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Doomsday and Simulation

Assume nothing is special about now, and you can forecast the end of anything from its age alone. Assume the same about your mind, and you are probably software.

The argument, in their words

Gott reasons that if nothing is special about the moment you look at something, that moment is uniformly distributed over its lifetime. So with 95% probability you are not in the first or last 2.5%, and the future duration lies between 1/39 and 39 times the past duration. The abstract of his 1993 Nature paper:

Making only the assumption that you are a random intelligent observer, limits for the total longevity of our species of 0.2 million to 8 million years can be derived at the 95% confidence level. — Gott, Nature 363 (1993), abstract

Homo sapiens is about 200,000 years old, so the future is between 5,100 and 7.8 million years. Gott has applied the same rule to the Berlin Wall, Stonehenge, Broadway plays, Nature itself, the Chicago White Sox and himself.

Musk’s version rests on Nick Bostrom’s simulation paper, which concludes only a disjunction:

This paper argues that at least one of the following propositions is true: (1) the human species is very likely to go extinct before reaching a “posthuman” stage; (2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof); (3) we are almost certainly living in a computer simulation. — Bostrom (2003), abstract

At Recode’s Code Conference in 2016, Elon Musk gave the version most people have heard. Games went from Pong to photorealism in forty years, so they will become indistinguishable from reality, and there will be billions of machines running them. Asked what he thought:

There’s a one in billions chance we’re in base reality. — Musk (2016), as transcribed by Ezra Klein, Vox

Rebuke 1: the rule is right on average and wrong for nearly everyone

Gott’s rule gives every phenomenon of age t a 50% chance of reaching age 2t, whatever the phenomenon is. Mark Brown sent Andrew Gelman the obvious test:

Apply the method to a 90 year old man and the 50% prediction confidence interval for his residual life will be (30,270). Apply it to his 9 year old great grandson and you get (3,27). — Mark Brown, quoted by Gelman (2018)

Gelman’s reply:

Hey, living to the age of 270—that’d be cool! I just have to make it to 90, then I’m most of the way there. — Gelman (2018)

There is a precise sense in which Gott is right, and it is worth seeing. Take a population with stable births and a realistic mortality curve, and pick a living person at random without looking at their age. Their age is then uniformly placed within their lifetime, and Gott’s intervals have exactly the coverage he claims. I checked this with a Gompertz–Makeham mortality model (hazard 0.0005 + 0.00003 e^0.095 age, life expectancy 77 years; the script is in the notes). Averaged over everyone alive, the 50% intervals cover 50.0% of the time and the 95% intervals cover 94.9%. Now condition on the age, which is the only thing Gott uses:

The 50% interval is right for almost nobody in particular. It averages to 50% because it is nearly always right for the middle-aged and nearly always wrong for the old and the young. The model is illustrative, but no plausible mortality curve changes the pattern. Gelman put the statistical point in 2005:

If you look carefully at classical statistical theory, you’ll see that it makes claims about averages, not about particular cases. However, this does not mean that there is a 95% chance that any particular interval will contain the true value. Especially not in this situation, where we have additional subject-matter knowledge. — Gelman (2005)

His summary: “The doomsday argument is pretty silly (and also, it’s not Bayesian).” Carlton Caves reached the same place by a different route. Gott’s rule is the Bayesian answer under one particular prior, proportional to 1/T² in total duration, which is the prior that has no time scale in it. Caves’s abstract:

Nothing licenses indiscriminate use of Gott’s rule as a predictor of future duration. It should only be used when the phenomenon in question has no identifiable time scales. — Caves (2008), abstract

Humans have a time scale. So do dogs, baseball careers and species.

Rebuke 2: Caves offered to bet

In 1999 Caves collected the names, breeds and birth dates of 24 dogs owned by people in his department and lodged a notarised list. For each of the six dogs over ten years old, Gott’s rule says a 50% chance of living to twice its age.

To sweeten the pot, I offered Gott 2:1 odds in his favor. Gott refused the bets on the grounds that “I don’t do bets.” If he had believed his own predictions, his expected gain would have been $3,000 US, and the probability that he would have been a net loser on the six bets was 7/64=0.11. I contacted the caretakers during May and June of 2008 and verified that all six dogs have died. Thus, as I fully expected, I would have won all the bets and been $6,000 US richer. — Caves (2008), sec. 4

Under Gott’s rule the chance that all six dogs fail to reach twice their age is 1/64. Caves would have put the chance near 1, because a dog of eleven is not going to see twenty-two.

