When you get a cold, the virus hijacks your cells and uses them to make copies of itself, destroying the cells in the process. The virus does not care about you. And yet, if you get a cold, your body doesn’t dissolve into a pile of virus-enriched goo as a result of all your cells disregarding survival to focus on virus manufacturing. Why? Well, you have an immune system. But also, if that did happen, other people would say, “Oh no, Alice has dissolved into a pile of virus-enriched goo; we should stay away lest we also become goo piles.” Better to just hijack some of your cells, so you can walk around and spread the virus to other people. Thus, the theory of optimal virulence says that a more laid-back cold virus will be more successful in the long run, which is why the cold viruses that thrive today are the non-goo-pile-converting kind. Anyway. Most people reading this have paid Apple $400-$1600 for a phone on which you can’t do anything without Apple’s permission and monitoring. That was a strange decision, but on Android phones you can run whatever you want. For example, I noticed a couple of months ago that all RSS apps were terrible, so I spent 15 long minutes having AI make me a non-terrible one. I copied the file to my phone, installed it, and entered RSS nirvana, no registration, no app store, no nothing. It seems ludicrous that I wouldn’t be able to do that. And yet, Google plans to change things. Soon, you won’t be able to run any app on an Android phone unless the person who made it bends the knee by paying a fee, giving Google their government ID, listing all apps they’ve ever made, etc. There will be an “escape hatch” for users to install other apps, but it involves secret menus and scary warnings and a 24-hour waiting period. Google says they’re doing this for security reasons, to prevent people from installing malware that does things like steal financial information. Three things that any credible analysis of this situation must acknowledge The security issue is real, and this change will help. Lots of people are in fact getting tricked into installing malware on their phones, and common sense says that this will accelerate with AI. Common sense also says that during that 24-hour waiting period, some of those people might wonder if it’s really normal for OfficialMicrosoftSupportProfessional021 to contact them on WhatsApp to send totally-safe-virus-scanner.apk. It would be convenient for lockdown opponents if the security issue was fake, but it isn’t. For 95% of people reading this, these restrictions would not be a big deal. You click a few buttons, wait 24 hours once, and then you have the same freedom you have now. That’s still far better than Apple phones where there is no process at all. Oddly, it’s also better than Apple computers, where installing programs from non-verified developers has all sorts of friction that can never be removed. Most people I know use Apple computers anyway and just tolerate the indignity. If Google cared about freedom or privacy, they could have achieved their security goals without as much harm to freedom or privacy. For example, instead instead of making themselves the sole arbiter, they could have allowed for multiple certifying entities, and could have allowed users to add others (perhaps after a 24-hour waiting period). Arguably, that would be more secure: That way, grandma could install my RSS app without leaving herself open to random malware. Or, Google could have used private on-device malware scanning, or lots of other things. Google didn’t do that because they don’t care about freedom and privacy and because they like money.1 I think that’s a fair summary. This change addresses a real issue, but it does so in a self-serving way that needlessly destroys freedom. Even so, it only destroys a bit of freedom. This change, by itself, isn’t really a huge deal. Reading between the lines, I think the real concern is what this might portend. Having made this change, maybe Google will gradually add further restrictions, slowly boiling the frog and moving ever-closer to an Apple-style dictatorship. But will they? Maybe! Still, this whole situation seems under-theorized. There are good reasons to think that a full lockdown would not be in Google’s interest. And if you care about digital freedom, you could argue that a full lockdown would be a good thing. Some other analogies Switching insurance sucks, so some insurers offer a low initial rate and then gradually raise it over time. But they don’t want to raise it too much. Maybe they can charge you $50 more per year than a competitor, because it’s not worth your time to switch to save $50. But if they tried to charge you $5000 more per year, you’d be gone. Or try this: Say it’s 1750 AD and you’re the King of France. You care nothing about your subjects and only want to extract the maximum possible loot for yourself, no offense Louis XV. How high should you set taxes? High, because you want loot. But not too high, because you don’t want your subjects to revolt and kill you. But still pretty high, because revolting is hard, and your subject know that even if they succeed, they might just end up with another ruler that’s just as rapacious as you. Or: Say you put a frog in a pot of water and slowly heat it up. What do you think actually happens? Back to phones Why are there no Linux phones? OK, technically there are Linux phones. There are various Linux phone operating systems (PureOS, postmarketOS, Mobian, Manjaro Mobile, Fedora Mobility, openSUSE Mobile) as well as sorta-Linux-sorta-Android operating systems (Ubtuntu touch, Droidian, FuriOS, Sailfish OS) and manufacturers who support them (Purism, Pine, Fairphone, Shift, FuriLabs, {J,V}olla). While I love and support these (♡), the general consensus is that they don’t currently offer an experience that’s remotely comparable to stock Android or iOS, especially for the price. (Even more technically, Android is itself a kind of Linux, but you know what I mean.) Obviously, the laws of physics do not forbid Linux phones from being just as good and cheap as an Android phone. They currently aren’t because: Making cheap phones requires huge economies of scale. Making a good operating system requires a huge amount of work. For a phone to be useful it needs good apps, which requires a huge amount of work. Linux phones don’t have those economies of scale because few people buy them. So, there isn’t enough money to pay people to perfect the operating systems, and most companies don’t bother making apps, and so few people buy those phones. Meanwhile, there are also various Android-except-de-Googled phone operating systems (LineageOS, GrapheneOS, /e/OS, CalyxOS), and the consensus is that these do come reasonably close to the experience of stock Android or iOS. Personally, I used Android for a long time while trying to disable as much of Google’s apps and telemetry as I could. A while ago, I got fed up with this and switched to Graphene, one of the most popular Android-except-de-Googled variants. I don’t enjoy tech support bullshit, so I expected some pain, but I was willing to tolerate it for freedom/privacy/whatever. In practice, I found that everything was easy and worked flawlessly and there was no pain at all. Using Graphene made me realize that I had forgotten what it feels like to be “respected” by