Crossposted on the EA Forum.

Displacement of other Space-Faring Civilizations (SFCs) may have macrostrategic impact for longtermists as hinted in a previous post. For example, our causal impact depends on how much value we produce relative to how much value would have been produced without us. In this post, we clarify the concept of Civ-Saturation, the metrics to describe it, and how to evaluate the Civ-Saturation Hypothesis^[1]. We finally present a simple simultaneous appearance displacement model. This model lets us compute how much marginal resources are grabbed when increasing the density of SFCs.

Sequence: This post is part 4 of a sequence investigating the longtermist implications of alien Space-Faring Civilizations. Each post aims to be standalone. You can find an introduction to the sequence in the following post.

Summary

This post formalizes the concept of Civ-Saturation (CS) and a few associated metrics, which help evaluate how the existence of Space-Faring Civilizations (SFCs) affects the distribution of cosmic resources. The key metric introduced is Marginal Civ-Saturation (MCS), which measures the marginal resources gained when adding new SFCs to a world. Three variants are presented: $MC{S}_h$ (within humanity's SFC potential space), $MC{S}_w$ (across an entire world), and $MC{S}_c$ (within correlated space). $MC{S}_h$ is the metric to use to make decisions, given our point of view. Using a simultaneous appearance displacement model, the post illustrates how MCS values vary with SFC density (D, the number of SFCs per reachable universe), showing that marginal resources drop by 20x between $D=0.01$ and $D=3$. The formalization helps classify worlds into three categories based on their MCS values: Non Civ-Saturated Worlds ($MC{S}_h>99\%$), Partially Civ-Saturated Worlds ($5\%\le MC{S}_h\le 99\%$), and Redundantly Civ-Saturated Worlds ($MC{S}_h<5\%$). These classes can be used for analyzing the longtermist macrostrategic implications of the Civ-Saturation Hypothesis^[1].

Types of SFCs: Actual, Potential, Precluded

First, let’s take some time to distinguish three types of SFCs:

• Actual SFCs are the SFCs that really exist in a world, thus excluding those that could have existed if they had not been precluded by the actual SFCs.

• Potential SFCs^[2] include the actual SFCs and the SFCs that would come to exist if other SFCs did not exist, i.e. if we ignored preclusion.

• Precluded SFCs include the potential SFCs that are precluded by the actual SFCs.

  

Notes:

• In summary: Potential SFCs = Actual SFCs $\cup$ Precluded SFCs
• In terms of density: Density(Potential SFCs) = Density(Actual SFCs) + Density(Precluded SFCs)
• If we assume that all potential SFCs would propagate at the same speed, and that preclusion happens when making contact, thus not at a distance, then the volume occupied by actual SFCs always includes the volume that could have been occupied by precluded SFCs. In terms of potential volume: PotentialVolume(Potential SFCs) = PotentialVolume(Actual SFCs) >= PotentialVolume(Precluded SFCs).
• Potential SFCs do not include Intelligent Civilizations (IC) that had the potential to become SFCs but disappeared endogenously (e.g., extinction caused by bioweapons, choosing not to spread in space).

Why make this distinction?

• When analyzing a world's Civ-Saturation (CS), the density of actual SFCs will be more relevant since they are the only SFCs that grab resources.
• However, when analyzing the Marginal Civ-Saturation (MCS), we will also have to look at the density of potential SFCs since removing an actual SFC may let precluded SFCs appear, or adding early actual SFCs may cause other previously late actual SFCs to become precluded.
• Finally, we are unsure if Humanity will become an actual SFC. But we know that Humanity is an Intelligent Civilization (IC). By existing today, we observe that Humanity on Earth was not precluded in the past. This could have consequences on how to compute the MCS.

Formalizing Civ-Saturation and the Civ-Saturation Hypothesis

Civ-Saturation of worlds

In Hinting at the macrostrategic consequences of Civ-Saturation, we introduced the Civ-Saturation Hypothesis: “Most resources will be claimed by Space-Faring Civilizations (SFCs) regardless of whether humanity creates an SFC.”^[1]

Formalizing Marginal Civ-Saturation (MCS). Let’s formalize Civ-Saturation (CS) and the Civ-Saturation Hypothesis^[3]. In Civ-Saturated worlds, how much resources are grabbed by all SFCs is mostly independent of the existence of one SFC. The marginal resources brought by the existence of an SFC are close to zero. The Civ-Saturation ($CS$) of a world can be expressed as a function of $R_g\left(D\right)$ the amount of resources grabbed by all SFCs in worlds with SFC density $D$, and $R_w$ the amount of resources available in each world, which we assume constant and normalize to one. What we care about for making decisions relative to prioritizing our existence is how much marginal resources are brought or lost when adding or removing an SFC, this is the Marginal Civ-Saturation ($MCS$), computed as the derivative of $C{S}_w$ by $D$, $w$ stands for World.

