From Simulated Majorities to "You Are Probably Simulated": Bostrom on the Missing Step in the Simulation Argument

Lex Fridman asks about the point where Bostrom's conversation with Joe Rogan seemed to stall: why does the possibility that someone eventually creates a simulation imply that we are likely in one? Bostrom explains that if you accept his third alternative, then counting all observers across time yields more simulated people with our kinds of experiences than non-simulated ones. But he notes there is still an extra step needed to get from that count to a claim about you in particular.

That step is indexical: what probability should you give to being one of the simulated ones? Bostrom calls his answer the bland principle of indifference. When there are two sets of observers, one much larger than the other, and no internal evidence tells you which one you belong to, your probability should be proportional to the sizes of the sets, so ten times as many simulated people makes you ten times more likely to be one of them.

Fridman finds the principle intuitive, since with too little information you should rationally spread probability evenly, and Bostrom agrees it seems pretty plausible to him. The principle is deliberately modest: it does not say you are special, only that in the absence of evidence you should weigh each set of observers by its size.

Fridman asks about holes in the argument, such as needing infinite time. Bostrom says you do not need infinity, only enough time for an intelligent civilization to emerge and run ancestor simulations. Different planets could reach that point at very different cosmological epochs, so the argument does not require them to be synchronized.

Both arguments use anthropic reasoning, Bostrom says, but the assumption needed for the simulation argument is much weaker. He introduces the Doomsday argument, first discovered by the physicist Brandon Carter and developed by the philosopher John Leslie, as a claim that we have systematically underestimated the probability of humanity going extinct soon. He adds that most people think something is wrong with it, though it has proved hard to say exactly what.

Bostrom stresses that the two arguments share a style of reasoning but not a strength of assumption. Anthropic reasoning is reasoning about indexical propositions, such as where or when you are, and the simulation argument gets by on far less than the Doomsday argument does, as he goes on to show.

He explains it through two urns, one with ten numbered balls and one with a million. If you draw ball number 7, a Bayesian update makes the ten-ball urn almost certain, even from a 50/50 prior. The Doomsday argument applies the same logic to humanity: suppose the total number of humans who will ever live is either 200 billion or 200 trillion, and treat your own birth rank, roughly 100 billion, as a random sample.

Under the 200 billion hypothesis, being human number 100 billion is unremarkable, but under 200 trillion you would be remarkably early. So you should shift probability toward the smaller total, meaning "doom soon". Bostrom stresses that this depends on the assumption that you should reason as if you were a random sample from all humans who will ever live, and the question is why you should accept that.

Bostrom calls the Doomsday premise the self-sampling assumption: reason as if you are a random sample from all observers in some reference class. Many people reject it because it concludes that we have almost no chance of reaching 200 trillion humans just from reflecting on when we were born. He says this reaction is understandable, since you would otherwise need sophisticated arguments about the impossibility of space colonization.

Yet in other domains something like this assumption seems necessary. In multiverse cosmology, a large enough universe contains observers who see every possible measurement, including statistical flukes, so we need to treat ourselves as typical observers to treat our evidence as support for a theory. Without a principle like this, inferences such as the temperature of the cosmic background or the fine-structure constant do not obviously go through.

That is why Bostrom describes a project of building a theory of anthropic reasoning, which he calls a difficult methodological area. For the simulation argument he emphasizes the contrast: the Doomsday argument asks you to reason as a random sample from all humans even though you know you are number 100 billion in the year 2020, whereas the simulation argument only says to be uniformly uncertain if you really cannot tell which group you are in. The relevant observers are the people in original histories and the people in simulations.

To make the principle of indifference intuitive, Bostrom suggests considering a sequence of cases where the fraction of simulated people grows. If everyone with your experiences is simulated, you can conclude with certainty that you are too. As the fraction moves from 90% to 99% to 99.9%, he says it would be strange for your probability not to approach one smoothly, and he mentions other supporting arguments.

Fridman raises the point that the fraction may not literally reach one, since even a mature civilization could run many simulations using only a small share of its resources, but not infinitely many. Bostrom agrees that physics sets limits. With a positive cosmological constant, the reachable universe is a finite sphere of matter, and there may be unavoidable computational losses and processes like proton decay that eventually lead to a heat death.

They then discuss stacked simulations, where one simulation spawns another. Bostrom acknowledges that we could not know which level we are on, but every computation done inside the tower must ultimately be powered by the computer in basement reality. If that machine has finite compute, and each level adds a large overhead, he suggests the tower might not be very tall, leaving most people nearer the bottom.