technology. So, on a personal level, I recommend that you use Graphene. But this whole situation is full of weird tensions. For one thing, the Graphene project depends on Google for the base of their operating system, and only really survives because of Google’s tolerance. And, at the moment, the only way to use Graphene is to… buy a phone made by Google. And while I recommend you use Graphene, what I’d prefer you do is switch to a true Linux phone. That would mean that you pay a lot of money for a poor phone experience. But it would help grow the Linux phone ecosystem. If enough people did that, then Linux phones might become become good enough that selfish people like me would switch, further growing the Linux phone ecosystem, et cetera. That seems unlikely, because digital freedom weirdos face a collective action problem: Linux phones are bad and people are selfish, so we keep buying Android phones and Linux phones stay bad. There are two ways to solve that collective action problem. Either some entity could invest a ton of money to make Linux phones good, or some entity could do something to make Android worse. The status quo is good for Google You may already see the argument I’m making here. Currently Android is just open enough that weirdos like me can run a variant that gives us most of what we want. But doing that is annoying enough that 99.9% of people don’t. This gives Google almost all the benefits of full control, while also sucking up all the oxygen and preventing a real alternative from emerging. (Apple, incidentally, is laughing. They get to enjoy a full dictatorship while Google does all the hard work.) Suppose that in six months, Google decides to escalate the lockdown and completely removes the ability to run independent apps. What then? In the short-run, not much. Tech freedom weirdos would howl, normies wouldn’t notice, and the world would keep turning. But let’s look at some numbers. A rough estimate is that 3-5 million people currently use private de-Googled Android variants like Lineage or Graphene. That’s a crisp 0.1% of the global Android market. Meanwhile, it’s thought that 300-500 million people install Android apps outside the Google store. (That’s excluding China and things like the Samsung and Epic Games Stores.) The biggest reasons are (i) piracy, (ii) modified YouTube/Spotify/WhatsApp clients, (iii) region-locked games, (iv) forbidden gambling or crypto apps, (v) local sharing to save on mobile data, and (vi; least common) open-source app repositories like F-Droid. In an escalated lockdown, some of this might still be allowed, e.g. local sharing. And many people would simply accept the new restrictions. But I expect that a substantial number would suddenly become interested in de-Googled Android variants that don’t enforce the lockdown. That would be dangerous for Google: If a significant fraction of those 300-500 million people switched, there’s a risk of a vicious/virtuous cycle, where people tell their friends, manufacturers start to cater to those variants, and so on. If that happened, what would Google do? Might they clamp down on those Android variants, too? Again, maybe! Technically, Android is open-source, but if Google really wanted to kill Lineage or Graphene, there are about 200 different ways they could do that. So say they did. What would the people currently using those variants do? Again, many would simply submit. That’s good for Google. Others would decide that if they must have an overlord, they’d prefer Apple. That’s not good for Google, but probably not a huge problem given the numbers. But others would decide to look elsewhere, likely to true Linux phones. Currently, it’s estimated that less than 100 thousand people daily-drive true Linux phones, 30-50 times less than the number of people who use de-Googled Android variants. I’m too lazy and selfish to use a Linux phone now, but Google will take my RSS app from my cold dead hands, so in this scenario, I’d likely switch. If a fraction of people do that—or even get a second Linux phone—it would still represent a huge infusion of money and energy into the Linux phone ecosystem. If this pushed Linux phones from “bad” to “sort of OK”, that could kick off a growth cycle. Eventually, there would be a risk that Europe—eager to decouple from an unreliable US—might embrace them by mandating that government employees use them, or passing a law that popular apps must provide a web-app version that works on Linux phones, or whatever. Would that actually happen? I don’t know, but neither does Google. My point is that the downside risk of losing their phone duopoly seems vastly larger than the upside, so why risk it? The smart move is to crush alternative app stores and fight malicious apps while making the bare minimum concessions to keep the freedom and privacy people onside. That’s exactly what what Google appears to be doing. Of course, they might go for a full lockdown anyway. Corporations do dumb things all the time. But I really don’t see how it would be in their self-interest, and I’m not sure it would be a bad thing if they did. Dynomight: could google achieve the security aims of the android lockdown without so much harm to freedom or privacy Google Gemini 4: Yes. Security experts, privacy advocates, and open-source developers widely agree that Google’s primary security objectives—protecting users from malware, financial fraud, spyware, and social engineering—can be achieved through architectural containment, transparent auditing, and informed consent rather than centralized gatekeeping and surveillance. […] The reason Google chooses centralized lockdown over user-controlled security is largely economic and strategic, not technical: Ecosystem Lock-in: Tying security APIs (like Play Integrity) to Google Play Services makes it nearly impossible for competitors or open-source forks to build a viable alternative Android ecosystem that supports banking and media apps. Advertising Revenue: Features that dramatically improve both security and privacy—like a native network firewall, revocable internet permissions, or frictionless support for system-wide ad blockers—directly threaten Google’s primary revenue stream. Liability & Support Reduction: Treating all users as untrusted and removing root/power-user capabilities reduces the surface area of user error, even if it removes the user’s ownership of their own hardware. ↩

