Quantifying anthropic effects on the Fermi paradox

Cross­posted to the EA fo­rum.

Sum­mary:

I ap­ply the self-in­di­ca­tion as­sump­tion (a the­ory of an­throp­ics) and some non-causal de­ci­sion the­o­ries to the ques­tion of how com­mon space­far­ing civil­i­sa­tions are in the Uni­verse. Th­ese the­o­ries push strongly to­wards civil­i­sa­tions be­ing very com­mon, and com­bin­ing them with the ob­ser­va­tion that we haven’t seen any ex­trater­res­trial life yields a quite spe­cific es­ti­mate of how com­mon such civil­i­sa­tions are. If you ac­cept the self-in­di­ca­tion as­sump­tion, you should be al­most cer­tain that we’ll en­counter other civil­i­sa­tions if we leave the galaxy. In this case, 95 % of the reach­able uni­verse will already be colon­ised when Earth-origi­nat­ing in­tel­li­gence ar­rives, in ex­pec­ta­tion. Of the re­main­ing 5 %, around 70 % would even­tu­ally be reached by other civil­i­sa­tions, while 30 % would have re­mained empty in our ab­sence. Even if you don’t ac­cept the self-in­di­ca­tion as­sump­tion, most non-causal de­ci­sion the­o­ries have the same prac­ti­cal im­pli­ca­tions. If you be­lieve that other civil­i­sa­tions colon­is­ing the Uni­verse is pos­i­tive, this pro­vides some rea­son to pre­fer in­ter­ven­tions that in­crease the qual­ity of the fu­ture over re­duc­ing non-AI ex­tinc­tion risk; if you think that other civil­i­sa­tions colon­is­ing the uni­verse is nega­tive, the op­po­site is true.

Introduction

There are billions of stars in the Milky Way, and billions of trillions of stars in the ob­serv­able uni­verse. The Fermi ob­ser­va­tion is the sur­pris­ing ob­ser­va­tion that not a sin­gle one of them shows any signs of life, named af­ter the Fermi para­dox. There are sev­eral pos­si­ble ex­pla­na­tions for the Fermi ob­ser­va­tion: per­haps life is very un­likely to arise from any par­tic­u­lar planet, per­haps life has just re­cently be­gun emerg­ing in the Uni­verse, or per­haps there is some rea­son that life never leaves the so­lar sys­tem in which it emerges. Without any fur­ther in­for­ma­tion, it may seem difficult to figure out which one it is: but the field of an­throp­ics has a lot to say about this kind of situ­a­tion.

An­throp­ics is the study of what we should do and what we should be­lieve as a con­se­quence of ob­serv­ing that we ex­ist. For ex­am­ple, we can ob­serve that life and civil­i­sa­tion ap­peared on Earth. Should we in­ter­pret this as strong ev­i­dence that life ap­pears fre­quently? After all, life is far more likely to arise on this planet if life ap­pears fre­quently than if it doesn’t; thus, life aris­ing on this planet is ev­i­dence for life ap­pear­ing fre­quently. On the other hand, some civil­i­sa­tions like ours is likely to ex­ist re­gard­less of how com­mon life is, and all such civil­i­sa­tions will ob­vi­ously find that life ap­peared on their planet: oth­er­wise they wouldn’t have ex­isted. No con­sen­sus has yet emerged on the cor­rect ap­proach to these ques­tions, but a num­ber of the­o­ries have been put for­ward.

This post is in­spired by the re­cent Dis­solv­ing the Fermi Para­dox (Sand­berg, Drexler and Ord, 2018), which doesn’t draw any spe­cial con­clu­sion from our ex­is­tence, and thus con­cludes that life is likely to be very un­com­mon.[1] Here, I in­ves­ti­gate the im­pli­ca­tions of the­o­ries that in­ter­pret our ex­is­tence as strong rea­son to act as if life ap­pears fre­quently. Speci­fi­cally, the self-in­di­ca­tion as­sump­tion im­plies that life should be quite com­mon, and de­ci­sion the­o­ries such as Ev­i­den­tial De­ci­sion The­ory, Func­tional De­ci­sion The­ory and Up­date­less De­ci­sion The­ory gives similar re­sults (an­throp­ics and de­ci­sion the­o­ries are ex­plained in Up­dat­ing on an­throp­ics). How­ever, the Fermi ob­ser­va­tion pro­vides a strong up­per bound on how com­mon life can be, given some as­sump­tions about the pos­si­bil­ity of space travel. The re­sult of com­bin­ing the self-in­di­ca­tion as­sump­tion with the Fermi ob­ser­va­tion gives a sur­pris­ingly spe­cific es­ti­mate of how com­mon life is in the Uni­verse. The de­ci­sion the­o­ries men­tioned above yields differ­ent prob­a­bil­ities, but it can be shown that they always give the same prac­ti­cal im­pli­ca­tions as the self-in­di­ca­tion as­sump­tion, given some as­sump­tions about what we value.

Short illus­tra­tion of the argument

In this sec­tion, I demon­strate how the ar­gu­ment works by ap­ply­ing it to a sim­ple ex­am­ple.

In the ob­serv­able uni­verse, there are ap­prox­i­mately stars. Now con­sider 3 differ­ent hy­pothe­ses about how com­mon civil­i­sa­tions are:

  • Civil­i­sa­tions are very com­mon: on av­er­age, a civil­i­sa­tion ap­pears once ev­ery stars.

  • Civil­i­sa­tions are com­mon: on av­er­age, a civil­i­sa­tion ap­pears once ev­ery stars.

  • Civil­i­sa­tions are un­com­mon: on av­er­age, a civil­i­sa­tion ap­pears once ev­ery stars.

Un­der the first hy­poth­e­sis, we would ex­pect there to be roughly civil­i­sa­tions in the ob­serv­able uni­verse. As­sum­ing that in­ter­galac­tic travel is pos­si­ble, this is very un­likely. The Sun seems to have formed fairly late com­pared with most stars in the Uni­verse, so most of these civil­i­sa­tions would have ap­peared be­fore us. All civil­i­sa­tions might not want to travel to differ­ent galax­ies, and some of these would have formed so far away that they wouldn’t have time to reach us; but at least one out of civil­i­sa­tions would al­most cer­tainly have tried to colon­ise the Milky Way. Since we haven’t seen any other civil­i­sa­tions, the first hy­poth­e­sis is al­most cer­tainly false.

The ques­tion then, is how we should com­pare the sec­ond and the third hy­poth­e­sis. Con­sider an ex­tremely large (but finite) part of the Uni­verse. In this re­gion, a large num­ber of copies of our civil­i­sa­tion will ex­ist re­gard­less of whether life is com­mon or un­com­mon. How­ever, if civil­i­sa­tions tend to ap­pear once ev­ery stars, we can ex­pect there to be more copies of our civil­i­sa­tion than if civil­i­sa­tions ap­pears once ev­ery stars. Thus, when think­ing about what civil­i­sa­tions such as ours should do, we must con­sider that our de­ci­sions will be im­ple­mented times as many times if civil­i­sa­tions are com­mon com­pared with if they’re un­com­mon (ac­cord­ing to some de­ci­sion the­o­ries). If we’re to­tal con­se­quen­tial­ists,[2] this means that any de­ci­sion we do will mat­ter times as much if civil­i­sa­tions are com­mon com­pared with if they’re un­com­mon (dis­re­gard­ing in­ter­ac­tions be­tween civil­i­sa­tions). Thus, if we as­sign equal prior prob­a­bil­ity to civil­i­sa­tions be­ing com­mon and civil­i­sa­tions be­ing un­com­mon, and we care equally much about each civil­i­sa­tion re­gard­less, we should act as if it’s al­most cer­tain that life is com­mon.

Of course, the same ar­gu­ment ap­plies to civil­i­sa­tions be­ing very com­mon: de­ci­sions are times more im­por­tant if civil­i­sa­tions ap­pear once ev­ery stars as if they ap­pear once ev­ery stars. How­ever, the first ar­gu­ment is stronger: given some as­sump­tions, the prob­a­bil­ity that we wouldn’t have seen any civil­i­sa­tion would be less than , if civil­i­sa­tions were very com­mon.

That civil­i­sa­tions ap­pears once ev­ery stars im­plies that there should be about civil­i­sa­tions in the ob­serv­able uni­verse. Look­ing at de­tails of when civil­i­sa­tions ap­pear and how fast they spread, it isn’t that im­plau­si­ble that we wouldn’t have seen any of these; so the ar­gu­ment against the first hy­poth­e­sis doesn’t work. How­ever, as time goes, and Earth-origi­nat­ing and other civil­i­sa­tions ex­pand, it’s very likely that they will en­counter each other, and that most space that Earth-origi­nat­ing in­tel­li­gence colon­ise would have been colon­ised by other civil­i­sa­tions if we didn’t ex­pand.

Cos­molog­i­cal assumptions

Be­fore we dive into the emer­gence and spread of life, there are some cos­molog­i­cal facts that we must know.

First, we must know how the emer­gence of civil­i­sa­tions varies across time. One strik­ingly rele­vant fac­tor for this is the rate of for­ma­tion for stars and hab­it­able planets. While there aren’t par­tic­u­larly large rea­sons to ex­pect the prob­a­bil­ity that life arises on hab­it­able planets to vary with time, cos­molog­i­cal data sug­gests that the rate at which such planets are cre­ated have varied sig­nifi­cantly.

Se­cond, we must know how fast such civil­i­sa­tions can spread into space. This is com­pli­cated by the ex­pan­sion of the Uni­verse, and de­pends some­what on what speed we think that fu­ture civil­i­sa­tions will be able to travel.

Planet formation

The rate of star for­ma­tion is a rel­a­tively well stud­ied area, with good ac­cess to past data. Star for­ma­tion rate peaked a few billion years af­ter the Big Bang, and has been ex­po­nen­tially de­clin­ing ever since. A fit to the past rates as a func­tion of red­shift (an as­tro­nom­i­cal ob­serv­able re­lated to time) is stars per year per mega­parsecs (Madau and Dick­in­son, 2014). It’s a bit un­clear how this rate will change into the fu­ture, but a de­cent guess is to sim­ply ex­trap­o­late it (Sand­berg, per­sonal com­mu­ni­ca­tion). Trans­lat­ing the red­shift z to time, the star for­ma­tion rate turns out to be log-nor­mal, with an ex­po­nen­tially de­creas­ing tail. This is de­picted in figure 1.

Figure 1: Num­ber of stars formed per year per cu­bic mega­parsec in the Uni­verse. The red line marks the pre­sent, ap­prox­i­mately 13.8 billion years af­ter the Big Bang.

