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:

f_{sim} = \frac{f_p f_I N_I}{f_p f_I N_I + 1}

where f_p is the fraction of human level civilizations that attain the capability to simulate a human populated civilization, f_I is the fraction of these civilizations interested in running civilization simulations, and N_I is the average number of simulations running in these interested civilizations. He then argues that if N_I is large, then either f_{sim}\approx 1 or f_p f_I \approx 0. Musk believes that it is highly likely that N_I is large and f_p f_I is not small so, ergo, we must be in a simulation. Bostrom says his gut feeling is that f_{sim} 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

f_{sim} = \frac{y}{x+y} = \frac{y/x}{y/x+1}

where x is the number of non-sim civilizations and y is the number of sim civilizations. (Re-labeling x and y 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 y can overwhelm x and ergo, you’re in a simulation. However, this is only true in a world where x is not growing or growing slowly. If x is also growing exponentially then we can’t say anything at all about the ratio of y to x.

I can give a simple example.  Consider the following dynamics

\frac{dx}{dt} = ax

\frac{dy}{dt} = bx + cy

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

x = \exp(at)

y = \frac{b}{a-c}\exp(at)

where I have chosen convenient initial conditions that don’t affect the results. Even though y is growing exponentially on top of an exponential process, the growth rates of x and y are the same. The probability of being in a simulation is then

f_{sim} = \frac{b}{a+b-c}

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.

Addendum: One of the predictions of this simple model is that there should be lots of pre-sim universes. I have always found it interesting that the age of the universe is only about three times that of the earth. Given that the expansion rate of the universe is actually increasing, the lifetime of the universe is likely to be much longer than the current age. So, why is it that we are alive at such an early stage of our universe? Well, one reason may be that the rate of universe creation is very high and so the probability of being in a young universe is higher than being in an old one.

Addendum 2: I only gave a specific solution to the differential equation. The full solution has the form Y_1\exp(at) + Y_2 \exp(ct).  However, as long as a >c, the first term will dominate.

Addendum 3: I realized that I didn’t make it clear that the civilizations don’t need to be in the same universe. Multiverses with different parameters are predicted by string theory.  Thus, even if there is less than one civilization per universe, universes could be created at an exponentially increasing rate.



5 thoughts on “The simulation argument made quantitative

  1. Maybe I really should have been a mathematician.

    (I did try taking a grad course in number theory once, and flunked, because they wanted me to do a computer calculation of dedekind sums and the entire reason i took the course was to get away from computers and doing simulations —and i hadnt even taken an uindergrad course in number theory.

    i had already done 2 simulations, on anti-body-antigen aggregation (one on a modification of Richard Goldberg’s 1952 J Am Chem Soc paper—‘the most probable size disttribution of antibodty-antigen complexes’ —-a calculus of variations problem which is basically a kind of multinomial generalization of the derivation of the Boltzmann distribution, except my case was for ‘finite systems’ so you don’t have a simple CLT (central limit theorem argument) ) and the other applying information theory to try to predict RNA secondary structure from primary structure—i used a ‘fact free science’ approach based on the idea that simply by looking at the ‘words’ in the RNA code one should be able to figure out what the meaning of the sentence was (how it folds up, and also how iot doesn’t—‘colorless green ideas sleep furiously’.) .

    I have a friend who says he is designing computer games after (getting a biology PhD from U Cal somewhere though I think he knows less biology than me—but he knows his computers) and I may ask him about this.

    That equation I find puzzling so i may be going senile. Something like z = x/(x+1 ) for the number of civilizations. In Bostrom’s paper it has originally the form like z = ab/ (ab +b) so it reduces to that. Reminds me of continued fractions .