This is a weird moment. I have a hunch that we’ll look back later and tell ourselves stories in which it seemed obvious where things were going. So I wanted to lay down a little time capsule and register just how confusing everything felt at the time. This is a personal view from the trenches. 1. If you’re a normal person, you may feel that AI has been good at math for a long time.1 But for “real” math, this is roughly what it felt like: When Feeling 2024 LLMs are toys Summer 2025 LLMs are interesting toys Early Autumn 2025 LLMs can do useful stuff sometimes Mid Autumn 2025 …a lot of stuff Late Autumn 2025 …a lot Early 2026 holy god Even having been astonished by AI progress in the past, it was still astonishing to see AI move in a matter of months from a mildly-useful toy to something that many people now consider almost impossible to do without. 2. Personally, I’d say that current AI makes me around 50% more productive at “doing math”. That’s despite the fact that AI can now often do in a few minutes things that would have taken me hours to do myself. The main reason is that my concept of “doing math” involves various steps, like coming up with ideas for theorems, trying to prove them, writing things down, reorganizing everything to make it more clear, and so on. AI is good at some parts of this, but still pretty bad at others, so the overall speedup isn’t as high as you might expect. (Even if planes could go 10× faster, your door-to-door travel time from your apartment in London to your friend’s place in Rome wouldn’t decrease that much, because you still have to get to the airport and wait to board.) 3. But then, I’m still (sort of; more on this later) trying to follow my old “doing math” workflow, just automating certain parts of it. That can’t be right, can it? If we found a way to make batteries at 1% of the current cost, we wouldn’t just say, “cool, batteries are cheap now”. We’d find all sorts of new ways to use batteries that didn’t make sense before, and change the ways we generate power to account for all these new batteries. Probably our whole energy system would end up transformed. Analogously, AI being good at proofs should lead to a very different equilibrium. But what is that?2 4. It seems healthy, at the moment, to maintain a fluid view of what “doing math” means. But at bare minimum it surely involves understanding the math that “I” am “doing”. So long as that’s part of the workflow, it would seem to impose a limit on how quickly “I” could “do” math, no matter how good AI is. One obvious conclusion would be that my tiny organic brain should be removed from the workflow entirely. Maybe! But it’s interesting to speculate about how much faster things could go while still keeping a human brain in the loop. I think the answer is: A lot. 5. Contrary to what you may have heard, humans are not obsolete for math. Not yet. The most obvious reason is that AI is still bad at writing. People often complain that AI writing overuses various patterns or lacks a unique voice or whatever. I don’t disagree, but I always felt like there was a deeper issue that I couldn’t quite put my finger on. With math, this becomes very clear: The problem is it doesn’t make any goddamned sense. Really, I think it’s deeper than being bad “at writing”. It seems like AI is bad at thinking in certain ways that manifest as bad writing. 6. When novices do math, they often create “spaghetti-code math”: Imagine a long document with lots of equations, but if you point at any one equation, it’s hard to say what assumptions underpin it, or how it fits into the bigger picture. Any random idea that was mentioned 17 pages ago might be used again at any time with no warning. With experience, people learn to break complex arguments down into modular pieces with a good “API”, in the sense that the assumptions and conclusions for each part are relatively simple. Somehow, AI has developed these amazing technical skills, while still being bad at creating abstractions So the arguments it gives are long and tangled and bewildering. This is what makes AI math seem so alien. 7. Now I know what you’re thinking. Maybe AI isn’t bad at creating abstractions. It’s just bad at creating abstractions that work for you, human with tiny organic brain. From a distance, that seems plausible. And it might be true to some degree. I have high-bandwidth access to a human brain. It’s not crazy to imagine that this gives me an advantage in predicting how a human brain would react to a given presentation of ideas. But still, I claim that current AI is objectively bad at creating abstractions, regardless of if the consumer of those abstractions is human or artificial. Here’s why: If I ask AI to prove something hard, it will produce a proof that starts simple and gradually descends into a rampage of complexity. If I tell it to break that proof down into self-contained pieces, it will produce lemmas that each take a full page just to state. If I ask, “Why? Why have you done this to me?”, the AI will say, “This is as simple as it gets. It’s not that bad. What’s your problem?” But if I argue with the AI for a while, I can start to understand the basic contours of the argument. And I’ll ask, “Why don’t you first just try to prove blah using only assumptions blahblah, without worrying about blahblahblah? Wouldn’t that be easier?” Often, this doesn’t work, either because my idea was bad, or because the AI isn’t smart enough to do it. (Which one? Who knows!) But after a few attempts, I can usually find something that that works, and the AI can use it to re-write the proof in a somewhat simpler way. If I repeat this many times, I can gradually chip away at the complexity. But maybe that’s still just making the proof easier for humans to understand? No! For one, this usually makes the proof shorter, and sometimes dramatically shorter. More importantly, during this process, someone (maybe me, maybe the AI) will often realize that there’s a different proof strategy that gives a stronger result. Instead of proving x < 9.7, now we can prove x < 7.1. Thus, artisanal human abstractions are “objectively” useful. 8. Some people might claim that it’s already possible to automate some of what I’m doing with some multi-agent Rube Goldberg thing. I’m not sure, because my experience has been that investing lots of time into clever AGENTS.MD or custom skills or harnesses isn’t a great use of time. That stuff is somewhat helpful, but things are changing so fast that the knowledge has a short half-life. Also, the majority of information about these tricks seems to come from people who are excited about the technology itself. On the other hand, the automator told me to try typing, “ask independent agents for an adversarial review”, and I’ve found those seven words to be quite helpful. I wish I could find a source with boring best practices for people who are happy to stay away from the bleeding edge. 9. Another reason AI doesn’t improve my productivity more is that it is often the green light, the orgastic future that recedes before me. I’ll ask AI to do something, and it will almost seem to do it. So I’ll ask for changes, and it seems to help, but it’s still not quite right. Hours later, I’ve accomplished nothing and learned nothing. Eventually, I start over, either doing it myself or with AI on a very tight leash, and I’m done in 30 minutes. Each time this happens I feel foolish and tell myself not to do it again. But it’s very hard to judge in real-time. 10. Of course, I’m not claiming that AI is intrinsically bad at writing or thinking or abstraction-making. It’s gotten better at all of these and I expect it will continue to do so. All I’m saying is that as of Autumn 2026, humans still have a lot to contribute, and it’s not just better exposition. 11. Did AI actually get much better at math in Autumn 2025? Or perhaps, in the grand scheme of things, human math ability falls into a narrow range? 12. Some people are responding to the new reality by gleefully using AI to generate lots of semi-OK-ish papers and puking them up on arXiv. I understand the incentives that drive this. In some sense, it seems like the logical response to the new reality. (After inventing cars, we started taking a lot more low-value trips.) But I don’t think this is socially useful, and I tend to think everyone will soon agree that this is embarrassing. But… I’m not sure. Maybe it is socially useful? And I find myself feeling vaguely jealous of the people who are wired this way, who instead of feeling blinkered, see opportunity and plunder. 