The Doomsday Argument in Many Worlds

You and I are highly unlikely to exist in a civilization that has produced only 70 billion people, yet we find ourselves in just such a civilization. Our circumstance, which seems difficult to explain, is easily accounted for if (1) many other civilizations exist and if (2) nearly all of these civilizations (including our own) die out sooner than usually thought, i.e., before trillions of people are produced. Because the combination of (1) and (2) make our situation likely and alternatives do not, we should drastically increase our belief that (1) and (2) are true. These results follow immediately when considering a many worlds version of the “Doomsday Argument” and are immune to the main criticism of the original Doomsday Argument.

Austin Gerig
Imagine you are sitting at a table, blindfolded, and that an urn is placed in front of you. You are told this urn can be one of two types:
it is either a small urn or a large urn.
If it is a small urn, it contains 10 balls numbered 1 through 10.
If it is a large urn, it contains 1 million balls numbered 1 through 1 million.
You currently do not know whether the urn is small or large, but would like to find out. Suppose you randomly draw one ball and find it numbered 7.
Given your draw, with what probability is the urn small?

Let T1 and T2 represent the theories that the urn is small and large respectively, and let D represent your data, i.e., that you have drawn the number 7.
Assuming you believe the urn equally likely to be small or large before you draw a number, then according to Bayes’ Law, your updated belief in T1 conditioned on observing your data is,
You therefore should believe the urn is small with almost certainty.
Notice that you have become confident in the size of the urn with only one draw.
Now, multiply the number of balls in each urn by 10^10 and imagine they correspond to birth numbers within our civilization, i.e., Adam drew number 1, Eve drew number 2, and so on (you and I have drawn numbers around 7 × 10^10).
T1 and T2 represent two competing theories, which we initially treat as equally likely. Under T1, the urn is small and contains only 10^11 numbers, which means only 10^11
people will ever exist and our civilization will die out within the next few centuries. Under T2, the urn is large and contains 10^16 numbers so that our civilization is large and will continue on for many years into the future.
As before, you would like to determine whether the urn is small or large, i.e., whether our civilization is small or large.
Given your data, D = 7 × 10^10, how should you update your belief in T1?
Without any compelling argument to the contrary, you should update it as before,
which means you should believe with almost certainty that our civilization is small and will die out within the next few centuries.
This argument is the Doomsday Argument (DA) as presented in Leslie (1989) and Leslie (1996)1.
The details of the DA can be restructured –the numbers can be changed, more urn types can be added, etc. – but the final result remains unchanged: when we condition on our birth number, we must drastically increase the probability that our civilization will soon die out.
There are many critiques of the DA, which I will not focus on here (see Leslie (1996) or Bostrom (2002) for a full treatment).
By most accounts, the DA has stood up to all criticisms except one.
As first mentioned in Dieks (1992) and expanded in Bartha and Hitchcock (1999) and Olum (2002), the DA fails to consider that you are more likely to exist in a large civilization than a small one.
This missing step exactly cancels the updating of your beliefs so that your original prior is retained.
In this paper, I intend two things: (1) to defend the counter-argument to the DA developed in Bartha and Hitchcock (1999) and Olum (2002),and (2) to show that this counter-argument does not work when the DA
is modified to allow for many worlds.
The take-home message is the following: given that we exist in a civilization that has produced 70 billion people so far, we should drastically increase our belief that many other civilizations exist and that nearly all of these civilizations (including our own) will die out before producing trillions of people.

II. The Devil’s Existence

The Devil’s Existence (DE) is a thought experiment where you are asked to determine whether or not the Devil exists (the Devil representing some doom event).
Suppose God creates 1 million rooms, each sequentially numbered, and that He initially intends to fill each room with one person.
He first goes to room 1 and generates a person inside.
He then moves on to room 2 and does the same, and so on, until the first 10 rooms have been filled.
At this point, if and only if the Devil exists, he arrives on the scene and destroys the remaining rooms so that no more people are created.
Suppose you have been created and find yourself in room 7.
Assuming you thought it equally likely that the Devil does or does not exist before
considering this information, how likely is it now that the Devil exists? …..
(….)

Conclusion
Consider our current circumstance: we exist in a civilization that has produced only 70 billion people.
At first sight, our data seems highly unlikely.
If our civilization dies out soon so that it is ultimately small, then not that many people exist and it is highly unlikely that you and I are alive.
If our civilization is ultimately very, very large, then our existence might be explained, but it is highly unlikely for you and I to have such low birth numbers.
Given the situation, we can either accept that we are atypical or we can seek plausible alternative theories that better explain our data.
Here, I have considered the following theory: there are many, many civilizations that exist and nearly all of these civilizations are small.
Under this theory, our existence is certain and our birth number is typical.
Because the theory makes our circumstance likely and alternatives do not, we should drastically increase our belief that the theory is true when conditioning on our data.
Read more: arxiv.org/pdf (http://arxiv.org/abs/1209.6251)

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