The for­ma­tion rate of planets similar to the Earth are a bit more un­cer­tain. In gen­eral, it seems like they should be similar to those of star for­ma­tion, but have some de­lay due to need­ing heav­ier met­als that weren’t available when the Uni­verse was young. Behroozi and Peeples (2015) think that this is neg­ligible, while Lineweaver (2001) thinks that it’s quite large. Lineweaver’s model as­sumes that the frac­tion of metal in the Uni­verse is pro­por­tional to the num­ber of stars that have already formed (since the pro­duc­tion of heavy met­als hap­pens in stars), and that the prob­a­bil­ity of form­ing an Earth-like planet is pro­por­tional to the log­a­r­ithm of this frac­tion.[3] Over­all, this re­sults in the peak of planet-for­ma­tion hap­pen­ing about 2 billion years af­ter the peak of star-for­ma­tion, as de­picted in figure 2.

Figure 2: Num­ber of Earth-like planets formed per year per cu­bic mega­parsec in the Uni­verse. The red line marks the pre­sent.

This is the planet-for­ma­tion rate I will be us­ing for the most part. I dis­cuss some other choices in Ap­pendix C.

Speed of travel

The uni­verse is ex­pand­ing: dis­tant galax­ies are re­ced­ing from us at fast pace. More­over, these galax­ies are ac­cel­er­at­ing away from us; as time goes by, most of these will ac­cel­er­ate to ve­loc­i­ties so fast that we will never be able to catch up with them. This im­plies that early civil­i­sa­tions can spread sig­nifi­cantly farther than late civil­i­sa­tions. A civil­i­sa­tion aris­ing 5 billion years ago would have been able to reach 2.7 times as much vol­ume as we can reach, and a civil­i­sa­tion aris­ing in an­other 10 billion years will only be able to reach 15 % of the vol­ume that we can reach (given the as­sump­tions be­low).

This grad­ual re­duc­tion of reach­able galax­ies will con­tinue un­til about 100 billion years from now; at that point galax­ies will only be able to travel within the groups that they are grav­i­ta­tion­ally bound to, all other galax­ies will be gone. The group that we’re a part of, the Lo­cal Group, con­tains a bit more than 50 galax­ies, which is quite typ­i­cal. Star for­ma­tion will con­tinue for 1 to 100 trillion years, af­ter this, but it will have de­clined so much that it’s largely neg­ligible.

Be­cause of the ex­pan­sion of the Uni­verse, probes will grad­u­ally lose ve­loc­ity rel­a­tive to their sur­round­ings. A probe start­ing out at 80 % of the speed of light would only be go­ing at 56 % of the speed of light af­ter 10 billion years. Due to rel­a­tivis­tic effects, very high ini­tial speeds can coun­ter­act this: light won’t lose ve­loc­ity at all, and a probe start­ing out at at 99.9 % of the speed of light would still be go­ing at 99.6 % af­ter 10 billion years. If there’s a limit to the ini­tial speed of probes, though, an­other way to coun­ter­act this de­cel­er­a­tion is to pe­ri­od­i­cally reac­cel­er­ate to the ini­tial speed (Sand­berg, 2018). This could be done by stop­ping at var­i­ous galax­ies along the way to gather en­ergy and ac­cel­er­ate back to ini­tial speed. As long as the stops are short enough, this could sig­nifi­cantly in­crease how far a probe would be able to go. Look­ing at the den­sity of galax­ies across the Uni­verse, there should be no prob­lem in stop­ping ev­ery billion light-years; and it may be pos­si­ble to reac­cel­er­ate even more fre­quently than that by stop­ping at the odd star in oth­er­wise empty parts of the Uni­verse, or by tak­ing minor de­tours (Sand­berg, per­sonal com­mu­ni­ca­tion).

Given some as­sump­tions about po­ten­tial fu­ture tech­nol­ogy, it seems plau­si­ble that most civil­i­sa­tions will be able to send out probes go­ing at 80 % of the speed of light (Arm­strong and Sand­berg, 2013). For most calcu­la­tions, this is the speed I’ve used. I’ve also as­sumed that probes can reac­cel­er­ate ev­ery 300 mil­lion years, which roughly cor­re­sponds to reac­cel­er­at­ing ev­ery 300 mil­lion light-years. Th­ese choices are by no means ob­vi­ous, but most rea­son­able al­ter­na­tives gives the same re­sults, as I show in Ap­pendix C.

To calcu­late the dis­tances reach­able at var­i­ous points in time, I use equa­tions from Arm­strong and Sand­berg (2013).[4] Th­ese are quite com­pli­cated, but when a probe moves at 80 % of the speed of light, any case where reac­cel­er­a­tion hap­pens more than once ev­ery few billion years will be very similar to con­tin­u­ous reac­cel­er­a­tion. Dur­ing con­tin­u­ous reac­cel­er­a­tion, some of the strange effects from the ex­pan­sion of the Uni­verse dis­ap­pear, and the probe’s ve­loc­ity rel­a­tive to its pre­sent sur­round­ings will always equal the ini­tial ve­loc­ity.

How­ever, to take into ac­count the ex­pan­sion of the rest of the Uni­verse, it is con­ve­nient to mea­sure the ve­loc­ity in co­mov­ing co­or­di­nates. Co­mov­ing co­or­di­nates use a co­or­di­nate frame that moves with the ex­pan­sion of the Uni­verse, such that the co­or­di­nates of galax­ies re­main con­stant even as they move away from the Earth. At any point in time, the co­mov­ing dis­tance be­tween two points is equal to the dis­tance be­tween those points to­day, re­gard­less of how far they have ex­panded from each other. As a con­se­quence, a probe with con­stant real ve­loc­ity will move slower and slower in co­mov­ing co­or­di­nates as the real dis­tance be­tween galax­ies grows, even if the probe is con­tin­u­ously reac­cel­er­at­ing. If it is reac­cel­er­at­ing of­ten, or mov­ing so fast that the de­cel­er­a­tion is neg­ligible, the ve­loc­ity mea­sured in co­mov­ing co­or­di­nates will at any time be , where is the time at which the probe was launched, is the ini­tial ve­loc­ity, and is the real length of one co­mov­ing unit at time . The dis­tance trav­el­led at time by a probe launched at time is , in co­mov­ing co­or­di­nates. Figure 3 de­picts this dis­tance as a func­tion of t, for a probe launched from the Earth around now.

Figure 3: Dis­tance that a probe leav­ing Earth around now could reach, if it reac­cel­er­ated to 80 % of the speed of light ev­ery 300 mil­lion years. The dis­tance cor­re­sponds to how far away any reached point is to­day, mea­sured in light-years. Since the Uni­verse is ex­pand­ing and ac­cel­er­at­ing, it will take dis­pro­por­tion­ally longer for the probe to reach points that are farther away.

Fur­ther refer­ences to dis­tance, vol­ume and ve­loc­ity should be in­ter­preted in co­mov­ing co­or­di­nates, un­less they are ex­plic­itly about the ex­pan­sion of the Uni­verse.

Civil­i­sa­tions in the reach­able universe

Prior prob­a­bil­ity distribution

In the clas­si­cal Drake equa­tion, the ex­pected num­ber of de­tectable civil­i­sa­tions in the Milky Way is pro­duced by mul­ti­ply­ing 7 differ­ent es­ti­mates to­gether:

  • , the rate of star for­ma­tion in the Milky Way
  • , the frac­tion of stars that have planets around them,
  • , the num­ber of Earth-like planets for each such system
  • , the frac­tion of such planets on which life ac­tu­ally evolves
  • , the frac­tion of life-filled planets where in­tel­li­gence even­tu­ally appears
  • , the frac­tion of in­tel­li­gent civil­i­sa­tions which are detectable
  • , the av­er­age longevity of such civilisations

I will use a mod­ified ver­sion of this equa­tion, where I con­sider the spread of in­ter­galac­tic civil­i­sa­tions across the Uni­verse rather than the num­ber of de­tectable civil­i­sa­tions in the Milky Way. In­stead of , , and , I will use the planet for­ma­tion rate de­scribed in Planet for­ma­tion. For the rest of this post, I will use “planet” and “Earth-like planet” in­ter­change­ably. Since what I care about is how in­tel­li­gent life ex­pands across the Uni­verse over time, rather than how many de­tectable civil­i­sa­tions ex­ist, I will re­place the frac­tion with a frac­tion , the frac­tion of in­tel­li­gent civil­i­sa­tions that even­tu­ally de­cide to send probes to other galax­ies. I as­sume that it’s im­pos­si­ble to go ex­tinct once in­ter­galac­tic colon­i­sa­tion has be­gun,[5] and will there­fore not use .

Thus, the re­main­ing un­cer­tain pa­ram­e­ters are , and . Mul­ti­ply­ing , and to­gether, we get the ex­pected frac­tion of Earth-like planets that even­tu­ally yields an in­ter­galac­tic civil­i­sa­tion, which I will re­fer to as .

For the prior prob­a­bil­ity of and , I will use the dis­tri­bu­tions from Sand­berg et al. (2018) with only small mod­ifi­ca­tions.

The frac­tion of planets that yields life is calcu­lated as us­ing the num­ber of times that life is likely to emerge on a given planet, , which is in turn mod­el­led as a log­nor­mal dis­tri­bu­tion with a stan­dard de­vi­a­tion of 50 or­ders of mag­ni­tude.[6] Sand­berg et al. (2018) use a con­ser­va­tively high me­dian of life aris­ing 1 time per planet. Since my ar­gu­ment is that an­throp­ics should make us be­lieve that life is rel­a­tively com­mon, rather than rare, I will use a con­ser­va­tively low me­dian of life ap­pear­ing once ev­ery planets.

I will use the same as Sand­berg et al., i.e., a log-uniform dis­tri­bu­tion be­tween and .

cor­re­sponds to the so called late filter, and is par­tic­u­larly in­ter­est­ing since our civil­i­sa­tion hasn’t passed it yet. While the an­thropic con­sid­er­a­tions fa­vor larger val­ues of and , the Fermi ob­ser­va­tion fa­vors smaller val­ues of at least one of , , and , mak­ing the one value which is only af­fected by one of them. The to­tal effect is that an­thropic ad­just­ments put a lot of weight on very small (which cor­re­sponds to a large late filter): Katja Grace has de­scribed how the self-in­di­ca­tion as­sump­tion strongly pre­dicts that we will never leave the so­lar sys­tem, and the de­ci­sion-the­o­retic ap­proaches ask us to con­sider that our ac­tions can be repli­cated across a huge num­ber of planets if other civil­i­sa­tions couldn’t in­terfere with ours (since this would make our ob­ser­va­tions con­sis­tent with civil­i­sa­tions be­ing very com­mon).

