The Simulation Argument Made Quantitative

Elon Musk, of Space X, Tesla, and Solar City fame, recently mentioned that he thought the the odds of us not living in a simulation were a billion to one. His reasoning was based on extrapolating the rate of improvement in video games. He suggests that soon it will be impossible to distinguish simulations from reality and in ten thousand years there could easily be billions of simulations running. Thus there are a billion more simulated universes than real ones.

This simulation argument was first quantitatively formulated by philosopher Nick Bostrom. He even has an entire website devoted to the topic (see here). In his original paper, he proposed a Drake-like equation for the fraction of all "humans" living in a simulation:

where

is the fraction of human level civilizations that attain the capability to simulate a human populated civilization, is the fraction of these civilizations interested in running civilization simulations, and is the average number of simulations running in these interested civilizations. He then argues that if is large, then either or . Musk believes that it is highly likely that is large and is not small so, ergo, we must be in a simulation. Bostrom says his gut feeling is that is around 20%. Steve Hsu mocks the idea (I think). Here, I will show that we have absolutely no way to estimate our probability of being in a simulation.

The reason is that Bostrom's equation obscures the possibility of two possible divergent quantities. This is more clearly seen by rewriting his equation as

where

is the number of non-sim civilizations and is the number of sim civilizations. (Re-labeling and as people or universes does not change the argument). Bostrom and Musk's observation is that once a civilization attains simulation capability then the number of sims can grow exponentially (people in sims can run sims and so forth) and thus can overwhelm and ergo, you're in a simulation. However, this is only true in a world where is not growing or growing slowly. If is also growing exponentially then we can't say anything at all about the ratio of to .

I can give a simple example. Consider the following dynamics

is being created by but both are both growing exponentially. The interesting property of exponentials is that a solution to these equations for is

where I have chosen convenient initial conditions that don't affect the results. Even though

is growing exponentially on top of an exponential process, the growth rates of and are the same. The probability of being in a simulation is then

and we have no way of knowing what this is. The analogy is that you have a goose laying eggs and each daughter lays eggs, which also lay eggs. It would seem like there would be more eggs from the collective progeny than the original mother. However, if the rate of egg laying by the original mother goose is increasing exponentially then the number of mother eggs can grow as fast as the number of daughter, granddaughter, great..., eggs. This is just another example of how thinking quantitatively can give interesting (and sometimes counterintuitive) results. Until we have a better idea about the physics underlying our universe, we can say nothing about our odds of being in a simulation.