$C{S}_w\left(D\right)=\frac{R_{\text{g}}\left(D\right)}{R_{\text{w}}}$, and $MC{S}_w\left(D\right)=\frac{\partial C{S}_w}{\partial D}=\frac{1}{R_{\text{w}}}\cdot \frac{\partial R_{\text{g}}\left(D\right)}{\partial D}$

Marginal Civ-Saturation of Humanity's SFC

We care about Humanity making a decision, which relies on $MC{S}_h$ instead of $MC{S}_w$. The Civ-Saturation Hypothesis^[1] is ultimately about whether Humanity should focus on "Extinction Risks" or "Alignment Risks/Future Value". An formal framing is that this hypothesis is about whether we should focus on increasing P(alignment AND Humanity creates an SFC)^[4] or P(alignment | Humanity creates an SFC). Moreover when making a decision, we need to take into account all information available to us, such as Humanity's SFC appearance time. Because of this, it is possible that we live in an almost fully Civ-Saturated world while the optimal choice for Humanity is still to optimize P(alignment AND Humanity creates an SFC)^[5]. An simple example is if we knew Humanity would be one of the last and very late potential SFCs to appear inside an almost fully Civ-Saturated world. In that case, most the resource we would grab first would be lost if we don't exist, because we would only grabbed resources not reachable by other SFCs. Then the optimal choice would be to focus on increasing P(alignment AND Humanity creates an SFC) and not P(alignment | Humanity creates an SFC), and thus to partially focus on creating an SFC. Overall, when making a decision from within Humanity, we will need to look at the Marginal Civ-Saturation within Humanity's SFC future grabbed space. Another way to understand it is that the space that Humanity's SFC will impact is special. It is NOT a random part of the world. It is a part of the world in which we were able to appear. 

We thus define three versions of the $MCS$ metric relative to three kinds of spaces. These marginal civ-saturations are computed within each of these spaces. 

• $MC{S}_h$ - The MCS within the space Humanity's SFC ($h$) would grab: $MC{S}_h(D_{\neg h}+D_h)=\frac{1}{R_{\text{h}}}\cdot \frac{\partial R_{\text{h}}(D_{\neg h}+D_h)}{\partial D_h}$ with $D=D_{\neg h}+D_h$. When Humanity's SFC exist, it is the only one to grab resources it its own space, and when it does not exist, then this space is either empty or occupied by other SFCs. $R_h$ is the amount of resources^[6] grabbed by SFCs in Humanity's SFC space assuming it exists. The $MC{S}_h$ metric can be used to reason about what Humanity should do.
• $MC{S}_w$ - The Marginal Civ-Saturation of a World ($w$):  $MC{S}_w\left(D\right)=\frac{1}{R_{\text{w}}}\cdot \frac{\partial R_{\text{g}}\left(D\right)}{\partial D}$. This metric is a characteristic of a world: which fraction of its maximum potential resources an additional SFC appearing at random, in time and space, can hope to be the only one to potentially reach. It can be used to reason about the state of the world, but not directly about what Humanity should do. A special case is if we knew Humanity's SFC was going to appear first in its reachable universe, then the $MC{S}_w$ would be equal to the $MC{S}_h$. Finally, absent scenarios in which the locations at which potential SFCs can appear are non-uniform and form clusters, the $MC{S}_w$ can be used as a lower bound for $MC{S}_h$.
• $MC{S}_c$ - The MCS within the space we are correlated ($c$) with, see appendix for details.

The metric to evaluate the Civ-Saturation Hypothesis. $MC{S}_h$ is a metric useful towards assessing whether to optimize for P(alignment AND Humanity creates an SFC) or P(alignment | Humanity creates an SFC). We can finally write a metric evaluating how much representative of reality (also understood as "how correct") the Civ-Saturation Hypothesis is: 

$CSHEvaluation=1-\left|MC{S}_h\right(D\left)\right|$

For illustration, the Civ-Saturation Hypothesis would be:

• 100% representative of reality if $MC{S}_h\left(D\right)$ is 0, which is equivalent to saying that changing the density of Humanity's SFC does not change how many resources are grabbed by all SFCs.
• 0% representative of reality if $MC{S}_h\left(D\right)$ is 1, which is similar to saying that all of Humanity's SFC future resources would be lost and never grabbed by other SFCs if Humanity does not create an SFC. This would be expected if the part of the universe Humanity lives in is not at all saturated in SFCs, i.e., if the universe is mostly empty OR if we knew that Humanity was among the last and very late SFCs to appear in a world.