    (Bostrom had a reasonable talk I heard on the radio, but this whole ‘superintelligence’ thing is beyond my field of vision. It seems quite popular among the ‘less wrong’ types ‘machine intelligence research institute’ (Wikipedia— funded by peter thiel who i think did paypal to make billions$, and also heard is an evangelical christian, also may fund a project on ‘universal basic income’ (see milton friedman, thomas paine etc.) and just funded a lawsuit by hulk Hogan to knock out gawker. most of these people I basically know nothing about except from the radio. oh, I see from Wikipedia that elon musk was a collaborator with thiel).

    this reminds me of a sort of ‘zeno’ argument about the mother goose and the babies—who reaches the finish mark first? (or is most fit, has most progeny? the tortoise or the hare?) Similar types of arguments seem to appear in questions like the approach to equilibrium (second law—time goes slower and slower as you approach equilibrium so you may never get there), what happens when you fall into a black hole, and maybe even issues in mathematical logic (eg Goodstein’s theorem , or the ‘hydra game’)—how do you compare convergence rates of different infinite limits. (This same issue came up when looking at the infinite versus finite cases for antibody-antigen aggregation —the finite case is very different because you run out of antibodies before you run out of antigens, while in the infinite case they both have the same cardinalities. This made me sympathetic to a ‘strict constructivism’ (reminds me of a term used by Scalia) viewpoint (or strict finitism—eg erret bish, or Edward nelson).

    (or see recent stuff (march 2016) by joel david hamkin (which I don’t understand, though I proposed what may be a related idea long ago—-and discussed it with Paul Cohen (independence of the continuum hypothesis, via forcing) at Stanford—he told me we don’t do it that way –I was relying on lowenheim-skolem’s theorem but it wasn’t rigorous)—-‘every function can be computable’. I also met Karl Pribram—he told me I could run his computer. He also had some ‘quantum brain’ theory using gabor transforms and a holograph metaphor.)

    (There is a physicist at U C Berkely named Richard muller, who has a book coming out about ‘Now—the physics of time’ which I gather says that actually time may be doing something like space—expanding or accelerating . This might be different from Julian Barbour’s ‘timeless universe’ view though possibly or likely compatible,. This might be good—there will be more time in a day. I may need this—my landlord showed yp today and told me they are going to back in here after the holiday so I better clean this joint up (specifically, my shower needs a new curtain because the water is going through the floor. This is my 3rd warning. Went out last nite to take out the trash and get the fresh air about 4 am to get ready for a hike in the mountains which leaves at 6—- saw 1 police car, then 2 then 10, and then a helicopter was flying overhead with a spotlight. I asked the police what was up and they told me to go inside, I looked it up on my local police report and apparently someone was running around with a gun but got caught. I later realized that my hike was tomorrow not today).

    ps I think the friedman or de-sitter universe .(a solution to Einstein’s field equations done by a young Russian around 1920, who also died in ww1) has an interpretation for a static universe—this was before hubble, zwicky, Lemaitre, etc. I think the idea was that because most planets and stars are moving away from earth, we think the universe is expanding but this is just due to sampling error.)


  2. Interesting, but I think the probability that the number of non-sim civilizations is growing exponentially is vanishingly small compared to the probability that the number of sim civilizations (conditional on there being sim civilizations) is growing exponentially.


  3. Not sure if it matters for your main point on the sim v real universe question but you’re missing a constraint on sim universes..dy/dt is more like

    tau_y dy/dt=(f(x)-c)y

    In other words sim universes only exist if real exist…more specifically if sufficiently advanced technology exists in real universe(s); f(x)>=c allows for sims to exist and self-replicate. I’d also add a third variable for base cases taking into account that base cases can go to zero (thus sim universes go to zero) for reasons other than real universe(s) collapsing.

    This also suggests an interesting imperative for simulants to stabilize the basecases. ..suggesting that at some point y may positively feedback on x.

    …further speculation… if so, then it seems that basecases would want to inform simulants of their state (that they are simulants) for mutual benefit.


  4. @shashaank In your case the number of sim universes would grow at rate ~x and thus grow much faster than x. It is also not clear why you would expect the rate of sim universe creation would grow exponentially but your point is valid. There are lots of possible models that are likely to be more plausible. My example was just to show that the rates of creation of both sim and non sim universes could have the same scaling even though sim universes arise from non-sim ones.


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