Look­ing at this from a to­tal con­se­quen­tial­ist per­spec­tive, the im­pli­ca­tions aren’t ac­tu­ally too big, if we be­lieve that the ma­jor­ity of all value and dis­value will ex­ist in the fu­ture. Con­sider the pos­si­bil­ity that no civil­i­sa­tion can ever leave their so­lar sys­tem. In this case, our ac­tions can be repli­cated across a vast num­ber of planets: since no civil­i­sa­tions ever leave their planet, life can be very com­mon with­out us notic­ing any­thing strange in the skies. How­ever, if this is true, no civil­i­sa­tion will ever reach farther than it’s so­lar sys­tem, strongly re­duc­ing the im­pact we can have on the fu­ture. This re­duc­tion of im­pact is roughly pro­por­tional to the ad­di­tional num­ber of repli­ca­tions, and thus, the to­tal im­pact that we can have is roughly as large in both cases.[7] To see why, con­sider that the main de­ter­mi­nant of our pos­si­ble im­pact is the frac­tion of the Uni­verse that civil­i­sa­tions like us will even­tu­ally con­trol, i.e. the frac­tion of the Uni­verse that we can af­fect. This frac­tion doesn’t vary much by whether the Uni­verse will be colon­ised by a large num­ber of civil­i­sa­tions that never leave their so­lar sys­tems, or a small num­ber of civil­i­sa­tions that colon­ise al­most all space they can reach. Since the amount of im­pact doesn’t vary much, the main thing de­ter­min­ing what we should fo­cus on is our an­thropic-naïve es­ti­mates of what sce­nar­ios are like­liest.[8]

I won’t pay much more at­ten­tion to the late filter in this ar­ti­cle, since it isn’t par­tic­u­larly rele­vant for how life will spread across the Uni­verse. To see why, con­sider that the Fermi ob­ser­va­tion and the an­thropic up­date to­gether will push the product to a rel­a­tively spe­cific value (roughly the largest value that doesn’t make the Fermi ob­ser­va­tion too un­likely, as shown in the next two sec­tions). As long as is sig­nifi­cantly larger than the fi­nal product should be, it doesn’t mat­ter what is: the up­dates will ad­just and so that the product re­mains ap­prox­i­mately the same. A 10 times larger im­plies a 10 times smaller , so the dis­tri­bu­tion of will look very similar. While our near term fu­ture will look differ­ent, the (an­thropic-ad­justed) prob­a­bil­ity that a given planet will yield an in­ter­galac­tic civil­i­sa­tion will re­main about the same, so the long term fu­ture of the Uni­verse will be very similar. If, how­ever, is so small that it alone ex­plains why we haven’t seen any other civil­i­sa­tions (which is plau­si­ble if we as­sign non-neg­ligible prob­a­bil­ity to space colon­i­sa­tion be­ing im­pos­si­ble) the long term fu­ture of the Uni­verse will look very differ­ent. The an­thropic up­date will push and to­wards 1, and the prob­a­bil­ity dis­tri­bu­tion of will roughly equal the dis­tri­bu­tion of . In this case, how­ever, hu­mans can­not (or are ex­traor­di­nar­ily un­likely to) af­fect the long term fu­ture of the Uni­verse, and whether the Uni­verse will be filled with life or not is ir­rele­vant to our plans. For this rea­son I will ig­nore sce­nar­ios where life is ex­traor­di­nar­ily un­likely to colon­ise the Uni­verse, by mak­ing lo­gu­niform be­tween and . How an­throp­ics and the Fermi ob­ser­va­tion should af­fect our be­liefs about late filters is an in­ter­est­ing ques­tion, but it’s not one that I’ll ex­pand on in this post.

All taken to­gether, the prior of is de­picted in figure 4. This and all sub­se­quent dis­tri­bu­tions were gen­er­ated us­ing Monte Carlo simu­la­tions, i.e., by gen­er­at­ing large amounts of ran­dom num­bers from the dis­tri­bu­tions and mul­ti­ply­ing them to­gether.

Figure 4: Prior prob­a­bil­ity dis­tri­bu­tion over the frac­tion of planets from which an in­ter­galac­tic civil­i­sa­tions will emerge. The prob­a­bil­ity den­sity mea­sures the prob­a­bil­ity per or­der of mag­ni­tude. Since ex­tends from 1 to be­low , while nei­ther nor varies more than 4 or­ders of mag­ni­tude, most of the shape of the graph is ex­plained by . The num­ber of times that life is likely to arise on a given planet, , has some prob­a­bil­ity mass on num­bers above . For all such , , which ex­plains the in­crease in prob­a­bil­ity just at the end.

My con­clu­sions hold for any prior that puts non-neg­ligible prob­a­bil­ity on life be­ing com­mon works, so the de­tails don’t ac­tu­ally mat­ter that much. This is dis­cussed in Ap­pendix C.

Up­dat­ing on the Fermi observation

In this anal­y­sis, the Fermi ob­ser­va­tion is the ob­ser­va­tion that no alien civil­i­sa­tion has reached our galaxy yet. I will take this ob­ser­va­tion at face value, and ne­glect the pos­si­bil­ity that alien civil­i­sa­tions are here but re­main un­de­tectable (known as the Zoo hy­poth­e­sis).

In or­der to un­der­stand how strong this up­date is, we must un­der­stand how many planets there are that could have yielded in­ter­galac­tic civil­i­sa­tions close enough to reach us, and yet didn’t. To get this num­ber, we need to know how many civil­i­sa­tion could have ap­peared a given year, given in­for­ma­tion about the planet for­ma­tion rate dur­ing all past years. Thus, we need an es­ti­mate of the time it takes for a civil­i­sa­tion to ap­pear on a planet af­ter it has been formed, on the planets where civil­i­sa­tions emerge. I will as­sume that this time is dis­tributed as a nor­mal dis­tri­bu­tion with mean 4.55 billion years (since this is the time it took for our civil­i­sa­tion to ap­pear) and stan­dard de­vi­a­tion 1 billion years, trun­cated (and renor­mal­ised) so that there is 0 % chance of ap­pear­ing in ei­ther less than 2 billion years or more than 8 billion years.[9] Thus, the num­ber of planets per vol­ume per year from which civil­i­sa­tions could arise at time is

where is the planet for­ma­tion rate per vol­ume per year at time and is the prob­a­bil­ity den­sity at t of a nor­mal dis­tri­bu­tion de­scribed as above, cen­tered around years.

Us­ing as the num­ber of pos­si­ble civil­i­sa­tions ap­pear­ing per vol­ume per year, all we need is an es­ti­mate of the vol­ume of space, , from which civil­i­sa­tions could have reached the Milky Way if they left their own galaxy at time . A probe leav­ing at time can reach a dis­tance of be­fore the pre­sent time, years af­ter the Big Bang, with defined as in Speed of travel. Probes sent out at time could there­fore have reached us from any point in a sphere with volume

Us­ing and , the ex­pected num­ber of planets from which in­ter­galac­tic civil­i­sa­tions could have reached us, if they ap­peared, is

What does this tell us about the prob­a­bil­ity that such a civil­i­sa­tion ap­pears on such a planet? Let de­note the prob­a­bil­ity that is the frac­tion of planets that yields in­ter­galac­tic civil­i­sa­tions, and de­note the ob­ser­va­tion that none of N planets de­vel­oped an in­ter­galac­tic civil­i­sa­tion. Bayes the­o­rem tells us that:

is the prior prob­a­bil­ity dis­tri­bu­tion de­scribed in the above seg­ment, and we can di­vide by by nor­mal­is­ing all val­ues of . Thus, the only new in­for­ma­tion we need is , i.e., the prob­a­bil­ity that none of the planets would have yielded an in­ter­galac­tic civil­i­sa­tion given that an ex­pected frac­tion of such planets yields such civ­i­liza­tions. Since each of the planets has a chance to not yield an in­ter­galac­tic civil­i­sa­tion .[10] In this case, this means that . Up­dat­ing the prior dis­tri­bu­tion ac­cord­ing to this, and nor­mal­is­ing, gives the pos­te­rior dis­tri­bu­tion in figure 5.

Figure 5: Prob­a­bil­ity dis­tri­bu­tion over the frac­tion of planets from which an in­ter­galac­tic civil­i­sa­tions will emerge, af­ter up­dat­ing on the Fermi ob­ser­va­tion. The prob­a­bil­ity den­sity mea­sures the prob­a­bil­ity per or­der of mag­ni­tude.

The pos­te­rior as­signs close to 0 prob­a­bil­ity to any frac­tion f larger than , since it’s very un­likely that no civil­i­sa­tion would have reached us if that was the case. Any frac­tion smaller than is mostly un­af­fected, how­ever, since us be­ing alone is the most likely out­come un­der all of them. In the next sec­tion, I will use “pos­te­rior” to re­fer to this new dis­tri­bu­tion.

Up­dat­ing on anthropics

As stated above, I will be us­ing an­thropic the­o­ries that weigh hy­pothe­ses by the num­ber of copies of us that ex­ists. This isn’t un­con­tro­ver­sial. If we con­sider a part of the Uni­verse that’s very large, but finite,[11] then some copy of us is likely to ex­ist re­gard­less of how im­prob­a­ble life is (as long as it’s not im­pos­si­ble). The ques­tion, then, is how we should rea­son about the fact that we ex­ist. I’d say that the three most pop­u­lar views on how to think about this are:

  • The self-sam­pling as­sump­tion[12] (SSA): We should rea­son as if we are a ran­domly se­lected ob­server from all ac­tu­ally ex­ist­ing ob­servers in our refer­ence class, and only up­date our be­liefs on the fact that at least one copy in that refer­ence class ex­ists.[13] Since at least one copy of us would ex­ist re­gard­less of how un­com­mon civil­i­sa­tions are, the fact that we ex­ist doesn’t provide us with any ev­i­dence. Thus, we should stick to our pri­ors.

  • The self-in­di­ca­tion as­sump­tion[14] (SIA): We should rea­son as if we are a ran­domly se­lected ob­server from all pos­si­ble ob­servers across all imag­in­able wor­lds, weighted by their prob­a­bil­ities. Prob­a­bil­ities are up­dated by mul­ti­ply­ing the prior prob­a­bil­ity of be­ing in each world with the num­ber of copies of us in each such world, and then nor­mal­is­ing. Thus, we are more likely to be in a world where life is com­mon, since life be­ing com­mon im­plies that more copies of arise across the mul­ti­verse.