About the normalization of MCS. The Marginal Civ-Saturation metrics are naturally normalized by the amount of resource in the space of interest (e.g., within the space occupied by Humanity's SFC if it exist for $MC{S}_h$). This is a result of differentiating the CS metric which is already normalized to the space of interest. This normalization is useful for separating the importance of a world, or a subspace of it, to how saturated it is. The CS and MCS metrics are thus not proportional to the importance of a world. When we will aggregate between several possible worlds, the Decision-Relevance of these worlds will be accounted separated using a method already described in a previous post: Decision-Relevance of worlds and ADT implementations.

Estimation methods in the appendix. In the appendix, we describe how to estimate these different metrics.

Simultaneous appearance displacement model

We are now going to illustrate how to compute the $MC{S}_h$ with a very simple model; we call it a simultaneous appearance displacement model.

We call $CS(1+)$ and $CS(2+)$ the fraction of resources reachable by at least one and respectively, at least two potential SFCs. $CS(1+)$ is equivalent to $CS$ described in the previous sections. $CS(2+)$ helps us express how much redundancy there is in reaching resources. We express them, in function of $D$ the density of potential SFCs per reachable universe. We assume for simplicity that:

• Potential SFCs appear independently.^[7]

• Potential SFCs have the exact same appearance time.

• Potential SFCs have the same propagation speed equal to the speed of light c.

Under these assumptions, the local density of SFC follows a Spatial Poisson Process (random independent events) in a 3D continuous space, the distributions of potential and actual SFCs are identical, and the $MC{S}_h$ equals both the $MC{S}_c$ and the $MC{S}_w$.

To compute the number of potential SFCs able to reach a point, we count the number of SFCs within a sphere of volume one since we normalized the potential SFC density by the volume of a reachable universe.

$CS(1+)=1-e^{-D}$

$CS(2+)=1-e^{-D}(1+D)$

$MC{S}_h=\frac{\partial CS(1+)}{\partial D}=e^{-D}$

Let’s report these values for a range of SFC densities:

Sharp transition from emptiness to civ-saturation around D = 1. Under the (incorrect) assumptions of this model, the marginal value of increasing the density $D$, of Humanity's SFC ($MC{S}_h$) or of any other SFCs, falls by around 20x between $D=0.01$ and $D=3$.

Civ-Saturation Classes

The longtermist macrostrategic implications of the Civ-Saturation Hypothesis can be classified by the level of Marginal Civ-Saturation ($MC{S}_h$). We are thus going to specify three classes of worlds relevant to describe these implications, which will be described in later posts:

• Non Civ-Saturated Worlds: The $MC{S}_h$ is above 99%.
  • In the case of the simultaneous appearance displacement model: At most 1% of all resources are grabbed and $D<{10}^{-2}$.
• Partially Civ-Saturated Worlds: The $MC{S}_h$ is equal to or between 99% and 5%.
  • In the case of the simultaneous appearance displacement model: More than 1% of resources are grabbed but less than 80% of resources are redundantly grabbed and ${10}^{-2}<D<3$.
• Redundantly Civ-Saturated Worlds: The $MC{S}_h$ is below 5%.
  • In the case of the simultaneous appearance displacement model: At least 80% of resources are grabbed by at least two potential SFCs and $D>3$.

Context

Evaluating the Existence Neutrality Hypothesis - Introductory Series. This post is part of a series introducing a research project for which I am seeking funding: Evaluating the Existence Neutrality Hypothesis. This project includes evaluating both the Civ-Saturation and the Civ-Similarity Hypotheses^[8] and their longtermist macrostrategic implications. This introductory series hints at preliminary research results and looks at the tractability of making further progress in evaluating these hypotheses.

Next steps: A first evaluation of the Civ-Saturation Hypothesis. In this post, the last two and the following one, we are introducing a first evaluation of the Civ-Saturation hypothesis. The next post will finally present the results from this first evaluation.

Acknowledgements

Thanks to Tristan Cook for having spent some of their personal time providing excellent feedback on this post and ideas, and Lukas Finnveden for initial discussions. Note that this research was done under my personal name and that this content is not meant to represent any organization's stance. 