Caves did the same arithmetic for baseball. In June 2008 the Chicago Cubs had not won the World Series since 1908. Gott’s rule gives a 50% chance that the drought lasts another 99 years. Giving every one of 30 clubs an equal chance each year gives (29/30)⁹⁹ = 3.5%. The Cubs won in 2016. Gott’s rule had given that about an 8% chance (8.4 more years on a 100-year drought); the equal-chance model gave it about 26%. Caves had done the White Sox too: in 1996 Gott’s rule gave a title by 2005 a 10% chance against 26% for equal chances, and they won in 2005.

Gott’s response

According to Caves, Gott replied that his rule applies to a dog drawn at random from the sample, not to particular dogs whose ages are known. Caves agrees, and that is the point:

It is obvious that in a large sample of dogs selected without regard to age, roughly half the dogs, within the inevitable statistical fluctuations, will be in the first half of their lives, with the rest in the second half. This is the trivial content of Gott’s Copernican principle. It is equally obvious that having a sample in which half the dogs are in the first half of their lives does not imply that any particular dog in the sample has a probability of 1/2 to survive beyond twice its present age. — Caves (2008), sec. 4

Humanity is one particular dog, of known age. Gott’s prediction for it is made on exactly the basis he declined to bet on for the dogs.

Rebuke 3: the answer depends on who “we” are

“A random intelligent observer” has to be random within some class. The class is not given by nature, and Gott’s own arithmetic moves with it. Here is his 95% rule applied to some reasonable choices of what we are, measured in years:

Each row is equally Copernican. The last row is the class that is actually doing the reasoning, and it predicts the end of Doomsday reasoners, not of people. Radford Neal, commenting on Gelman’s 2005 post:

One problem with the Doomsday Argument is that there isn’t just one of them – there are several, some Bayesian, some not, propounded with varying degrees of statistical literacy. None of them are precise (due to the “reference class” problem), which is one argument for why they’re invalid. — Neal, comment on Gelman (2005)

Bostrom, who is sympathetic to Doomsday reasoning and wrote the book on it, agrees: “Gott’s version is incorrect.”

We found shortcomings in the versions that have been presented in the literature, we argued that John Leslie’s proposal for solving the reference class problem is unworkable, and we showed that DA has alternative interpretations and is inconclusive. — Bostrom, Anthropic Bias (2002), ch. 10

Rebuke 4: Musk deleted two of Bostrom’s three options

Bostrom’s argument is close to arithmetic. Let f_P be the fraction of human-level civilisations that reach a posthuman stage, f_I the fraction of those interested in running ancestor simulations, and N_I the number such a civilisation runs, each roughly as populous as the real history. The fraction of human-type observers who are simulated is

f_sim = f_Pf_IN_I / (f_Pf_IN_I + 1).

If N_I is enormous, either f_Pf_I is tiny or f_sim is close to 1. That is the trilemma, and given substrate-independence and the premise about computing power, it is valid. Bostrom then declines to say which horn is true:

In the dark forest of our current ignorance, it seems sensible to apportion one’s credence roughly evenly between (1), (2), and (3). — Bostrom (2003), sec. VII

That puts his credence that we are simulated at something like one in three. Musk’s “one in billions” for base reality needs f_Pf_IN_I of about a billion. In other words it needs the first two horns to be false with near certainty, which is to assume that civilisations like ours survive and want to run simulations. Musk’s premise was that games keep improving. Bostrom’s first horn, extinction before maturity, is the Doomsday question again, and nobody knows its answer. Bostrom’s FAQ describes the move:

I note that people who hear about the simulation argument often react by saying, “Yes, I accept the argument, and it is obvious that it is alternative n that obtains.” But different people pick a different n. Some think it obvious that (1) is true, others that (2) is true, yet others that (3) is true. Many if not all of these people are overconfident. — Bostrom, The Simulation Argument FAQ (2025), Q2

The FAQ also has an entry titled “What if one is Elon Musk?”, which begins: “This is not actually a frequently asked question, but perhaps it should be.”