  • An­thropic De­ci­sion The­ory[15] (ADT): Epistem­i­cally, we should only up­date on the fact that at least one copy of us ex­ist, as SSA does. How­ever, we should take into ac­count that any ac­tion that we perform will be performed by all of our copies as well; since there is no way that our copies will do any­thing differ­ently than we do, we’re effec­tively mak­ing a de­ci­sion for all of our copies at once. This means that we should care pro­por­tion­ally more about mak­ing the right de­ci­sions in the case where there are more copies of us, if we ac­cept some form of to­tal con­se­quen­tial­ism. For each pos­si­ble world, the ex­pected im­pact that we will have is the prior prob­a­bil­ity of that world mul­ti­plied with the num­ber of copies in that world.[16] De­spite their differ­ences, SIA and ADT will always agree on the rel­a­tive value of ac­tions and make the same de­ci­sions,[17] given to­tal con­se­quen­tial­ism (Arm­strong, 2017).

Per­son­ally, I think that ADT is cor­rect, and I will as­sume to­tal con­se­quen­tial­ism in this post.[18] How­ever, since SIA and ADT both mul­ti­ply the prior prob­a­bil­ity of each pos­si­ble world with the num­ber of copies in that world, they will always agree on what ac­tion to take. The fact that SIA nor­mal­ises all num­bers and treats them as prob­a­bil­ities is ir­rele­vant, since all we’re in­ter­ested in is the rel­a­tive im­por­tance of ac­tions.

So how does an­throp­ics ap­ply to this case? Dis­re­gard­ing the effects from other civil­i­sa­tions in­terfer­ing, we should ex­pect the num­ber of copies of us to be pro­por­tional to the prob­a­bil­ity that we in par­tic­u­lar would arise from a given Earth-like planet. This is true as long as we hold the num­ber of planets con­stant, and con­sider a very large uni­verse. We can dis­re­gard the effects from other civil­i­sa­tions since any copy of us would ex­pe­rience an empty uni­ver­sity, as well, which we’ve already ac­counted for in the above sec­tion.

The ques­tion, then, is how the sought quan­tities , and re­late to the prob­a­bil­ity that we in par­tic­u­lar arise from a given planet. As de­tailed in Ap­pendix A, and are pro­por­tional to the prob­a­bil­ity that our civil­i­sa­tion in par­tic­u­lar ap­pears on Earth. On the other hand, we shouldn’t ex­pect to have any spe­cial re­la­tion to the prob­a­bil­ity of us ex­ist­ing (ex­cept that it af­fects the prob­a­bil­ity of the Fermi ob­ser­va­tion, which we’ve already con­sid­ered). is con­cerned about what hap­pens af­ter this mo­ment in time, not be­fore, so the prob­a­bil­ity that a civil­i­sa­tion like ours is cre­ated should be roughly the same no mat­ter what is. As­sum­ing that the num­ber of copies of us are ex­actly pro­por­tional to and , and in­de­pen­dent of , we can call the num­ber of copies of you in a large, finite world , where is some con­stant.

Thus, the an­thropic ad­justed prob­a­bil­ity that some val­ues of , and are cor­rect is the prior prob­a­bil­ity that they’re cor­rect mul­ti­plied with . If we use SIA, dis­ap­pear af­ter nor­mal­is­ing, since we’re just mul­ti­ply­ing the prob­a­bil­ity of ev­ery event with the same con­stant. If we use the de­ci­sion the­o­retic ap­proach, we’re only in­ter­ested in the rel­a­tive value of our ac­tions, so doesn’t mat­ter.

Us­ing this to up­date on the pos­te­rior we got from up­dat­ing on the Fermi ob­ser­va­tion, we get the dis­tri­bu­tion in figure 5.

Figure 5: An­thropic ad­justed prob­a­bil­ity dis­tri­bu­tion over the frac­tion of planets from which an in­ter­galac­tic civil­i­sa­tions will emerge, tak­ing into ac­count the Fermi ob­ser­va­tion. If you en­dorse the self-in­di­ca­tion as­sump­tion, the An­thropic-ad­justed prob­a­bil­ity den­sity cor­re­sponds to nor­mal prob­a­bil­ity den­sity. If you en­dorse one of the de­ci­sion-the­o­retic ap­proaches, it mea­sures the product of prob­a­bil­ity den­sity and the num­ber of times that your ac­tions are repli­cated across the Uni­verse.

As you can see, the an­thropic up­date is very strong, and the com­bi­na­tion of the Fermi ob­ser­va­tion and the an­thropic up­date yields a rel­a­tively small range for non-neg­ligible val­ues of . A more zoomed-in ver­sion shows what value of should be ex­pected. To see the in­ter­ac­tion be­tween the Fermi ob­ser­va­tion and the an­thropic up­date, the origi­nal pos­te­rior and the an­thropic-ad­justed prior are also de­picted in figure 6.

Figure 6: The red line de­picts the prob­a­bil­ity dis­tri­bu­tion over the frac­tion of planets from which an in­ter­galac­tic civil­i­sa­tions will emerge, af­ter up­dat­ing on the Fermi ob­ser­va­tion, zoomed in on the smaller in­ter­val. The solid blue line de­picts the same dis­tri­bu­tion af­ter ad­just­ing for an­throp­ics. The dashed blue line de­picts the an­thropic ad­justed prior dis­tri­bu­tion, with­out tak­ing into ac­count the Fermi ob­ser­va­tion.

Ad­just­ing for an­throp­ics strongly se­lects for a large val­ues of and , which would im­ply that al­most all planets de­velop civil­i­sa­tion. There­fore, the an­thropic-ad­justed prior prob­a­bil­ity that a given planet yields an in­ter­galac­tic civil­i­sa­tion (the dashed blue line in figure 6) is very similar to the prob­a­bil­ity that an ex­ist­ing civil­i­sa­tion be­comes in­ter­galac­tic: . Thus, the dis­tri­bu­tion looks very similar to the log uniform dis­tri­bu­tion of . How­ever, for val­ues of smaller than , ei­ther or has to be smaller than . Since the an­thropic ad­just­ment is made by mul­ti­ply­ing with and , this means that the an­thropic ad­justed prob­a­bil­ity de­clines pro­por­tion­ally be­low . This looks like an ex­po­nen­tial de­cline in figure 6, since the x-axis is log­a­r­ith­mic.

Up­dat­ing the prior on the Fermi ob­ser­va­tion with­out tak­ing an­throp­ics into ac­count yields the pos­te­rior (the red line in figure 6) that strongly pe­nal­ises any hy­poth­e­sis that would make in­ter­galac­tic civil­i­sa­tions com­mon. The up­date from the Fermi ob­ser­va­tion is pro­por­tional to . Since is very large, and the an­thropic up­date is pro­por­tional to , the up­date from the Fermi ob­ser­va­tion is sig­nifi­cantly stronger than the an­thropic up­date for mod­er­ately large val­ues of . Thus, up­dat­ing the an­thropic ad­justed dis­tri­bu­tion on the Fermi ob­ser­va­tion yields the blue line in figure 6.

The me­dian of that dis­tri­bu­tion is that each planet has a prob­a­bil­ity of yield­ing an in­ter­galac­tic civil­i­sa­tion; the peak is around .

Si­mu­lat­ing civil­i­sa­tions’ expansion

In­so­far as we trust these num­bers, they give us a lot of in­for­ma­tion about whether in­ter­galac­tic civil­i­sa­tions are likely to arise from the planets that we haven’t di­rectly ob­served. This in­cludes both planets that will be cre­ated in the fu­ture and planets so far away that any po­ten­tial civil­i­sa­tions haven’t been able to reach us, yet. In or­der to find out this im­plies for our fu­ture, I run a simu­la­tion of how civil­i­sa­tions spread across the Uni­verse.

To do this, I con­sider all galax­ies close enough that a probe sent from that galaxy could some­day en­counter a probe sent from Earth. Probes sent from Earth to­day could reach galax­ies that are presently about light-years away; since civil­i­sa­tions that start colon­i­sa­tion ear­lier would be able to reach farther, I con­sider all galax­ies less than light-years away.

I then run the simu­la­tion from the Big Bang to about 60 billion years af­ter the Big Bang, at which point planet for­ma­tion is neg­ligible (about as many planets per year as now). For ev­ery point in time, there is some prob­a­bil­ity that a civil­i­sa­tion arises from among a group of galax­ies, calcu­lated from and the pre­sent planet for­ma­tion rate. If it is, it spreads out­wards at 80 % of the speed of light, colon­is­ing each galaxy that it passes. The de­tails of how this is simu­lated is de­scribed in Ap­pendix B.

By the end of the simu­la­tion, each group of galax­ies ei­ther has a time at which they were first colon­ised, or they are still empty. We can then com­pare this to the time at which Earth-origi­nat­ing probes would have reached the galax­ies, if they were to leave now. As­sum­ing that Earth claims ev­ery point which it reaches be­fore other civil­i­sa­tions, we get an es­ti­mate of the amount of space that Earth would get for a cer­tain . Ad­di­tion­ally, we learn what frac­tion of that space would be colon­ised by other civil­i­sa­tions in our ab­sence, and what frac­tion would have re­mained empty.

For , the me­dian from the pre­vi­ous sec­tion, Earth-origi­nat­ing probes ar­rives first to only 0.5 % of the space that they’re able to reach. Other civil­i­sa­tions even­tu­ally ar­rive at al­most all of that space; probes from Earth doesn’t lead to any ex­tra space be­ing colon­ised.

How­ever, we can do bet­ter than us­ing point es­ti­mates for . Figure 7 is a graph of the frac­tion of the reach­able uni­verse that Earth-origi­nat­ing probes get as func­tion of .

Figure 7: The red line de­picts the ex­pected frac­tion of the reach­able uni­verse that will only reached by Earth-origi­nat­ing in­tel­li­gence. The yel­low line de­picts the ex­pected frac­tion of the reach­able uni­verse that Earth-origi­nat­ing in­tel­li­gence reaches some time be­fore other civil­i­sa­tions ar­rive. The blue line is the sum of these: it de­picts the to­tal ex­pected frac­tion of the reach­able uni­verse that Earth-origi­nat­ing in­tel­li­gence will find empty, when they ar­rive.

As you can see, Earth-origi­nat­ing probes get a sig­nifi­cantly larger share of the Uni­verse for smaller . The an­thropic-ad­justed prob­a­bil­ity mass on those es­ti­mates isn’t neg­ligible, so they must be taken into ac­count. Sum­ming over the prob­a­bil­ity and the frac­tion of the Uni­verse that we get for differ­ent val­ues of , Earth-origi­nat­ing in­tel­li­gence get 5 % of the reach­able uni­verse on av­er­age. 64 % of this is space that would have oth­er­wise been oc­cu­pied by alien civil­i­sa­tions, while 36 % is space that would have re­mained empty in our ab­sence.