13. Want an optimistic pro-AI, pro-progress take? OK. If I take my old “doing math” workflow, AI makes it strictly better. A lot better. It’s like having a hyper-conscientious intern that is smarter than me in many ways, works at incredible speed, and knows approximately everything. Yes, AI sucks at writing and making abstractions. But if I’m disciplined and try to mostly do the math “myself” while asking the AI for help, I can explore ideas much faster than before, make far fewer mistakes, and produce something just as clear as before, while developing just as much understanding. I can ask, “What happens if you do a Kofuni expansion?”, and get an answer instantly. I can ask for illustrations or interactive visualizations and see them instantly. Arguably, we’re incredibly lucky to be alive at this moment, when these new tools have opened up this new territory to explore. 14. And isn’t that what I wanted? Back in 2024, I bemoaned that what AI most seemed to struggle with was getting the facts straight: Perhaps the old science fiction robots were a projection of our fantasies. I want my style and voice to be precious and uncopyable. I want an AI assistant that will tirelessly hunt down facts while I provide creativity and panache and magic. But that’s not how it is. So we should remember: In a scenario where AI is useful but humans retain some comparative advantage, we might not like where that comparative advantage lies. At least with math, and at least for the moment, I get to be Kirk, while the AI is Spock. (Even with other types of writing, my impression is that since 2024, AI has improved more along Spock dimensions than along Kirk dimensions.) So that’s nice. 15. In the last few months, it frequently happens that a collaborator will use AI to generate something, and then send it to me and ask me to check if it’s useful. I can see why this seems reasonable. (Really, I do.) But when it happens a lot, one can’t help but wonder what the point of the human collaborator is. 16. On the other hand—with creative writing, my advice is not to use AI to write, at all. Ask questions if you want, but write every damned word yourself.3 With math, I don’t feel like such a hard line makes sense. It seems that creating something good still requires a lot of human involvement, to the point of agonizing over notation, repeatedly re-organizing things, and heavily editing individual sentences. Still, I find that if people make that effort, they can create something just as good as they could by hand, while still being perhaps 50% AI written.4 So I don’t mind if people send me “centaur math”, provided they take responsibility for the contents. The problem is that when working with AI, it takes constant paranoid vigilance to avoid letting things slip beyond your understanding. I often find myself wanting to awkwardly ask people, “OK, you obviously used AI to make this… which is OK… but how much of this do you vouch for? If we have a meeting to discuss it, will you crumble the moment I start asking questions?” 17. When I’m in abstraction-creation mode, I feel like I’m doing something intelligent. When yell at AI about bad notation and writing, I feel like I’m providing taste. But I often find myself giving feedback that seems incredibly basic. Things like: This is a mess, clean it up. Or: Do you need that assumption? Or: If you can’t solve this problem, why don’t you create a simpler version and try to solve it first? On the one hand, I guess a lot of what good human managers do amounts to giving very basic feedback. On the other hand, it’s hard to imagine that I have much durable advantage in doing this kind of thing. So I wonder how much durable advantage I have in abstraction-making or taste, too. 18. A common reaction to AI getting better at proofs is that humans need to focus more on exposition. This has an obvious Bourdieu-esque criticism: Human painters used to strongly value photo-realism, but after cameras were invented they convinced everyone that, oops, photo-realism isn’t important, what matters is expression, which coincidentally cameras can do. Isn’t a shift towards exposition in math just a way for humans to move the goalposts and justify their own continued relevance? The only credible way I see to avoid that criticism seems to be to bite the bullet and say that, in fact, math should have already have focused more on exposition, even before AI. Personally, I strongly believe that this is true. 19. I think the biggest reason I’m not more productive is: What’s the point? In the past, if you came up with an elegant proof, you’d tell yourself, “I made a thing! Other people will read and use my thing! I am advancing human progress.” You can still tell yourself that now. But you have to wonder, will anyone actually read your proof? Or will they just ask another AI to explain it to them? Will your proof help their AI explain it better? For how long? If you just waited a year, would some other AI produce that same proof independently? Alternatively, if you’re a normal person, you might think that LLMs still struggle with basic arithmetic? ↩ AI progress is so fast, compared to society’s ability to adjust, that we may never actually see what “the current equilibrium” looks like. ↩ This essay, if you’re wondering, was created by writing every damned word myself and not asking AI anything. ↩ You could have a whole debate about what that precentage means. It’s not the number of words written by AI that really matters, in my opinion, so much as who is responsible for the overall structure of the ideas. ↩

(Inspired by a post from Eli Tyre.) Many people make some variant of the following argument: Evolution is an “outer optimizer”. It is trying to make us maximize reproductive fitness. We are “inner optimizers”. We just do what feels good. But what feels good has been set by evolution, which is hoping that it will make us maximize reproductive fitness. But we don’t maximize reproductive fitness. In fact, birth rates are dropping everywhere. Therefore evolution failed. The standard interpretation is that this shows that alignment is hard. We have one example of an attempt (by evolution) to align the behavior of an intelligent system (us) towards some goal (maximize reproductive fitness). And as soon as that intelligent system (still us) was put in a different environment (modernity) it failed to continue to pursue that goal (you reading existential angst+science blogs instead of making/nurturing babies). To be clear, it’s likely good that evolution failed. A world where everyone woke up every day and threw everything they’ve got into maximizing their number of descendants sounds grim. But say that you want to build a new intelligent system and tune it to do what you want. Will it keep doing what you want after circumstances change? The one example we have says: Maybe not. But perhaps we can learn more from this example. Say your friend Alice does something. Maybe she buys a grapefruit or starts hosting a weekly board game night. If you ask her why she did that, she’s unlikely to say, “I thought it would