Bostrom also separates his principle from Gott’s kind:

The Doomsday argument rests on a much stronger and more controversial premiss, namely that one should reason as if one were a random sample from the set of all people who will ever have lived (past, present, and future) even though we know that we are living in the early twenty‑first century rather than at some point in the distant past or the future. The bland indifference principle, by contrast, applies only to cases where we have no information about which group of people we belong to. — Bostrom (2003), sec. V

That is the right distinction. Gott knows our position in the sequence and throw the knowledge away. Bostrom’s principle applies only when nothing you can observe tells simulated and unsimulated minds apart.

Scope

None of this says humanity is safe. Extinction risk is real, and the actual hazards deserve estimates. The rebuke is to the idea that our species’ age supplies one for free.

Nor does it say we are not simulated. Bostrom’s disjunction may well be true, and I cannot say which horn holds. The rebuke to the simulation side is narrow: to the step from “at least one of three” to “almost certainly the third”.

The birth-rank version of the Doomsday argument, due to Carter and Leslie, is a Bayesian update rather than a confidence rule, and it is not judged here.

In their favour

Gott’s first prediction came true. He visited the Berlin Wall in 1969, when it was eight years old, and told a friend there was a 50% chance it would last between 2⅔ and 24 more years. It came down twenty years later. On 27 May 1993, the day the Nature paper appeared, he gave 95% intervals for the 44 Broadway and off-Broadway shows then running, and he reports that they did well. Olum checked Broadway runs from 1970 to 1990 and found why: for shows past their first few months, the share running a given length falls off roughly as T^−1.7, close to the 1/T² that makes Gott’s rule exact. For long-running shows the rule had the right prior by luck or instinct. Olum also found it fails for new shows: a show open 44 days gets a 97.5% chance of closing within 156 days from Gott, against 78% in the data.

That is the honest case for the rule. Where a phenomenon has no time scale you know of, the scale-free answer is sensible, and Caves says so. It resembles the Lindy heuristic that old books and old firms will last longer than new ones, which is often a good bet. The error is to use it where you do know something.

Musk’s premise, that simulations will become indistinguishable from reality, is not absurd, and Bostrom’s argument needs something like it too. The error is in the number, not the premise.

Disclosure: this page was researched and drafted by a language model, which also ran the mortality calculation in Rebuke 1. Check the quotations.

References

• Gott, J. R. III (1993). “Implications of the Copernican principle for our future prospects.” Nature 363, 315–319. doi:10.1038/363315a0.
• Caves, C. M. (2008). “Predicting future duration from present age: Revisiting a critical assessment of Gott’s rule.” arXiv:0806.3538. Caves, C. M. (2000). “Predicting future duration from present age: A critical assessment.” Contemporary Physics 41, 143–153. arXiv:astro-ph/0001414. The 2000 paper quotes Gott’s own account of the Berlin Wall prediction from Timothy Ferris, “How to Predict Everything”, The New Yorker, 12 July 1999.
• Gelman, A. (2005). “Doomsday and Bayes.” Statistical Modeling, Causal Inference, and Social Science, 10 November, with Radford Neal’s comment. Gelman, A. (2018). “Doomsday! Problems with interpreting a confidence interval when there is no evidence for the assumed sampling model.” Same blog, 4 May.
• Leslie, J. (1996). The End of the World: The Science and Ethics of Human Extinction. Routledge. Leslie, J. (1990). “Is the end of the world nigh?” Philosophical Quarterly 40(158), 65–72.
• Bostrom, N. (2002). Anthropic Bias: Observation Selection Effects in Science and Philosophy. Routledge. Full text online; chapters 4, 6, 7 and 10 quoted.
• Olum, K. D. (2002). “The doomsday argument and the number of possible observers.” Philosophical Quarterly 52(207), 164–184. arXiv:gr-qc/0009081. Dieks, D. (1992). “Doomsday—or: the dangers of statistics.” Philosophical Quarterly 42(166), 78–84.
• Bostrom, N. (2003). “Are You Living in a Computer Simulation?” Philosophical Quarterly 53(211), 243–255. PDF. Bostrom, N. (2025). The Simulation Argument FAQ, version 2.0.
• Klein, E. (2016). “Elon Musk believes we are probably characters in some advanced civilization’s video game.” Vox, 2 June. Transcript of Musk at Recode’s Code Conference.