Th­ese num­bers de­note the ex­pected val­ues for a ran­dom copy of Earth cho­sen from among all pos­si­ble copies of Earth weighted by prior prob­a­bil­ity. Us­ing SIA, this is equiv­a­lent to the ex­pected value of what will hap­pen on our Earth. Us­ing ADT, this is the ex­pected value weighted by the num­ber of copies that our de­ci­sions af­fect.

Ap­pendix C de­scribes how these num­bers vary for differ­ent choices of pa­ram­e­ters. The re­sults are sur­pris­ingly ro­bust. For the sce­nar­ios I con­sider, the av­er­age frac­tion of space that would be oc­cu­pied by other civil­i­sa­tions in our ab­sence varies from roughly 50 % to roughly 80 %.

Implications

Th­ese re­sults af­fect the value of colon­is­ing the Uni­verse in a num­ber of ways.

Most ob­vi­ously, the fact that hu­mans won’t get to colon­ise all the galax­ies in reach diminishes the size of our fu­ture. This effect is mostly neg­ligible. Ac­cord­ing to these as­sump­tions, we will get about 5 % of the reach­able uni­verse in ex­pec­ta­tion. Since un­cer­tainty about the size of the far fu­ture spans tens of or­ders of mag­ni­tude, a fac­tor of 20 isn’t re­ally rele­vant for cause pri­ori­ti­sa­tion to­day.

The effect that our space colon­i­sa­tion might have on other civil­i­sa­tions is more rele­vant.

Dis­plac­ing other civilisations

Us claiming space will lead to other civil­i­sa­tions get­ting less space. If is the frac­tion of space that we are likely to take from other civil­i­sa­tions, rather than take from empty space, one can ex­press the ex­pected value of the far fu­ture as , where is the to­tal vol­ume of space that we are likely to get, is the prob­a­bil­ity that Earthly life will sur­vive long enough to get it, is the value of Earth-origi­nat­ing in­tel­li­gence ac­quiring one unit of vol­ume and is the value of an alien civil­i­sa­tions ac­quiring one unit of vol­ume.[19] For the rea­sons men­tioned above, I doubt that us dis­plac­ing aliens would change any­one’s mind about the value of fo­cus­ing on the long term. How­ever, it might play a small role in de­ter­min­ing where long-ter­mists should al­lo­cate their efforts.

In a sim­ple model, most long-ter­mist causes fo­cus on ei­ther in­creas­ing , the prob­a­bil­ity that Earth-origi­nat­ing in­tel­li­gence sur­vives long enough to colon­ise space, or on in­creas­ing , the value of space colon­i­sa­tion. Some ex­am­ples of the former is work against risks from biotech­nol­ogy and nu­clear war. Some ex­am­ples of the lat­ter is AI-al­ign­ment (since un­al­igned AI also is likely to colon­ise space) and spread­ing good val­ues. The value of in­creas­ing is un­af­fected by the fact that we will dis­place other civil­i­sa­tions, since in­creas­ing by one unit yields value, which is in­de­pen­dent of . How­ever, in­creas­ing gen­er­ates value, which de­creases as in­creases.

How much one should value Earth-origi­nat­ing and alien civil­i­sa­tions is very un­clear. If you ac­cept moral anti-re­al­ism, one rea­son to ex­pect aliens to be less valuable than Earth-origi­nat­ing civil­i­sa­tions is that hu­mans are more likely to share your val­ues, since you are a hu­man. How­ever, there might be some con­ver­gence among goals, so it’s un­clear how strong this effect is.

Con­sider the case where , i.e., both Earth-origi­nat­ing and alien civil­i­sa­tions are net-pos­i­tive, but Earth-origi­nat­ing civil­i­sa­tions are bet­ter. The ex­tremes are at , where we don’t care at all that we’re dis­plac­ing other civil­i­sa­tions, and , where we value all civil­i­sa­tions equally, in ex­pec­ta­tion. With the es­ti­mate of , work to re­duce ex­tinc­tion is 36 % as good if as it is if . If is in the mid­dle, say at , re­duc­ing the risk of ex­tinc­tion is about 70 % as valuable as oth­er­wise.

Now con­sider the case where . If you be­lieve that alien civil­i­sa­tions are more likely to cre­ate harm than good, the value of in­creas­ing is greater than it would be oth­er­wise. If you think that alien civil­i­sa­tions cause about as much harm as hu­mans cre­ate good, for ex­am­ple, then in­creas­ing is 1.64 as good as it would have been oth­er­wise.

A more in­ter­est­ing case is where you be­lieve that and , ei­ther be­cause you’re pes­simistic about the fu­ture or be­cause you hold val­ues that pri­ori­tise the re­duc­tion of suffer­ing. If you also be­lieve that , the fact that we’re dis­plac­ing aliens could make space colon­i­sa­tion net-pos­i­tive, if aliens colon­is­ing space is more than 1.4 times as bad as Earth-origi­nat­ing in­tel­li­gence colon­is­ing space (and if you are con­fi­dent in the as­sump­tions and ac­cu­racy of these re­sults). Whether this is likely to be the case is dis­cussed by Jan Brauner and Fried­er­ike Grosse-Holz in sec­tion 2.1 of this ar­ti­cle and by Brian To­masik here.

Co­op­er­a­tion or conflict

If we loosen the as­sump­tion that who­ever gets to a galaxy first gets to keep it, we can see that there are pos­si­bil­ities for con­flict, which seems very bad, and co­op­er­a­tion, which seems valuable.

It’s un­clear ex­actly how likely con­flict would be on an in­ter­galac­tic scales, in the cases where differ­ent civil­i­sa­tions en­counter each other. To a large de­gree, this rests on whether defense or offense is likely to dom­i­nate on an in­ter­galac­tic scale. Phil Tor­res seems to think that most civil­i­sa­tions will want to at­tack their neigh­bours just to make sure that their neigh­bours doesn’t at­tack them first, while An­ders Sand­berg sug­gests that fu­ture tech­nol­ogy might en­able a perfect scorched earth strat­egy, re­mov­ing any in­cen­tive to at­tack. Fur­ther­more, there is a ques­tion of how bad con­flict would be. There are mul­ti­ple ways in which con­flicts could plau­si­bly waste re­sources or lead to suffer­ing, but I won’t list them all here.

The pos­i­tive ver­sion of this would be to trade and co­op­er­ate with neigh­bour­ing civil­i­sa­tions, in­stead. This could take the form of trad­ing var­i­ous re­sources with each other, for mu­tual gain, though this in­crease in re­sources seems un­likely to dom­i­nate the size of our fu­ture. Another case is where neigh­bour­ing civil­i­sa­tions are do­ing some­thing that we don’t want them to do. If, for ex­am­ple, an­other civil­i­sa­tion is ex­per­i­ment­ing on var­i­ous minds with no re­gard for their suffer­ing, there might be a mu­tu­ally benefi­cial deal where we pay them to use the fu­ture equiv­a­lent of anaes­the­sia.

So how much space would these con­sid­er­a­tions af­fect? Here, con­sid­er­a­tions of how ex­actly an ex­tra civil­i­sa­tion might im­pact the in­ter­galac­tic game­board gets quite com­pli­cated. Of the 5 % of reach­able space that Earth-origi­nat­ing in­tel­li­gence would be able to claim, all of the 64 % that we get to be­fore other civil­i­sa­tions will be space that the other civil­i­sa­tions might want to fight about when they even­tu­ally get there, so such con­sid­er­a­tions might be quite im­por­tant. Similarly, the 95 % of space that Earth-origi­nat­ing in­tel­li­gence won’t ar­rive at un­til af­ter other civil­i­sa­tions have claimed it is space that the fu­ture rulers of Earth might de­cide to fight about, or space that the fu­ture rulers of Earth might benefit from trad­ing with.

If you be­lieve that con­flict is likely to cre­ate large dis­value re­gard­less of whether the fight is be­tween Earth-origi­nat­ing civil­i­sa­tions and alien civil­i­sa­tions, or be­tween alien civil­i­sa­tions, one po­ten­tially in­ter­est­ing met­ric is the amount of space that both we and oth­ers will even­tu­ally ar­rive to, that would oth­er­wise only ever have been reached by one civil­i­sa­tion. Un­der the base as­sump­tions, about 1.7 % of all the space that we would be able to reach is space that would only have been reached by one civil­i­sa­tion, if Earth hadn’t ex­isted. This is roughly 36 % as large as all of the space that we’ll get to first (which is about 4.7 % of the reach­able uni­verse).

Fur­ther research

Th­ese re­sults also sug­gests that a few other lines of re­search might be more valuable than ex­pected.

One po­ten­tially in­ter­est­ing ques­tion is how other ex­pla­na­tions of the Fermi para­dox changes the re­sults of these calcu­la­tions, and to what ex­tent they’re af­fected by an­thropic up­dates. In gen­eral, most ex­pla­na­tions of the Fermi para­dox ei­ther don’t al­low a large num­ber of civil­i­sa­tions like us to ex­ist (e.g. a berserker civil­i­sa­tion kill all nascent life), which would im­ply a weak an­thropic up­date, or don’t give us par­tic­u­larly large power over the reach­able uni­verse (e.g. the Zoo hy­poth­e­sis). As a re­sults, such hy­poth­e­sis wouldn’t have a par­tic­u­larly large im­pact on the ex­pected im­pact we’ll have, as long as their prior prob­a­bil­ities are low.

How­ever, there are some ex­cep­tions. Hy­pothe­ses which posits that early civil­i­sa­tions are very un­likely, while civil­i­sa­tions emerg­ing around now are likely would sys­tem­at­i­cally al­low a larger frac­tion of the Uni­verse to go to civil­i­sa­tions like ours. An ex­am­ple of a hy­poth­e­sis like this is neo­catas­trophism, which as­serts that life has been hin­dered by gamma ray bursts up un­til now. As gamma ray bursts get less com­mon when the star for­ma­tion rate de­clines, neo­catas­trophism could ac­count for the lack of life in the past while al­low­ing for a large amount of civil­i­sa­tions ap­pear­ing around now. I haven’t con­sid­ered this ex­plic­itly here (al­though vari­a­tions with later life-for­ma­tion touch on it) since it seems un­likely that a shift from life be­ing very un­likely to life be­ing likely could hap­pen par­tic­u­larly fast across galax­ies: there would be large vari­a­tion in gamma ray bursts be­tween differ­ent types of galax­ies. How­ever, I might be wrong about the prior im­plau­si­bil­ity, and the an­thropic up­date seems like it might be quite large.[20] In any case, there may be similar ex­pla­na­tions of the Fermi para­dox which would be more prob­a­ble on pri­ors, while re­ceiv­ing a similarly strong an­thropic up­date. There are also some ex­pla­na­tions that doesn’t neatly fit into this frame­work: I dis­cuss the im­pli­ca­tions of the simu­la­tion hy­poth­e­sis in Ap­pendix D.