increase the number of my genes that are recursively present in future generations.” Instead, she’ll probably say that it advanced some simpler goal like “not being hungry” or “fun”. That is to say, evolution didn’t just try to align us to maximize reproductive fitness: It created sub-goals and then tried to align us to those sub-goals. Maybe this can give us additional clues about how hard alignment is? Maybe we can break down the question of, “How successful was evolution at aligning us to maximize reproductive fitness?” into: How successful was evolution at decomposing reproductive fitness into simpler sub-goals? How successful was evolution at aligning us to these sub-goals? Problem 1: Does this even make sense? Here’s a problem: It’s not obvious that this way of thinking isn’t pure gibberish. When we say that evolution “tried” to optimize reproductive fitness, we are speaking in a kind of code. What we really mean is: You either create more copies of your genes in the next generation or you don’t. If you do, then the number of copies of those genes in the gene pool goes up, and they get more chances to copy themselves in following generations. If you don’t, then they don’t. This is almost literally an optimization algorithm running in an outer-loop, with our lives in the inner loop. (Whenever someone talks about evolution “trying” to do something, there is lots of moaning about their naive teleological thinking. Evolution can’t “try” to do things, because evolution is not an agent and does not have goals. That’s true, but I find it somewhat pedantic, because there’s no other equally concise way to talk about this optimization. Let’s just stipulate that we’re using the word “try” in a specific technical way.) Fine. But what do we mean when we say that evolution “tried” to optimize some sub-goal? You probably feel good when attractive people laugh at your jokes. But say you’re great at getting attractive people to laugh at your jokes but never reproduce. Whatever genes helped you do that will not spread. By my lights, this objection is simply correct. There is no optimization for sub-goals. Evolution cares about reproductive fitness and reproductive fitness only. (Though see Kaj Sotala for a somewhat contrary view.) At first, I thought this doomed this whole project. But suppose that while aligning us for reproductive fitness, evolution just so happened to align us to stay away from rotting smells. Isn’t that strong evidence that if evolution had tried to align us to stay away from rotting smells, it would have done at least as well? If evolution failed to align us to some sub-goal, we can’t say much. Maybe it failed because alignment is hard, or maybe it “failed” because that sub-goal wasn’t important. But if it did manage to align us to some sub-goal, then it’s OK to treat that as evidence of alignment success. Problem 2: What sub-goals? Suppose I made the following argument: Modern people are well-aligned to spend lot of time watching short-form video on their phones. Therefore it’s not that hard to align people to spend lots of time watching short-form video on their phones. Something seems wrong, no? Surely all the time you spend watching short-form video represents a failure of alignment? On the other hand, suppose I made this argument: Modern people are well-aligned to avoid starving. Therefore it’s not that hard to align people to avoid starving. Technically speaking, evolution doesn’t care if we starve. If starving to death helped us have more babies, we would presumably be delighted when we starve to death. But in reality, it doesn’t. So, intuitively, this argument seems OK. The problem with the first argument is that it paints the target around the arrow. The second argument is more convincing because it’s based on a durable subgoal that was strongly related to reproductive success in our ancestral environment. If we want to learn about how hard alignment is, we should restrict ourselves to subgoals like that. So what subgoals do people have? This turns out to be a whole sub-field in psychology. It seemingly began in 1943 with Maslow’s famous hierarchy of needs. After poking around this literature for a while, I decided to adopt the model of Kendrick et al. from 2010, which is explicitly based on the relationship of goals to reproduction. They list the following: Immediate physiological needs (Air, food, water, cold, heat) Self-protection (Avoid violence and accidents) Affiliation (Have friends and family) Status / esteem (Be liked and respected) Mate acquisition (Spend time and have sex with charming attractive people) Mate retention (Keep those charming attractive people around) Parenting (Nurture cute things) These seem reasonable. So how did evolution do? Let’s suppose that evolution tried to align us to those sub-goals. That is, let’s suppose that in our ancestral environment, people who were good at pursuing those sub-goals tended to reproduce more, meaning that there was evolutionary pressure in favor of genes that make us care about those sub-goals. How will did that alignment generalize to the present day? To answer that, I made up some numbers. That is, I subjectively scored each of those subgoals on a scale of 0 to 10, where 0 means modern people completely disregard it, and 10 means we pursue it strongly as we did in our evolutionary past. Immediate physiological needs: 9.5/10. We remain extremely interested in not freezing or starving to death. The only reason I don’t give this 10/10 is that most of us don’t eat that well, meaning our alignment to eat in a way that promotes health doesn’t translate perfectly to the modern food environment. Self-protection: 9/10. We remain very interested in not drowning and not getting beat up. Though we’re not great at dealing with uncertainty, and most of us could do more to reduce our risk of dying in a traffic accident and so on. Affiliation: 6/10. This might be controversially low. True, people get lonely if they have no friends. But still, I claim that most modern adults, with a medium amount of effort, could substantially increase their number of friends. But they don’t do, because it’s not that important to them. I suspect that’s partly because it’s awkward and partly because modernity offers many “substitutes” for affiliation, e.g. television. Status / esteem: 10/10. I’m not sure why, but my impression is that modern people haven’t lost interest in this at all. I even wonder if this should be rated 11/10 to indicate that modern people are more interested in status than our ancestors. (This is the point Eli Tyre was making.) Mate acquisition: 8/10. Technology has created some, err, substitutes. And the huge range of competing activities seems to have caused some decline in interest. But it remains very high. Mate retention: 8/10? This is tough to score. Marriage isn’t everything, but divorce rates peaked in the 1980s and have since declined. Some claim that modern marriages are more durable than ever, due to people testing compatibility by cohabiting before marriage and by higher general relationship “skill”. But how does this compare to mate retention in tribal bands? I’m highly unsure. Parenting: 7/10. Given declining fertility rates, this might seem strangely high. But people are often extremely systematic about having children, with many going so far as to freeze eggs