Appendix

Definition of $MC{S}_c$

 $MC{S}_c(D_{\neg c}+D_c)=\frac{1}{R_{\text{c}}}\cdot \frac{\partial R_{\text{c}}(D_{\neg c}+D_c)}{\partial D_c}$ with $D=D_{\neg c}+D_c$, and $R_c$ the amount of resources grabbed by SFCs in our correlated space (see the post Decision-Relevance of worlds and ADT implementations for details about the correlated space). This metric can be used to reason about what the group of SFCs we are correlated with should collectively do, but not directly what Humanity should do, and neither what each SFC in this group should do. A special case is when our impact is dominated by our correlation with our exact copies, then the $MC{S}_c$ is equal to the $MC{S}_h$.

Estimating $MC{S}_h$

How can we estimate $MC{S}_h\left(D\right)$? Either we have access to the appearance and propagation models of SFCs and we can directly compute it, or we can use the following approximation. We know that:

$MC{S}_h(D_{\neg h}+D_h)=\frac{1}{R_{\text{h}}}\cdot \frac{\partial R_{\text{h}}(D_{\neg h}+D_h)}{\partial D_h}$

$D=D_{\neg h}+D_h$,  $D_h<<1$, and $D_h<<D_{\neg h}$

Thus, we can neglect interactions between exact copies of Humanity's SFC and approximate the derivative by the slope between two arbitrary points at low $D_h$. For illustration, we use $D_h={10}^{-6}$and $D_h=0$.

$R_h$ is the size of Humanity's SFC if it exists, and $R_o$ is the size of the Other SFCs that would existing within Humanity's SFC potential space if it does not exist.

$MC{S}_h(D_{\neg h}+D_h)=R_{\text{h}}^{-1}\cdot (R_{\text{h}}\cdot {10}^{-6}-R_o\cdot {10}^{-6})\cdot {10}^6$

$MC{S}_h(D_{\neg h}+D_h)=(R_h-R_o)/R_{\text{h}}=1-R_o/R_h$

$CSHEvaluation=R_o/R_h$

In the end, the $MC{S}_h$ is equal to the share of Humanity's SFC resources lost if it does not exist, which matches our intuition. How representative of reality the Civ-Saturation Hypothesis is can be approximated by the share of Humanity's SFC resources not lost if it does not exist.

Alternatively, if we know $MC{S}_w$, we can use it as a likely lower bound for $MC{S}_h$.

Estimating $MC{S}_w$

Again, either we have access to the appearance and propagation models of SFCs, or we can use the following approximation: $MC{S}_w\left(D\right)\approx 1-CS\left(D\right)$. We also know that $CS\left(D\right)$ is monotonously increasing with D and transitions pretty quickly from 0 to 1 when $D$ increases from 0.01 to 3 (see the results from the simultaneous appearance displacement model). Such that when we are aggregating this metric over a likelihood distribution over $D$, we can use the probabilities $P(D>3)$ and $P(D>0.01)$ to bound the metric: $[1-P(D>0.01\left)\right]<MC{S}_w\left(D\right)<[1-P(D>3\left)\right]$. 

Estimating $MC{S}_c$

The $MC{S}_c$ is equal to the $MC{S}_h$ when our impact is dominated by ourselves (no copies) or by our perfect copies. When our impact is dominated by our approximate copies or by our correlation with SFC Shapers (see Decision-Relevance of worlds and ADT implementations for the definitions of these groups), then $MC{S}_c$ may be almost equal to the $MC{S}_w$.

1. ^{^}

   The Civ-Saturation Hypothesis is introduced in Longtermist implications of aliens Space-Faring Civilizations - Introduction. The Civ-Saturation Hypothesis posits that when making decisions, we should assume most of Humanity's SFC future resources will eventually be grabbed by SFCs regardless of whether Humanity's SFC exists or not.

2. ^{^}

    Potential SFCs could alternatively have been called Preclusion-Excluded SFCs.

3. ^{^}

   In this post we assume we have no uncertainty, meaning that we know the perfect description of the world in which we exist. In a following post, we will account for uncertainty, but not here.

4. ^{^}

   By P(Alignment), I mean the probability that the SFC Humanity would create is aligned with some kind of ideal moral value (e.g., CEV), and has the ability to optimize it strongly. This requires some degree of success at both technical alignment and AI governance.

5. ^{^}

   Even assuming the Civ-Similarity Hypothesis is almost 100% true.

6. ^{^}

    $R_h$ and resources in general can be approximated by volumes of space given the universe is either homogeneous on cosmic scale or indistinguishable from it.

7. ^{^}

    Which, ignoring panspermia, is plausibly correct for potential SFCs (but would not be correct for actual SFCs).

8. ^{^}

   The Civ-Similarity Hypothesis posits that the expected utility efficiency of Humanity's future Space-Faring Civilization (SFC) would be similar to that of other SFCs.