It might also be more im­por­tant than ex­pected to figure out what might hap­pen if we en­counter other civil­i­sa­tions. Speci­fi­cally, re­search could tar­get how valuable other civil­i­sa­tions are likely to be when com­pared with Earth-origi­nat­ing civil­i­sa­tions; whether in­ter­ac­tions with such civil­i­sa­tions are likely to be pos­i­tive or nega­tive; and whether there is some­thing we should do to pre­pare for such en­coun­ters to­day.[21]

There are also a few large un­cer­tain­ties re­main­ing in this anal­y­sis. One big un­cer­tainty con­cerns how an­throp­ics in­ter­acts with in­finites. Ad­di­tion­ally, all of this anal­y­sis as­sumes to­tal con­se­quen­tial­ism with­out diminish­ing marginal re­turns to re­sources, and I’m un­sure how much ap­plies to other, more com­pli­cated the­o­ries.

Acknowledgments

I’m grate­ful to Max Daniel and Hjal­mar Wijk for com­ments on the fi­nal draft. Thanks also to Max Dal­ton, Parker Whit­fill, Ai­dan Goth and Cather­ine Scan­lon for dis­cus­sion and com­ments on ear­lier ver­sions, and to An­ders Sand­berg and Stu­art Arm­strong for hap­pily an­swer­ing ques­tions and shar­ing re­search with me.

Many of these ideas were first men­tioned by Brian To­masik in the fi­nal sec­tion of Rank­ing Ex­pla­na­tions of the Fermi Para­dox, which served as a ma­jor source of in­spira­tion.

Most of this pro­ject was done dur­ing a re­search in­tern­ship at the Cen­tre for Effec­tive Altru­ism. Views ex­pressed are en­tirely my own.

References

Arm­strong, S. (2017). An­thropic De­ci­sion The­ory. arXiv preprint arXiv:1110.6437.

Arm­strong, S. and Sand­berg, A. (2013). Eter­nity in 6 hours: in­ter­galac­tic spread­ing of in­tel­li­gent life and sharp­en­ing the Fermi para­dox. Acta Astro­nau­tica, 89, 1-13.

Behroozi, P., & Peeples, M. S. (2015). On the his­tory and fu­ture of cos­mic planet for­ma­tion. Monthly No­tices of the Royal Astro­nom­i­cal So­ciety, 454(2), 1811-1817.

Bostrom, N. (2011). In­finite Ethics. Anal­y­sis & Me­ta­physics, 10.

Lineweaver, C. H. (2001). An es­ti­mate of the age dis­tri­bu­tion of ter­res­trial planets in the uni­verse: quan­tify­ing metal­lic­ity as a se­lec­tion effect. Icarus, 151(2), 307-313.

Madau, P., & Dick­in­son, M. (2014). Cos­mic star-for­ma­tion his­tory. An­nual Re­view of Astron­omy and Astro­physics, 52, 415-486.

Sand­berg, A. (2018). Space races: set­tling the uni­verse fast. Tech­ni­cal Re­port #2018-01. Fu­ture of Hu­man­ity In­sti­tute. Univer­sity of Oxford.

Sand­berg, A., Drexler, E., & Ord, T. (2018). Dis­solv­ing the Fermi Para­dox. arXiv preprint arXiv:1806.02404.

Ap­pendix A: How and re­lates to our existence

In this ap­pendix, I dis­cuss whether it’s jus­tified to di­rectly up­date and when up­dat­ing on our ex­is­tence. de­notes the prob­a­bil­ity that life ap­pears on an Earth-like planet, and de­notes the prob­a­bil­ity that such life even­tu­ally be­comes in­tel­li­gent and de­vel­ops civil­i­sa­tion.

In or­der to di­rectly up­date and on our ex­is­tence, the prob­a­bil­ity of us ap­pear­ing needs to be k times as high if ei­ther or is k times as high, for all rele­vant val­ues of and . If this is the case, then there should be k times as many copies of us in wor­lds where or is k times as high, for all val­ues of and . Be­cause of this, we can mul­ti­ply with the num­ber of planets to get the ex­pected num­ber of copies of us on those planets, up to a con­stant fac­tor. We can then use this value to perform the an­thropic up­date, as de­scribed in Up­dat­ing on an­throp­ics.

This does not mean that the prob­a­bil­ity of us ex­ist­ing needs to be pro­por­tional to , in ev­ery pos­si­ble world. Every value of and in­cludes a large num­ber of pos­si­ble wor­lds: some of those wor­lds will have copies of us on a larger frac­tion of planets, while some will have copies of us on a smaller frac­tion of planets. All we need is that for ev­ery value of and , the weighted av­er­age prob­a­bil­ity of our ex­is­tence across all pos­si­ble wor­lds is pro­por­tional to . The prob­a­bil­ity that we weigh the differ­ent wor­lds by is the prior prob­a­bil­ity dis­tri­bu­tion, i.e., the prob­a­bil­ity dis­tri­bu­tion that we have be­fore up­dat­ing on the fact that we ex­ist.

Be­fore tak­ing into ac­count that we ex­ist, it doesn’t seem like our par­tic­u­lar path to civil­i­sa­tion is in any way spe­cial. If is k times as high, the av­er­age path to civil­i­sa­tion must be k times as likely. If we have no rea­son to be­lieve that our path to civil­i­sa­tion is differ­ent from the av­er­age, then our par­tic­u­lar path to life should also be k times as likely. For ev­ery value of and , is there­fore pro­por­tional to the ex­pected num­ber of copies of us that ex­ist (as long as we sum over all wor­lds with those par­tic­u­lar val­ues of and ). Thus, we can di­rectly up­date and when up­dat­ing on our ex­is­tence.

Note that this isn’t the only con­clu­sion we can draw from our ex­is­tence. The fact that life emerged through RNA is good ev­i­dence that life is likely to emerge through RNA on other planets as well, and the fact that we’ve had a lot of wars is some ev­i­dence that other in­tel­li­gent species are also likely to en­gage in war, etc.[22] As a con­trast, our ex­is­tence is weak or no ev­i­dence for life be­ing able to ap­pear through, for ex­am­ple, a sili­con-base in­stead of a car­bon-base. When we up­date to­wards life be­ing more com­mon, most of the prob­a­bil­ity mass comes from wor­lds where life is likely to emerge in broadly similar ways to how it emerged here on Earth. This is a rea­son to ex­pect that we’ll en­counter civil­i­sa­tions that arose in a similar fash­ion to how we arose, but it doesn’t change how the ba­sic up­dat­ing of and works.

In gen­eral, the mechanism of the an­thropic up­date isn’t that weird, so it shouldn’t give par­tic­u­larly counter-in­tu­itive con­clu­sions in mun­dane cases.[23] With the self-in­di­ca­tion as­sump­tion, up­dat­ing on our ex­is­tence on Earth is ex­actly equal to notic­ing that life ap­peared on a spe­cific planet, and up­dat­ing on that. In that case, we’d con­clude that it must be rel­a­tively com­mon for life to ap­pear in roughly the way that it did, so that is ex­actly what the an­thropic up­date should make us be­lieve. As always, An­thropic De­ci­sion The­ory gets the same an­swer as the self-in­di­ca­tion as­sump­tion: the most likely world that con­tains the most copies of us is go­ing to be the wor­lds in which it’s rel­a­tively com­mon for life to ap­pear in roughly the way that it did here.

Ap­pendix B: De­tails about the simulation

The simu­la­tion simu­lates the space in­side a sphere with ra­dius light years, which con­tains roughly 400 billion galax­ies. Keep­ing track of that many galax­ies is com­pu­ta­tion­ally in­tractable. Thus, I split the space into a much smaller num­ber of points: about 100 000 points ran­domly dis­tributed in space.

I then run the simu­la­tion from the Big Bang to about 60 billion years af­ter the Big Bang. For ev­ery time in­ter­val , cen­tered around , each point has a prob­a­bil­ity of gen­er­at­ing an in­ter­galac­tic civil­i­sa­tion equal to

where is the frac­tion of planets that yields in­ter­galac­tic civil­i­sa­tions, is the rate of civil­i­sa­tions ap­pear­ing de­scribed in Up­dat­ing on the Fermi ob­ser­va­tion, and is the vol­ume that each point rep­re­sents, equal to the to­tal vol­ume di­vided by the num­ber of points.

Any point which yields an in­ter­galac­tic civil­i­sa­tion at time is deemed to be colon­ised at time . For ev­ery other point close enough to to be reach­able, I then calcu­late the time at which a probe leav­ing at reaches . If is ear­lier than the ear­liest known time at which be­comes colon­ised, is stored as the time at which be­comes colon­ised.

This pro­cess is then re­peated for , for ev­ery point that isn’t already colon­ised at or ear­lier. The simu­la­tion con­tinues like this un­til the end, when ev­ery civil­i­sa­tion ei­ther has a first time at which it was colon­ised, or re­mains empty. Th­ese times can then be com­pared with the times at which Earth-origi­nat­ing probes would reach the var­i­ous points, to calcu­late the vari­ables that we care about.

For most vari­a­tions men­tioned in Ap­pendix C, I’ve run this simu­la­tion 100 times for each value of f, and taken the av­er­age of the in­ter­est­ing vari­ables. I’ve used val­ues of from to , with 0.4 be­tween the log­a­r­ithms of ad­ja­cent val­ues of . To get the stated re­sults, I take the sum of the re­sults at each , weighted by the nor­mal­ised an­thropic ad­justed pos­te­rior. For the base case, I’ve run 500 simu­la­tions for each , and used a dis­tance of 0.1 be­tween the log­a­r­ithms of the val­ues of .

Ap­pendix C: Variations

This ap­pendix dis­cusses the effects of vary­ing a num­ber of differ­ent pa­ram­e­ters, to as­sess ro­bust­ness. Some parts might be difficult to un­der­stand if you haven’t read all sec­tions up to and in­clud­ing Si­mu­lat­ing civil­i­sa­tions’ ex­pan­sion.

Speed of travel

Un­der the as­sump­tion of con­tin­u­ous reac­cel­er­a­tion, the ini­tial ve­loc­ity doesn’t change the fi­nal re­sults. This is be­cause the ini­tial ve­loc­ity both af­fects the frac­tion of planets that yields in­ter­galac­tic civil­i­sa­tions, and af­fects the num­ber of planets that are reach­able by Earth. A 10 times greater ve­loc­ity means that the vol­ume from which planets could have reached us is time greater. If the prior over is similar be­tween to , this means that the es­ti­mate of will be times lower. Dur­ing the simu­la­tion, the vol­ume reach­able by Earth will be times greater, and at any point, the num­ber of planets that could have reached any given point will be times greater. This ex­actly can­cels out the differ­ence in , yield­ing ex­actly the same es­ti­mate of the frac­tion of the Uni­verse that probes from Earth can get.