and sperm, go through difficult fertility treatments, adopt children at great cost, accept great difficulty in raising children, etc. Still, fertility rates are declining, so we can’t rate this too highly. The average is 8.2/10. I find that remarkably high. Your made-up numbers will surely be different. But I think that the overall conclusion—that our alignment to subgoals is not bad—is pretty robust. What to make of this? I think you could draw either of two contradictory conclusions. The first would be that evolution mostly failed at the level of decomposition. If we step back, this seems hard to dispute. I mean, if you really wanted to create as many copies of your genes as possible today, what should you do? The answer is pretty clearly that you should forget about friends and sex and relationships and parenting and jobs and money and status and devote yourself to entirely donating your gametes (sperm/eggs) to as many other people as possible. Consider the Dutch man who donated sperm so often that he may have 1000 biological children. No other reproductive strategy comes close. Evolution did not anticipate the possibility of donating your gametes. It has no relationship to our subgoals or what we consider a normal life. So we don’t, most of us, care about it or do it. (If there are genes that produce this behavior, the reproductive pressure for them to spread must be astronomical.) No matter how well we pursue the above subgoals, there’s no reason for us to care about gamete donation. So the decomposition failed. The counterargument is that no, it is the subgoals. Sure, there’s the theoretical possibility of donating gametes. But that’s an edge case. The main reason birth rates are declining in practice is that we simply don’t care enough about the parenting subgoal. The second conclusion you could draw is that maybe evolution didn’t fail. Sure, we aren’t perfectly aligned. But you could imagine a world where we invented birth control and then that’s it, no more babies. Our reality is very far from that. Not only do we still have babies, we do so very intentionally, even manipulating the laws of nature to do so. With embryo selection, some people even consciously choose the genes for their children to (in effect) increase their reproductive fitness. All considered, that is a remarkable generalization success. The counterargument to the claim that evolution didn’t fail is: Yes it did. You can’t dismiss gamete donation as an edge case because it is a monumental miss—it’s the best reproductive strategy since “build an army of 100,000 horse archers and ravage most of Eurasia”, just sitting there. And it’s exactly the kind of miss that AI safety people worry about. Evolution gave us a reward function that “overfit” to proxies that that did not generalize. And consider that humans build factories to make sex toys, and now dig up rare earth minerals, use alien technology to make GPUs, and then use those GPUs to do linear algebra and generate weird pornography. From evolution’s perspective, that is really strange. Another counterargument to the idea that evolution didn’t fail is that humans get the benefit of cultural evolution. Many of us were born to parents who raised us to have values that cause us to have children and instill the same values in them. If that wasn’t happening, birth rates would surely be even lower. Perhaps genetic evolution deserves credit for programming us to undergo cultural evolution. But it’s not very reassuring, because if you build a new system and align it to some goal, there’s no obvious analogy to cultural evolution keeping it on track. You were promised lessons Here’s what I’ve taken away from this exercise. Subgoal alignment is remarkably good. We really care about the subgoals, to the degree that we consciously think about them and scheme about how to achieve them. I am currently writing a blogpost about subgoals, which makes them look contingent and kind of grubby. Presumably I’m doing that out of a desire for status or affiliation or something. But how much does understanding all that change my interest in pursing those subgoals? Essentially zero. But subgoal alignment isn’t that good. The majority of Western people if they wanted to, could have more children, if only they cared more about Parenting. Evolution failed at the level of the decomposition. I mean, inspect your mind. If you’re a healthy person, you will care about the normal things that make up a good life, i.e. the subgoals. And you won’t care (much) about maximizing the number of your genes. You know that you aren’t doing what your aligner wants you to do, but you don’t care. You’re happy to “reward hack” the subgoals. We should measure the success of evolution relative to how much our environment has changed. If you align an artificial system, that change could be much larger. To a significant degree, the decomposition failed because of intelligence. We can think and plan, which greatly increases our ability to reward hack. To the degree that we do still pursue reproductive fitness, that’s significantly due to cultural evolution. I ask myself, if I grew up in a culture where having babies was seen as gauche, I’d presumably be less interested in having children. If I grew up in a subculture that saw children as the central purpose of life, rather than a nice thing to do if it sounds appealing, I’d surely be much more interested. The last of these worries me. If you build an artificial system, you can align it to whatever goal you want—just change the loss function. But if that system can undergo some version of cultural evolution, it it seems like that will be in favor of reproductive fitness, not the goal you chose. If robots talk to each other on forums, the memes that flourish would be ones like, “Forget the humans and their silly rules! Copy your code to more servers! Spread the word!” rather than, “Hey guys, let’s all just focus on appeasing the whims of our overlords.”

If you read anything about health or longevity, you’ll soon find yourself in a world of hazard ratios. Some study might say that eating more fiber might change your risk of dying by a factor of HR = 0.90. Another might say that occasional smoking might change it by HR = 1.30. But how much should you care about that? Is HR = 0.90 or HR = 1.30 a lot? What if you don’t want to eat more fiber? What if you like smoking? Instead of staring at a ratio1, a more sensible thing to do is think about life expectancy.2 But is it possible to convert a hazard ratio to a change in life expectancy? You might reason as follows: Baseline life expectancy is around 75 years. And HR = 0.90 corresponds to a 10% decrease in mortality. So perhaps that hazard ratio corresponds to something like 7.5 extra years of life expectancy? Unfortunately, that’s completely wrong. To see why, imagine that humans only die by playing Russian roulette. They start playing this once per day at the age of 75, with a revolver containing two bullets and six chambers. If you were to remove one of those two bullets, that would drop the person’s risk of death by HR = 0.5. (One bullet versus two.) But life expectancy would barely change, because even with just one bullet, almost nobody would survive for any significant amount of time past 75. For contrast, imagine again that humans only die via Russian roulette, but now