A Primer on the Doomsday Argument

Rarely does philosophy produce empirical predictions. The Doomsday argument is an important exception. From seemingly trivial premises it seeks to show that the risk that humankind will go extinct soon has been systematically underestimated. Nearly everybody's first reaction is that there must be something wrong with such an argument. Yet despite being subjected to intense scrutiny by a growing number of philosophers, no simple flaw in the argument has been identified.

It started some fifteen years ago when astrophysicist Brandon Carter discovered a previously unnoticed consequence of a version of the weak anthropic principle. Carter didn't publish his finding, but the idea was taken up by philosopher John Leslie who has been a prolific author on the subject, culminating in his monograph The End of the World (Routledge, 1996). Versions of the Doomsday argument have also been independently discovered by other authors. In recent years, there have been numerous papers trying to refute the argument, and an approximately equal number of papers refuting these refutations.

Here is the doomsday argument. I will explain it in three steps:

Step I

Imagine a universe that consists of one hundred cubicles. In each cubicle, there is one person. Ninety of the cubicles are painted blue on the outside and the other ten are painted red. Each person is asked to guess whether she is in a blue or a red cubicle. (And everybody knows all this.)

Now, suppose you find yourself in one of these cubicles. What color should you think it has? Since 90% of all people are in blue cubicles, and since you don’t have any other relevant information, it seems you should think that with 90% probability you are in a blue cubicle. Let’s call this idea, that you should reason as if you were a random sample from the set of all observers, the self-sampling assumption.

Suppose everyone accepts the self-sampling assumption and everyone has to bet on whether they are in a blue or red cubicle. Then 90% of all persons will win their bets and 10% will lose. Suppose, on the other hand, that the self-sampling assumption is rejected and people think that one is no more likely to be in a blue cubicle; so they bet by flipping a coin. Then, on average, 50% of the people will win and 50% will lose. – The rational thing to do seems to be to accept the self-sampling assumption, at least in this case.

Step II
Now we modify the thought experiment a bit. We still have the hundred cubicles but this time they are not painted blue or red. Instead they are numbered from 1 to 100. The numbers are painted on the outside. Then a fair coin is tossed (by God perhaps). If the coin falls heads, one person is created in each cubicle. If the coin falls tails, then persons are only created in cubicles 1 through 10.

You find yourself in one of the cubicles and are asked to guess whether there are ten or one hundred people? Since the number was determined by the flip of a fair coin, and since you haven’t seen how the coin fell and you don’t have any other relevant information, it seems you should believe with 50% probability that it fell heads (and thus that there are a hundred people).

Moreover, you can use the self-sampling assumption to assess the conditional probability of a number between 1 and 10 being painted on your cubicle given how the coin fell. For example, conditional on heads, the probability that the number on your cubicle is between 1 and 10 is 1/10, since one out of ten people will then find themselves there. Conditional on tails, the probability that you are in number 1 through 10 is one; for you then know that everybody is in one of those cubicles.

Suppose that you open the door and discover that you are in cubicle number 7. Again you are asked, how did the coin fall? But now the probability is greater than 50% that it fell tails. For what you are observing is given a higher probability on that hypothesis than on the hypothesis that it fell heads. The precise new probability of tails can be calculated using Bayes’ theorem. It is approximately 91%. So after finding that you are in cubicle number 7, you should think that with 91% probability there are only ten people.

Step III

The last step is to transpose these results to our actual situation here on Earth. Let’s formulate the following two rival hypotheses. Doom Early: humankind goes extinct in the next century and the total number of humans that will have existed is, say, 200 billion. Doom Late: humankind survives the next century and goes on to colonize the galaxy; the total number of humans is, say, 200 trillion. To simplify the exposition we will consider only these hypotheses. (Using a more fine-grained partition of the hypothesis space doesn’t change the principle although it would give more exact numerical values.)

Doom Early corresponds to there only being ten people in the thought experiment of Step II. Doom Late corresponds to there being one hundred people. Corresponding the numbers on the cubicles, we now have the "birth ranks" of human beings – their positions in the human race. Corresponding to the prior probability (50%) of the coin falling heads or tails, we now have some prior probability of Doom Soon or Doom Late. This will be based on our ordinary empirical estimates of potential threats to human survival, such as nuclear or biological warfare, a meteorite destroying the plant, runaway greenhouse effect, self-replicating nanomachines running amok, a breakdown of a metastable vacuum state due to high-energy particle experiments and so on (presumably there are dangers that we haven’t yet thought of). Let’s say that based on such considerations, you think that there is a 5% probability of Doom Soon. The exact number doesn’t matter for the structure of the argument.

Finally, corresponding to finding you are in cubicle number 7 we have the fact that you find that your birth rank is about 60 billion (that’s approximately how many humans have lived before you). Just as finding you are in cubicle 7 increased the probability of the coin having fallen tails, so finding you are human number 60 billion gives you reason to think that Doom Soon is more probable than you previously thought. Exactly how much more probable will depend on the precise numbers you use. In the present example, the posterior probability of Doom Soon will be very close to one. You can with near certainty rule out Doom Late.

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That is the Doomsday argument in a nutshell. After hearing about it, many people think they know what is wrong with it. But these objections tend to be mutually incompatible, and often they hinge on some simple misunderstanding. Be sure to read the literature before feeling too confident that you have a refutation.

If the Doomsday argument is correct, what precisely does it show? It doesn’t show that there is no point trying to reduce threats to human survival "because we’re doomed anyway". On the contrary, the Doomsday argument could make such efforts seem even more urgent. Working to reduce the risk that nanotechnology will be abused to destroy intelligent life, for example, would decrease the prior probability of Doom Soon, and this would reduce its posterior probability after taking the Doomsday argument into account; humankind’s life expectancy would go up.

There are also a number of possible "loopholes" or alternative interpretations of what the Doomsday argument shows. For instance, it turns out that if there are many extraterrestrial civilizations and you interpret the self-sampling assumption as applying equally to all intelligent beings and not exclusively to humans, then another probability shift occurs that exactly counterbalances and cancels the probability shift that the Doomsday argument implies. Another possible loophole is if there will be infinitely many humans; it’s not clear how to apply the self-sampling assumption to the infinite case. Further, if the human species evolves into some vastly more advanced species fairly soon (within a century or two), maybe through the use of advanced technology, then it is not clear whether these posthumans would be in the same reference class as us, so it’s not clear how the Doomsday argument should be applied then. Yet another possibility is if population figures go down dramatically – it would then take much longer before enough humans are been born that your birth rank starts looking surprisingly low. And finally, it may be that the reference class needs to be relativized so that not all observers, not even all humans, will belong to the same reference class.

The justification for relativizing the reference class would have to come from a general theory of observational selection effects, of which the self-sampling assumption would be only one element. A theory of observational selection effects - of how to correct for biases that are introduced by the fact that our evidence has been filtered by the precondition that a suitably positioned observer exists to "have" the evidence - would have applications in a number of scientific fields, including cosmology and evolutionary biology.

So although the Doomsday argument contains an interesting idea, it needs to be combined with additional assumptions and principles (some of which remain to be worked out) before it can be applied to the real world. In all likelihood, even when all details are filled in, there will be scope for differing opinions about our future. Nonetheless, a better understanding of observational selection effects will rule out certain kinds of hypotheses and impose surprising constraints on any coherent theorizing about the future of our species and about the distribution of observers in the universe.