longer equals just the expected value of f with respect to P(f), because you're more likely to be in a LongLived civilization than a ShortLived one inasmuch as the LongLived ones have more inhabitants.

We can compute P(L) as follows:

P(L) = ∫01 P(L|f) P(f) df.

If there are f LongLived civilizations each with nX observer-moments and 1-f ShortLived civilizations each with X observer-moments, then the probability you'll be in a LongLived civilization is (fnX)/[fnX + (1-f)X] = fn/[f(n-1)+1].

Substituting into (eqn. A):

P(S|obs) / P(L|obs) = { n [1-∫01 P(f) fn/[f(n-1)+1] df] } / { [∫01 P(f) fn/[f(n-1)+1] df] }.       (eqn. D)

Note that this is the same as (eqn. B) except with an extra factor of n/[f(n-1)+1] inside the integrand, which weights the average based on the fact that you're more likely to be in civilizations with more observer-moments.

Case: P(f) = 1

For concreteness, suppose we took P(f) = 1. Because

∫ fn/[f(n-1)+1] df = [n/(n-1)2][(n-1)f-ln[(n-1)f+1]] + const

we have

01 fn/[f(n-1)+1] df = [n/(n-1)2][(n-1)-ln(n)]

and

1-∫01 fn/[f(n-1)+1] df = (n-1)2/(n-1)2 - [n/(n-1)2][(n-1)-ln(n)] = (1 + n [ln(n) - 1])/(n-1)2.

Substituting back into (eqn. D):

P(S|obs)/P(L|obs) = n [{(1 + n [ln(n) - 1])/(n-1)2}/{[n/(n-1)2][(n-1)-ln(n)]}]
= [(1 + n [ln(n) - 1])/[(n-1)-ln(n)].

For big n, this is approximately {n [ln(n) - 1]}/n = ln(n)-1. In other words, the anthropic update doesn't favor S over L that much even for big n.

Case: P(f) = (1-y) f-y for y in [0,1)

Maybe our prior P(f) = 1 was too inclined toward high values of f. Intuitively it seems plausible that more civilizations are ShortLived than LongLived, even before considering doomsday updates. We can represent this by the class of probability distributions P(f) = (1-y) f-y for y in [0,1).

Integrating these analytically is too hard, so I used Wolfram Alpha to compute numerical answers for a few y and n values. For example, for y = 2/3 and n = 77, here's the equation for computing the ratio (1-P(L))/P(L).

I compiled the P(S|obs)/P(L|obs) numbers into the following table. In addition, I observed a pattern for how the numbers were trending. P(S|obs)/P(L|obs) for y=2/3 seemed to be behaving a lot like n2/3 for large n, so I put n2/3 as a column next to it. Similarly, P(S|obs)/P(L|obs) for y=1/2 seemed to be behaving a lot like n1/2. And of course, when y = 0, P(f) = 1, which we saw above is approximately ln(n)-1, so I listed those values in the last column. We can see how close these approximations are for large n.

n P(S|obs)/P(L|obs) for y=2/3 n2/3 P(S|obs)/P(L|obs) for y=1/2 n1/2 P(S|obs)/P(L|obs) for y=0 ln(n)-1
5 6 3 4 2 1.7 0.6
10 9 5 5 3 2.1 1.3
20 13 7 7 5 2.6 2
100 33 22 16 10 3.8 3.6
1000 136 100 50 32 6 5.9
106 1.2 * 104 104 1571 1000 12.8 12.8
1015 1.2 * 1010 1010 5.0 * 107 3.2 * 107 33.6 33.5

In general, the pattern is that for large n, P(S|obs)/P(L|obs) is roughly ny. For y close to 0, we can improve this approximation as ny [ln(n)-1].

This means that even for plausible probability distributions that place heavy weight on low values of f, the reduction in probability that we're in LongLived vs. ShortLived is not linear in n. This means the doomsday effect is less than I suggest it to be in the remainder of the piece! I need to think about this more. A cheating approach is to take y = 0.99 and reproduce basically a linear doomsday, but it's dubious that such a prior is warranted. For instance, it places only 2.2% probability on f being bigger than 0.1.

Maybe something like y=3/4 is more reasonable. This says that it's 71% likely that no more than 25% of civilizations are LongLived. In this case, we would have P(S|obs)/P(L|obs) being approximately n3/4, and since P(S|obs) is almost 1 for large n, this implies that roughly P(L|obs) = n-3/4. If we're in LongLived, then our influence is n times greater than in ShortLived, so now in Pascalian terms, it seems (n-3/4)(n) = n1/4 times as important to take actions targeted toward a long-term future, although this multiple needs to be discounted by the reduced probability of success for long-term work.

What's a reasonable estimate for n? As noted in the main article, the number of humans so far is about X = 1011. In Astronomical Waste, Bostrom suggests as a high bound about 1038 potential human lives per century in the Virgo Supercluster and as a low bound about 1023 human lives per century. Suppose the civilization lasted 10 billion years, or 108 centuries. Then this would imply nX as high as 1046 or as low as 1031, which means n lies between 1035 and 1020. As a point estimate, say the factor was n = 1028. Then the multiple by which the long-term future is more important in expectation is n1/4 = 107. Even after a big discount factor for the much greater difficulty of making a lasting difference to the very long-term future, this multiplier may remain greater than 1.

Many vs. few civilizations compared

What I've done so far has been to compute P(S|obs)/P(L|obs) for many civilizations, but its value depends on the prior for f that's chosen. To highlight the difference that it makes for us to have many civilizations instead of just one, it would be illuminating to examine P(S|obs)/P(L|obs) for many civilizations compared to its value for one civilization.

For large n, P(S|obs) is nearly 1, so P(L|obs) is approximately P(L|obs)/P(S|obs). For one civilization, (eqn. C) tells us this value is (1-y)/n. For many civilizations, it's n-y. Thus,

[P(L|obs) for many civ's]/[P(L|obs) for one civ] = n1-y/(1-y).

In other words, being one among many civilizations makes it on the order of n1-y times more likely we're long-lived than if we used the doomsday argument for a single civilization.

Gerig, Olum, and Vilenkin paper

After writing this Appendix, I came across a paper that presents the same analysis: Universal Doomsday: Analyzing Our Prospects for Survival (2013) by Gerig, Olum, and Vilenkin. The authors use precisely the same set-up as above, with their R being my n and their D being my obs. In section 3, they too find that in the case of a uniform prior P(f) = 1, P(L|obs) is asymptotically 1/ln(n).

The paper notes:

We can compare these results with the traditional doomsday argument. Using the uniform prior, the prior chance that our civilization would be long-lived is 1/2, and the posterior chance about 1/ln(R). If we took a prior chance of survival P(L) = 1/2 in the traditional doomsday argument, (1.1) would give our chance of survival as only 1/(R + 1). Thus, at least in this case, taking account of the existence of multiple civilizations yields a much more optimistic conclusion [assuming doomsday is bad].