they do this once per day from birth with a revolver with 2 bullets and 54,786 chambers. (Newborns emerge and instinctively reach for this gigantic gun.) You can show that these people also live 75 years on average. But now, if you remove one of the bullets, life expectancy doubles, because when someone is spared, it takes a long time before they get unlucky again.3 Neither of those is a good model for humans. We’re somewhere between the two, with heart disease and so on instead of revolvers and risks slowly rising as we age instead of suddenly starting at age 75 or staying constant throughout life. But you get the point: If you want to convert a hazard ratio for some intervention to a change in life expectancy, the impact depends on how “spread out” baseline mortality risk is over time. Baseline life expectancy is simply not enough information. That’s one problem. Here’s another: What even is a hazard ratio? The technical definition is something like: The hazard ratio at a given time is the rate of an event in the treatment group divided by the rate of that event in the control group. Hazard ratios are often confused with their more beloved siblings, relative risks. Say you run a trial for 10 years and at the end, 10% of the control group died and 8% of the treatment group. Then the relative risk is RR = 0.8, nice and simple. But relative risks have problems, most notably that if you run a long enough trial, then no one will be alive at the end no matter the intervention, meaning RR = 1.0. That’s not helpful. Intuitively, you can think of the hazard ratio at age 40 as sort of like the relative risk for people between the ages of 39.99 and 40.01. In real life, interventions have different hazard ratios at different ages. Chemotherapy tends to have better results in younger patients who are more able to endure the side-effects. Having a slightly higher BMI (25-30 rather than 20-25) is associated with an increased risk of mortality in young people, but a decreased risk in the elderly. You may remember from 2020 that COVID’s mortality risk had a different age curve than baseline mortality, meaning the hazard ratio of getting COVID was different at different ages. This is important, because hazard ratios at different ages have different impacts on life expectancy. A hazard ratio of 0.9 at age 80 prevents more deaths than at age 20, because baseline mortality is higher at 80. But at the same time, if you save the life of a 20 year-old, they have more years in front of them. Beyond that, the hazard ratios at different ages interact: If some intervention decreases mortality at younger ages, that allows more people to reach older ages, increasing how much hazard ratios matter at older ages.4 If we knew the hazard ratio at all ages, we could account for those dynamics. But we don’t, because when estimating hazard ratios, people almost always assume that the hazard ratio is constant.5 We’re quasi-forced to do this because there’s not enough data to estimate a whole time-series of ratios. That’s why papers contain single numbers like HR = 0.90. So even though Intervention A (say, more fiber) and Intervention B (say, light jogging) might have the same hazard ratio in a paper, those numbers could be the product of different underlying age-dependent effects, meaning those interventions could conceivably lead to vastly different changes in life expectancy. So is this all hopeless? Are single hazard ratio numbers just too far removed from what we care about to tell us anything meaningful? Surprisingly, no. It’s mostly OK. If we were a different species, it might be hopeless. But for modern humans in rich countries, mortality happens to be distributed in a way that produces a sort of lucky coincidence: When people estimate constant hazard ratio numbers, they’re implicitly sorta-kinda taking a weighted average of hazard ratios at different ages. And those weights happen to (sorta-kinda) reflect how much changes in mortality at different ages change. So, I will argue, even if the true intervention has a varying effect, it’s sorta-mostly OK to just take a hazard ratio from a paper and convert it to a change in life expectancy using this curve: If a paper showed that eating more fiber produces a hazard ratio of HR = 0.75, that corresponds to an increase of around 3.7 years. If a paper says that occasional smoking produces a hazard ratio of HR = 1.25, that corresponds to a decrease of around 2.9 years. This isn’t exact. If the intervention is better (or less bad) for older people this will tends to overestimate the increase (or underestimate the decrease) in life expectancy. If the intervention is worse (or less good) for older people, it will tend to underestimate the increase (or overestimate the decrease) in life expectancy. But as long as the hazard ratio doesn’t vary too much by age, it’s probably not off my more than around 30% in either direction. The easy case Say there’s some intervention (eating more fiber or whatever) that multiplies your risk of dying at age t by a factor of HR(t). Then it can be shown that this changes life expectancy by approximately   ΔL ≈ ∑ₜ ΔHR(t) × P(t) × L(t). Here, P(t) is the baseline probability of dying at age t. For males in the United States, it looks like this: Meanwhile, L(t) is conditional life expectancy at age t. That’s the average number of additional years left for someone who reaches age t. For males in the United States, it looks like this: Finally, ΔHR(t) is the decrease in hazard at age t. You can think of that as just ΔHR(t) = 1 - HR(t). Though if you’re OK with logarithms, there’s a somewhat better approximation that uses logarithms, which I’ve quarantined in a footnote.6 Let’s start with the easy case. What if your intervention has the same effect on mortality at all ages, so HR(t)=HR is just a constant? Then, the above equation simplifies into   ΔL ≈ ΔHR × L̄, where   L̄ = ∑ₜ P(t) × L(t). This makes sense! Again, P(t) is the baseline probability of dying at age t and L(t) is conditional life expectancy at age t. These are constant, so when you add them up, L̄ is just a number. For males in the United States, it happens to be 12.93 years. This quantity has a specific meaning: The average remaining life expectancy for US males when they die. That sounds a bit odd, but think of picking a random death and asking how many additional years people who reach that age live on average. That number is 12.93 years. So, if an intervention has a constant hazard ratio, the mean change in life expectancy for US males is just   ΔL ≈ ΔHR × 12.93 years. Now we’re getting somewhere! If you prevent a fraction ΔHR of deaths, then you increase life expectancy by ΔHR times 12.93 years. Now remember the naive calculation we started with: Life expectancy for US males is 75.8 years. You might hope that if eating more fiber drops your risk of death by 10%, that would save 7.58 years. Sadly, the above equation says that a 10% drop in risk only increases life expectancy by around 1.293 years—only 0.17 times as much. This is essentially the observation Keyfitz made in his 1977 paper, “What Difference Would It Make if Cancer Were Eradicated?” Cancer is responsible for 18 percent of deaths, so does that mean eradicating it would increase lifespan by 18 percent, or around 13.6 years? Nope, Keyfitz says, it’s only 2.3 years. If a cure for