This animation is based on an article (in preparation, with Dustin Wehr), whose goal is to show a much more detailed analysis of Bostrom's thought experiment. One of the results, visible above, is that the resulting trilemma is by no means clear-cut, i.e. the probabilities are not necessarily close to 0 or 1.

The beauty of the simple reasoning is still here, though: the outcome does not depend on the particular model of civilization dynamics but only on some means and frequencies. Still, to understand the animation, I will roughly describe how this universe was constructed.

• The system has discrete time in arbitrary units, but it could correspond to a century or millennium.
• Likewise, the population units are arbitrary, perhaps millions of individuals. The areas of the bubbles are proportional to the population sizes.
• It is the animation speed and screen size that dictate those arbitrary units, for your viewing pleasure. One could rescale it, after lots of speculative astro-biology, to conform to our universe, but that would not change what's important: the probabilities.
• Civilizations are born randomly, at a fixed rate (first input) per time unit. All start as blue.
• All civilizations face doom, and the input fields above specify probabilities of extinction per time unit for both young and old (advanced) civilizations. Quantities such as the overall probability of going extinct or a mean lifetime depend on how the transition is modeled, and the lower table simply gives the chance of ascension (becoming super-advanced technologically) based on the actual random run being visualized.
• Young civilizations grow exponentially and once they are past a certain age (250) they have a chance of becoming demiurgic, i.e. far more advanced than we are. If that happens, their populations decrease exponentially to some fixed size, and their chance of extinction changes (lowers by default). Demiurgic civilizations are green.
• Some fraction of the demiurgic civilizations are “evil”, meaning that they run ancestral simulations. This is the fourth input field. The pale yellow bubble shows the simulated populations.
• Each simulation includes all of the civilization's ancestors. There are about 4 simulations per time unit, and with the present probability of their extinction equal to 0.004 that means 250 units of mean lifetime and roughly 1000 simulations per demiurgic civilization.
• A parameter crucial in the analysis is the ratio of cumulative ancestors for civilizations that don't run simulations to the corresponding number for evil civilizations. The model allows some blue civilizations to grow without becoming green, so d could be greater than 1, but most of the time blue civilizations go extinct early on, bringing d below 1. This means that advanced civilization simulate more beings than there are ancestors (on average).
• Finally, a direct count gives the chance (ratio) of living in a simulation.

For the default settings, with 2% of evil civilizations, after about 100000 time steps, the chance of not going extinct before ascension was 92%, d was around 1.31, and the chance of being simulated was about 60%. Not as shocking, as Bostrom's reasoning suggested.

Yes, the parameters were chosen specifically to undermine the intuitive result, but that's how it goes in math: one counter-example is enough. Here of course there's an infinite family of counter-examples, but I will leave the details for the paper.

The last thing I want to stress is that the central result is not the probability of living in a simulation per se, but the constraint tying it to other quantities (extinction, being evil, ancestral populations etc.). As the experiment shows, it's not a simple choice between “astronomically large” or “insignificantly small”.