How­ever, if we as­sume no reac­cel­er­a­tion, the speed does mat­ter.

Suffi­ciently high ini­tial ve­loc­i­ties give the same an­swer as con­tin­u­ous reac­cel­er­a­tion. The mo­men­tum is so high that the probes never ex­pe­rience any no­tice­able de­cel­er­a­tion, so the lack of reac­cel­er­a­tion doesn’t mat­ter. This situ­a­tion is quite plau­si­ble. If there’s no prac­ti­cal limit to how high ini­tial ve­loc­i­ties you can use, the op­ti­mal strat­egy is sim­ply to send away probes at a speed so fast that they won’t need to stop be­fore reach­ing their des­ti­na­tion.

Lower ini­tial ve­loc­i­ties means sub­stan­tially smaller mo­men­tum, which means that probes will slow down af­ter a while. This af­fects the num­ber of probes that will reach our part of the uni­verse in the next few billion years sub­stan­tially more than it af­fects the num­ber of probes that should have been able to reach us be­fore now, since probes slow­ing down be­comes more no­tice­able dur­ing longer times­pans. Thus, there will be some­what fewer other civil­i­sa­tions than in other sce­nar­ios. With an ini­tial ve­loc­ity of 80 % of the speed of light, only 51 % of the space that Earth-origi­nat­ing in­tel­li­gence get to is space that other civil­i­sa­tions would even­tu­ally reach. This sce­nario is less plau­si­ble, since pe­ri­od­i­cally reac­cel­er­at­ing would provide large gains with lower speeds, and I don’t know any spe­cific rea­sons why it should be im­pos­si­ble.

Visi­bil­ity of civilisations

So far, I have as­sumed that we wouldn’t no­tice other civil­i­sa­tions un­til their probes reached us. How­ever, there is a pos­si­bil­ity that any in­ter­galac­tic civil­i­sa­tions would try to con­tact us by send­ing out light, or that they would oth­er­wise do things that we would be able to see from afar. In this case, the fact that we can’t see any civil­i­sa­tion has stronger im­pli­ca­tions, since there are more planets from which light have reached us than there are planets from which probes could have reached us. This would make the Bayesian up­date stronger and yield a lower es­ti­mate of : we would con­trol more of the reach­able uni­verse and the Uni­verse would be less likely to be colon­ised with­out us. This effect isn’t no­tice­able on the re­sults if civil­i­sa­tions travel at 80 % of light speed (or faster) while con­tin­u­ously reac­cel­er­at­ing. How­ever, in this case the speed of travel mat­ters some­what. If civil­i­sa­tions could only travel at 50 % of light speed, 56 % of the space that Earth-origi­nat­ing civil­i­sa­tion get is space where other civil­i­sa­tions would even­tu­ally ar­rive.

Priors

Since the Bayesian and the an­thropic up­date is so strong, any prior that as­signs non-neg­ligible prob­a­bil­ity to life ap­pear­ing on to of planets will yield broadly similar re­sults. Given to­day’s great un­cer­tainty about how life and civil­i­sa­tions emerge, I think all rea­son­able pri­ors should do this.

One thing that can af­fect the con­clu­sions some­what is the rel­a­tive prob­a­bil­ities in­side that in­ter­val. For ex­am­ple, a log-nor­mal dis­tri­bu­tion with stan­dard de­vi­a­tion 50 and with cen­ter around as­signs greater prob­a­bil­ities to life be­ing less com­mon, in the rele­vant in­ter­val, while a similar dis­tri­bu­tion cen­tered around 100 as­signs greater prob­a­bil­ities to life be­ing more com­mon. This differ­ence is very small: the me­dian f in the former case is , the me­dian in the lat­ter case is . There­fore, it doesn’t sig­nifi­cantly af­fect the con­clu­sion.

Time to de­velop civilisation

The time needed for a civil­i­sa­tion to ap­pear af­ter a planet has been formed mat­ters some­what for the re­sults. Most the­o­ries of an­throp­ics agree that we have one dat­a­point tel­ling us that 4.55 billion years is likely to be a typ­i­cal time, but we can’t de­duce a dis­tri­bu­tion from one dat­a­point. To take two ex­tremes, I have con­sid­ered a uniform dis­tri­bu­tion be­tween 2 billion years and 4.55 billion years, as well as a uniform dis­tri­bu­tion be­tween 4.55 billion years and 8 billion years. In the first case, where civil­i­sa­tions ap­pears early, only 60 % of the space that we can claim would have oth­er­wise been claimed by other civil­i­sa­tions. In the lat­ter case, the num­ber is in­stead 73 %.

Planet for­ma­tion rate

Mul­ti­ply­ing the planet for­ma­tion rate across all ages of the Uni­verse with the same amount does not change the re­sults. The Fermi ob­ser­va­tion im­plies that in­ter­galac­tic civil­i­sa­tions can’t be too likely to arise within the vol­ume close to us, so more planets per vol­ume just means that a lower frac­tion of planets yield in­ter­galac­tic civil­i­sa­tions, and vice versa. As long as our prior doesn’t sig­nifi­cantly dis­t­in­guish be­tween differ­ent val­ues of , the num­ber of civil­i­sa­tions per vol­ume of space will be held con­stant.

How­ever, vary­ing the planet for­ma­tion rate at par­tic­u­lar times in the his­tory of the Uni­verse can make a large differ­ence. For ex­am­ple, if the num­ber of hab­it­able planets were likely to in­crease in the fu­ture, rather than de­crease, the Uni­verse would be very likely to be colon­ised in our ab­sence. Similarly, if the Uni­verse be­came hos­pitable to life much ear­lier than be­lieved, we would have much stronger ev­i­dence that in­ter­galac­tic civil­i­sa­tions are un­likely to arise on a given planet, and life might be much less com­mon in the Uni­verse. Th­ese effects can also act to­gether with un­cer­tainty about the time it takes for civil­i­sa­tions to emerge from hab­it­able planets, to fur­ther vary the timing of life.

One case worth con­sid­er­ing is if planet for­ma­tion rate fol­lows star for­ma­tion rate, i.e., there is no de­lay from metal­lic­ity.[24] In or­der to con­sider an ex­treme, we can com­bine this with the case where civil­i­sa­tions take some­where be­tween 2 and 4.55 billion years to form. In this case, we have good ev­i­dence that in­ter­galac­tic civil­i­sa­tions are un­likely to ap­pear, and only 49 % of the space that Earth-origi­nat­ing in­tel­li­gence claims is likely to be colon­ised in its ab­sence.

The op­po­site sce­nario is where the planet for­ma­tion rate peaks 2 billion years later than Lineweaver’s (2001) model pre­dicts, i.e., the need for metal­lic­ity in­duces a de­lay of 4 billion years in­stead of 2 billion years. To con­sider the ex­treme, I have com­bined this with the case where civil­i­sa­tions take be­tween 4.55 and 8 billion years to form. In this case, we are at the peak of civil­i­sa­tion for­ma­tion right now, and ex­trater­res­trial life is likely to be quite com­mon. In ex­pec­ta­tion, 81 % of the space that Earth-origi­nat­ing in­tel­li­gence gets to first would have been colon­ised in its ab­sence.

An­thropic up­dates on variations

Just as an­thropic con­sid­er­a­tions can af­fect our be­liefs about the like­li­hood of civil­i­sa­tions ap­pear­ing, they can af­fect our be­liefs about which of these vari­a­tions should re­ceive more weight. Similarly, the vari­a­tions im­plies differ­ences in how much space Earth-origi­nat­ing in­tel­li­gence is likely to af­fect, and thus how much im­pact we can have. Tak­ing both of these into ac­count, the vari­a­tions that get more weight are, in gen­eral, those that lead to civil­i­sa­tions ap­pear­ing at the same time as us (emerg­ing 13.8 billion years af­ter the Big Bang, on a planet that has ex­isted for 4.55 billion years) get­ting a larger frac­tion of the Uni­verse.[25] Most im­por­tantly, this im­plies that the­o­ries where planet for­ma­tion hap­pens later are fa­vored; in the cases I’ve con­sid­ered above, the hy­poth­e­sis that planet for­ma­tion hap­pens later gets about 7.5 times as much weight as the base hy­poth­e­sis, which gets about 6 times as much weight as the case where planet for­ma­tion hap­pens early. Whether this is a dom­i­nant con­sid­er­a­tion de­pends on how strong the sci­en­tific ev­i­dence is, but in gen­eral, it’s a rea­son to lend more cre­dence to the case where ex­trater­res­trial life is more com­mon.

Ap­pendix D: In­ter­ac­tions with the simu­la­tion hypothesis

The simu­la­tion hy­poth­e­sis is the hy­poth­e­sis that our en­tire civil­i­sa­tion ex­ists in­side a com­puter simu­la­tion de­signed by some other civil­i­sa­tion in the real world.[26]

For simu­lated civil­i­sa­tions, a nat­u­ral ex­pla­na­tion of the Fermi para­dox is that the ones simu­lat­ing us would pre­fer to watch what we do with­out oth­ers in­terfer­ing. I ex­pect most simu­lated civil­i­sa­tions to ex­ist in simu­la­tions where space is faked, since this seems much cheaper (al­though just a few simu­la­tions that prop­erly simu­lates a large amount of space might con­tain a huge num­ber of civil­i­sa­tions, so this point isn’t ob­vi­ous).

If we think that there’s at least a small prob­a­bil­ity that civil­i­sa­tions will cre­ate a very large num­ber of simu­lated civil­i­sa­tions (e.g. a 1 % chance that 1 % of civil­i­sa­tions even­tu­ally cre­ate a billion simu­la­tions each), a ma­jor­ity of all civil­i­sa­tions will ex­ist in simu­la­tions, in ex­pec­ta­tion. Since the ma­jor­ity of our ex­pected copies ex­ists in simu­la­tions, our an­thropic the­o­ries im­plies that we should act as if we’re al­most cer­tain that we are in a simu­la­tion. How­ever, tak­ing into ac­count that the copies liv­ing in the real world can have a much larger im­pact (since an­ces­tor simu­la­tions can be shut down at any time, and are likely to con­tain less re­sources than base­ment re­al­ity), a ma­jor­ity of our im­pact might still come from the effects that our real-world copies have on the fu­ture (as­sum­ing to­tal con­se­quen­tial­ism). This ar­gu­ment is ex­plored by Brian To­masik here.

