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I was reading this paper, and was having trouble with the equations in part IV.

The notation is simple enough to understand, however I don't understand where he drawing his concluding equation from, the one immediately after the notation.

Would somebody please be able to simply the math in that paper (what little there is of it), and explain in plain English why it is valid?

Here is Section IV in its entirety:

We shall develop this idea into a rigorous argument. Let us introduce the following notation:

$f_P$: Fraction of all human-level technological civilizations that survive to reach a posthuman stage

$\overline{N}$: Average number of ancestor-simulations run by a posthuman civilization

$\overline{H}$: Average number of individuals that have lived in a civilization before it reaches a posthuman stage

The actual fraction of all observers with human-type experiences that live in simulations is then $f_{sim}=\frac{f_P\overline{N}\overline{H}}{(f_P\overline{N}\overline{H})+\overline{H}}$

Writing $f_I$ for the fraction of posthuman civilizations that are interested in running ancestor-simulations (or that contain at least some individuals who are interested in that and have sufficient resources to run a significant number of such simulations), and $\overline{N_I}$ for the average number of ancestor-simulations run by such interested civilizations, we have $\overline{N}=f_I\overline{N_I}$ and thus: $f_{sim}=\frac{f_Pf_I\overline{N_I}}{(f_Pf_I\overline{N_I})+1}\tag{a}$ Because of the immense computing power of posthuman civilizations, $\overline{N_I}$ is extremely large, as we saw in the previous section. By inspecting (a) we can then see that at least one of the following three propositions must be true:

(1) $f_P\approx 0$
(2) $f_I\approx 0$
(3) $f_{sim}\approx 1$

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    @Asaf Karagila I cannot copy the equations in the paper in order to do that though?2011-11-12

1 Answers 1

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The argument is essentially as follows. Suppose that on average $\overline{H}$ individuals live in a civilization before it becomes posthuman; then a posthuman ancestor-simulation will on average contain $\overline{H}$ individuals. Say that on average each posthuman civilizations runs $\overline{N}$ ancestor-simulations. Then each of them will on average run ancestor-simulations containing $\overline{N}\overline{H}$ individuals: $\overline{H}$ individuals in each of $\overline{N}$ ancestor-simulations.

Now suppose that $f_P$ is the fraction of all human civilizations that survive to reach the posthuman stage. If we start with a grand total of $M$ human civilizations, then $f_PM$ of them will survive to the posthuman stage. On average each of those $f_PM$ posthuman civilizations will produce $\overline{N}\overline{H}$ simulated individuals, for a grand total of $f_PM\overline{N}\overline{H}$ simulated individuals. There are also $M\overline{H}$ real individuals: an average of $\overline{H}$ of them for each of $M$ human civilizations. Thus, the total number of individuals, real and simulated, is $f_PM\overline{N}\overline{H}+M\overline{H} = M\overline{H}(f_P\overline{N}+1)$. The fraction of these who are simulated is therefore $f_{sim}=\frac{f_PM\overline{N}\overline{H}}{M\overline{H}(f_P\overline{N}+1)}=\frac{f_P\overline{N}}{f_P\overline{N}+1}\tag{1}$ after cancelling $M\overline{H}$.

Perhaps not all posthuman civilizations run ancestor-simulations; let $f_I$ be the fraction of them that do run a significant number, and let $\overline{N_I}$ be the average number of ancestor-simulations run by the these civilizations. The difference between $\overline{N}$ and $\overline{N_I}$ is similar to the difference between average lifespan and average lifespan of those who survive infancy: if many children die in infancy, the former will be much lower than the latter. Similarly, if $f_I$ is much smaller than $1$, $\overline{N}$ will be much smaller than $\overline{N_I}$.

In fact, $\overline{N}=f_I\overline{N_I}$. To see this, recall that we have $f_PM$ posthuman civilizations, so we have $f_If_PM$ posthuman civilizations that run ancestor-simulations. We assume that the remaining $M-f_If_PM$ posthuman civilizations run a negligible number of ancestor-simulations, so altogether we have $f_If_PM\overline{N_I}$ ancestor-simulations: $\overline{N_I}$ on average for each of $f_If_PM$ posthuman civilizations. Divide this total by $f_PM$, the number of posthuman civilizations, to get $\overline{N}$, the average number of ancestor-simulations per posthuman civilization: $\overline{N} = \frac{f_If_PM\overline{N_I}}{f_PM}=f_I\overline{N_I}\;.$

Substituting this value of $\overline{N}$ into $(1)$, we get $f_{sim}=\frac{f_Pf_I\overline{N_I}}{f_Pf_I\overline{N_I}+1}\;.\tag{2}$

The author then claims that $\overline{N_I}$, the average number of ancestor-simulations run by each posthuman civilization that runs them in any significant numbers, must be extremely large, since such civilizations will have immense computing power. Thus, either $f_Pf_I$ is very small, so as to compensate for the very large factor $\overline{N_I}$, or $f_Pf_I\overline{N_I}$ is large in comparison with $1$, and the fraction in $(2)$ has the form $f_{sim}=\frac{\text{very large number}}{\text{same very large number}+1},$ in which case $f_{sim}$ is very nearly $1$: think of a fraction like $\frac{1000}{1001}$. This is proposition (3) at the end of Section IV.

The alternative, as I said, is that $f_Pf_I$ is very small, so that $f_Pf_I\overline{N_I}$ isn’t a very large number (even though $\overline{N_I}$ is). That can be true only if one or both of the fractions $f_P$ and $f_I$ are very small $-$ practically $0$, in fact, since $\overline{N_I}$ is ‘extremely large’. These are propositions (1) and (2) at the end of Section IV.

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    wow, thank you so much!2011-11-13