That line of rea­son­ing is mostly un­af­fected by the anal­y­sis in this post, but there might be some in­ter­ac­tions, de­pend­ing on the de­tails. If we think that the ma­jor­ity of our simu­lated copies are simu­lated by civil­i­sa­tions very similar to us, which also emerged around 13.8 billion years af­ter the Big Bang, the fact that we’ll only get 5 % of the reach­able uni­verse is com­pen­sated by the fact that our simu­la­tors also only have 5 % as much re­sources to run simu­la­tions with, in ex­pec­ta­tion. How­ever, if we think that most species will simu­late an­ces­tor civil­i­sa­tions in pro­por­tion to how com­mon they are nat­u­rally, most of our simu­la­tions are run by civil­i­sa­tions who ar­rived early and got a larger frac­tion of the Uni­verse. Thus, the fact that we only get 5 % of the reach­able uni­verse isn’t nec­es­sar­ily com­pen­sated by a cor­re­spond­ing lack of re­sources to simu­late us. Since we’re rel­a­tively late, I’d ex­pect us to have rel­a­tively lit­tle power com­pared with how fre­quently we ap­pear, so this would strengthen the case for fo­cus­ing on the short term. I haven’t quan­tified this effect.

Footnotes

[1] No an­thropic the­o­ries are named in the pa­per, but one way to get this re­sult would be to use the self-sam­pling as­sump­tion on a large, finite vol­ume of space.

[2] To­tal con­se­quen­tial­ism de­notes any eth­i­cal the­ory which as­serts that moral right­ness de­pends only on the to­tal net good in the con­se­quences (as op­posed to the av­er­age net good per per­son) (https://​​plato.stan­ford.edu/​​en­tries/​​con­se­quen­tial­ism/​​). It also ex­cludes bounded util­ity func­tions, and other the­o­ries which as­signs differ­ent value to iden­ti­cal civil­i­sa­tions be­cause of ex­ter­nal fac­tors.

[3] The frac­tion of met­als are nor­mal­ised to to­day’s lev­els. For ev­ery star in the same weight-class as the Sun (calcu­lated as 5 % of the to­tal num­ber of stars), the prob­a­bil­ity of form­ing an Earth-like planet is cho­sen to be 0 if the frac­tion of met­als is less than of that of the Sun, and 1 if the frac­tion is more than 4 times that of the Sun. For re­ally high amount of metal, there is also some prob­a­bil­ity that the Earth-like planet is de­stroyed by the for­ma­tion of a so-called hot Jupiter.

[4] Speci­fi­cally, I use the differ­en­tial equa­tions from sec­tion 4.4.1. I calcu­late the dis­tance (mea­sured in co­mov­ing co­or­di­nates) trav­el­led by a probe launched at as , where .

[5] Some au­thors, such as Phil Tor­res (https://​​www.sci­encedi­rect.com/​​sci­ence/​​ar­ti­cle/​​pii/​​S0016328717304056) have ar­gued against the com­mon as­sump­tion that space colon­i­sa­tion would de­crease ex­is­ten­tial risk. How­ever, it seems par­tic­u­larly un­likely that species that has started in­ter­galac­tic colon­i­sa­tion would go ex­tinct all at once, since the dis­tances in­volved are so great.

[6] is the quan­tity calcu­lated as in Sand­berg et al. (2018).

[7] If you’re a to­tal con­se­quen­tial­ist who only cares about presently ex­ist­ing peo­ple and an­i­mals, how­ever, this isn’t true. In that case, us never leav­ing the So­lar Sys­tem doesn’t make much of a differ­ence, but the ad­di­tional num­ber of repli­ca­tions would still in­crease your im­pact (al­though this might de­pend on the de­tails of your an­thropic and eth­i­cal be­liefs). As a re­sult, I think that the ma­jor­ity of your ex­pected im­pact comes from sce­nar­ios where we’re very un­likely to leave the So­lar Sys­tem, ei­ther be­cause of large ex­tinc­tion risks, or be­cause space colon­i­sa­tion is ex­tremely difficult. I have no idea what the prac­ti­cal con­se­quences of this would be.

[8] There does, how­ever, ex­ist some vari­a­tion in im­pact, and those vari­a­tions can mat­ter when we’re not con­fi­dent of the right an­swer. For ex­am­ple, the case for the pos­si­bil­ity of in­ter­galac­tic travel isn’t air­tight, and there may be some mechanism that makes civil­i­sa­tions like ours get a larger frac­tion of the Uni­verse if in­ter­galac­tic travel is im­pos­si­ble: one ex­am­ple would be that we seem to have emerged fairly late on a cos­molog­i­cal scale, and late civil­i­sa­tions gets a larger dis­ad­van­tage if in­ter­galac­tic travel is pos­si­ble, since the ex­pan­sion of the Uni­verse means that we can’t reach as far as ear­lier civil­i­sa­tions. On the other hand, if in­ter­galac­tic travel is im­pos­si­ble, it’s likely to be be­cause tech­nolog­i­cal progress stopped ear­lier than ex­pected, which prob­a­bly means that the fu­ture con­tains less value and dis­value than oth­er­wise. I haven’t tried quan­tify­ing any of these effects.

[9] This choice is quite ar­bi­trary, but the re­sults aren’t par­tic­u­larly de­pen­dent on the de­tails of the dis­tri­bu­tion. See Ap­pendix C for more dis­cus­sion.

[10] Since is very tiny and is very large, this is ap­prox­i­mately equal to . When run­ning the calcu­la­tions in mat­lab, re­sults in a smaller er­ror, so that’s the ver­sion I’ve used.

[11] As far as I know, there is no the­ory of an­throp­ics that works well in in­finite cases (ex­cept pos­si­bly UDASSA, which I’m not sure how to ap­ply in prac­tice). Since the recom­men­da­tions of ADT and SIA are the same un­der any large finite size, the sim­plest thing to do seems to be to ex­trap­o­late the same con­clu­sions to in­finite cases, hop­ing that fu­ture philoso­phers will figure out whether there’s any more rigor­ous jus­tifi­ca­tion for that. This is some­what similar to how many eth­i­cal the­o­ries don’t work in an in­finite uni­verse (Bostrom, 2011); the safest thing to do for now seems to be to act ac­cord­ing to their finite recom­men­da­tions.

[12] Known as the ‘halfer po­si­tion’ when speak­ing about the Sleep­ing Beauty Prob­lem.

[13] There is no clear an­swer to ex­actly how we should define our refer­ence class, which is one of the prob­lems with SSA.

[14] Known as the ‘thirder po­si­tion’ when speak­ing about the Sleep­ing Beauty Prob­lem.

[15] An­thropic De­ci­sion The­ory is sim­ply non-causal de­ci­sion the­o­ries such as Ev­i­den­tial De­ci­sion The­ory, Func­tional De­ci­sion The­ory, and Up­date­less De­ci­sion The­ory ap­plied to an­thropic prob­lems. The an­thropic rea­son­ing fol­lows di­rectly from the premises of these the­o­ries, so if you en­dorse any of them, this is prob­a­bly the an­thropic the­ory you want to be us­ing (un­less you pre­fer UDASSA, which han­dles in­fini­ties bet­ter).

[16] Note that ADT (in its most ex­treme form) never up­dates on any­thing epistem­i­cally, it it’s in a large enough uni­verse. If you perform an ex­per­i­ment, there will be always be some copy in the Uni­verse that hal­lu­ci­nated each pos­si­ble out­come, so you can’t con­clude any­thing from ob­serv­ing an out­come. How­ever, there are more copies of you if your ob­ser­va­tion was the most com­mon ob­ser­va­tion, so you should still act ex­actly like some­one who up­dated on the ex­per­i­ment.

[17] Note that this is only true for an­thropic dilem­mas. Since ADT is a non-causal de­ci­sion the­ory, it may recom­mend en­tirely differ­ent ac­tions in e.g. New­comb’s prob­lem.

[18] If you use ADT or SIA and are an av­er­age util­i­tar­ian, you will act similarly to a to­tal con­se­quen­tial­ist us­ing SSA, in this case. If you use SSA and are an av­er­age util­i­tar­ian, you should prob­a­bly act as if life was ex­tremely un­com­mon, since you have a much greater con­trol over the av­er­age in that case. I have no idea what a bounded util­ity func­tion would im­ply for this.

[19] This model as­sumes a lin­ear re­la­tion­ship be­tween civil­i­sa­tions’ re­sources and value; thus, it won’t work if you e.g. care more about the differ­ence be­tween hu­man­ity hav­ing ac­cess to 0 and 1000 galax­ies than the differ­ence be­tween 1000 and 3000 galax­ies. It’s a some­what stronger as­sump­tion than to­tal con­se­quen­tial­ism, since to­tal con­se­quen­tial­ism only re­quires you to have a lin­ear re­la­tion­ship be­tween num­ber of civil­i­sa­tions and value.

[20] If es­sen­tially all space went to civil­i­sa­tions aris­ing in a 2 billion year pe­riod around now, I’d guess that the an­thropic up­date in fa­vor of that the­ory would be one or two or­ders of mag­ni­tude.

[21] While this last ques­tion might seem like the most im­por­tant one, there’s a high prob­a­bil­ity that any fu­ture rulers of Earth who care about the an­swer to such re­search could do it bet­ter them­selves.

[22] At the ex­treme, our ex­is­tence is ev­i­dence for some very par­tic­u­lar de­tails about our planet. For ex­am­ple, the fact that we have a coun­try named Egypt is some ev­i­dence that other civil­i­sa­tions are likely to have coun­tries named Egypt. How­ever, the prior im­prob­a­bil­ity of civil­i­sa­tions hav­ing a coun­try named Egypt out­weighs this, so we don’t ex­pect most other civil­i­sa­tions to be that similar to us.

[23] This has pre­vi­ously been as­serted by Stu­art Arm­strong. Note that I don’t use his method of as­sum­ing a medium uni­verse in this post, but my method should always yield the same re­sult.

[24] From what I can gather, this view isn’t un­com­mon. The metal­lic­ity-de­lays that Behroozi and Peeples (2015) con­sider isn’t sig­nifi­cantly differ­ent from no de­lay at all.

[25] To see why, con­sider that twice as many civil­i­sa­tions ap­pear­ing at the same time as us im­plies twice the an­thropic up­date; and that each civil­i­sa­tions like us get­ting twice as much space im­plies that we will af­fect twice as much space. Mul­ti­ply­ing these with each other, the amount of space that we will af­fect is pro­por­tional to the amount of space that civil­i­sa­tions ap­pear­ing at the same time as us will af­fect.

[26] This should not be con­fused with the simu­la­tion that I use to get my re­sults, which (hope­fully) con­tains no sen­tient be­ings.