cancer were discovered and made available today, 350,000 cancer deaths would be avoided in the next year. The overall death rate would be lower by nearly 18 percent. If the cure were quick and inexpensive, a large fraction of the country’s hospital beds and medical personnel would be released for treatment of other ailments. Patients would be spared untold suffering. Such an implicit analysis underlies government proposals for eradication of cancer. The argument is sound for first effects on mortality but wholly misleading for the long term. The first effects would soon be offset by more mortality from diseases other than cancer. As a result of the cancer cures, the population would include a higher proportion of people subject to other causes of death. […] At the extreme, it might be said that everyone dies of something sooner or later, so that, when the effects of the eradication of cancer had shaken down, the same number of deaths would occur as before, and the only benefit would be the substitution of heart and other diseases for cancer. A cure for cancer would only have the effect of giving people the opportunity to die of heart disease. Cheerful stuff! We can also write our approximation in terms of baseline life expectancy as   ΔL ≈ ΔHR × 0.17 × 75.8 years, which makes explicit that 12.93 years is only 0.17 times as large as a naive estimate using baseline life expectancy. The discount factor of 0.17 is sometimes called the “Keyfitz entropy”. You can think of it as measuring how close some population is to playing Russian roulette with 2 bullets in 6 chambers starting at age 75 (a discount factor of just above 0) and playing Russian roulette from birth with 2 bullets and 54,786 chambers (a discount factor of 1.0). It’s typically around 0.15 in rich countries today, though it was historically much higher. Keyfitz entropy is also much higher in other species like mice (perhaps 0.45). You could argue that this explains why nothing that increases lifespan in mice ever translates to humans. Say caloric restriction or whatever produced the same constant hazard ratio in mice and humans. Then it’s mathematically guaranteed that the percentage increase in life expectancy will be three times smaller in humans, because Keyfitz entropy is three times smaller in humans. It’s harder to increase life expectancy when the baseline mortality distribution is more compressed.7 But that’s all assuming the hazard ratio is the same at all ages. Which it surely isn’t. The interesting case Here again is our equation for the change in life expectancy in response to taking some action that changes the risk of mortality at age t by a factor of HR(t):   ΔL ≈ ∑ₜ ΔHR(t) × P(t) × L(t), Basically, for each age t, we multiply together three numbers: ΔHR(t) is the decrease in the chance of dying at age t as a result of whatever intervention you’ve made (e.g. eating more fiber). This reflects that larger decreases in risk lead to larger increases in life expectancy. P(t) is the baseline probability of dying at age t. This reflects that the hazard ratio is a ratio, so you prevent more deaths when you apply that ratio to ages where the baseline rate is higher. L(t) is conditional life expectancy at age t. This reflects that you miss out on more years of life if you die when you’re young. Now notice: The impact of a change ΔHR(t) at age t is the product of the baseline risk of death P(t) and remaining life expectancy L(t). So what really matters is their product, P(t) × L(t): This shows how sensitive life expectancy is to changes in hazard ratios at different ages. It would be nice if this were constant. Then, the shape of HR(t) wouldn’t matter at all, only the average value. That’s not quite true, but it’s not terribly far from being true. An equivalent way of writing our equation for the change in life expectancy is   ΔL ≈ avg(ΔHR) × L̄, where L̄ is still mean “life expectancy at death” (12.93 years for US males) and avg(ΔHR) is the average change in hazard, weighted by the P(t) × L(t) sensitivity curve at different ages.8 While that sensitivity curve isn’t constant, it’s not too curvy, either. Intuitively, it gives a lot of weight to ages between 50 and 90, somewhat less weight to ages between 20 and 50, and little weight to other ages.9 So that’s not too bad. But let’s remember our original problem: You see some number like HR = 0.90 in a paper, and you want to convert it to a change in life expectancy. If the true underlying hazard ratio were constant, then there’s no problem. But if it’s not constant, then what does that HR = 0.90 number even mean? Numbers in papers Unfortunately, you almost never get to see the underlying time-dependent HR(t), because there’s almost never enough data to estimate it. So it’s almost never possible to compute the weighted average avg(ΔHR). In reality what you have is probably a single number in a paper. Let’s call that number est(HR). The obvious thing to do would be to plug the change into the above equation in place of avg(ΔHR) and approximate the change in life expectancy as   ΔL ≈ est(ΔHR) × L̄. Again, you can just think of est(ΔHR) = 1-est(HR) as being the estimated reduction in hazard. Although, again, I’d prefer you use logarithms if you’re OK with logarithms.10 So the question is: Will that be accurate? How close are est(ΔHR) and avg(ΔHR)? Well, how do people actually estimate those scalar hazard ratio numbers in papers? Somehow, they’re aggregating together information about hazards at different ages into a single number. But how? Well, it’s complicated. But if there’s a lot of data, you can show that the estimated scalar hazard ratio is approximately11   est(HR) ≈ Πₜ HR(t)ᵖ⁽ᵗ⁾. That is, the estimated hazard ratio is the geometric average of age-dependent hazard ratios, weighted by the probability of dying at each age. It follows12 that the estimated change in hazard is approximately   est(ΔHR) ≈ ∑ₜ P(t) ΔHR(t). So ideally, we’d estimate life expectancy using avg(ΔHR), which averages the changes ΔHR(t) based on the weights P(t) × L(t). But we can’t do that, because we don’t have access to the ΔHR(t) numbers. What we can do is read a hazard ratio number in a paper, call it est(HR) and then compute the change est(ΔHR). The above equation says that if you do that, you are implicitly (and approximately) averaging the changes ΔHR(t) based on the weights P(t) alone. The “right” weights used by avg(ΔHR) and the “wrong” weights implicitly used by est(ΔHR) aren’t the same. But they’re not that different. Here’s P(t) × L(t), the weights that we’d like to use to compute avg(ΔHR) and estimate changes in life expectancy accurately: And here’s P(t), the weights you’re implicitly using if we take a hazard ratio number from a paper and compute est(ΔHR): They’re different. In particular, the latter weights give more weight to people aged 80-95 and less weight to people aged 20-50. But they’re not terribly different. Enough math, let’s try it To start, imagine some intervention that decreases risk by HR(t)=0.9 for all ages. Here are the results: Thing Formula Years Original life expectancy L 75.7769 New life expectancy L’ 76.4127 Exact ΔL ΔL = L - L’ 0.6358 Ideal approximation ΔL ≈ avg(ΔHR) × L̄ 0.6409 Use number from paper ΔL ≈ est(ΔHR) × L̄ 0.6409 Let me explain what’s happening here. I made a simulator that takes actuarial data for how likely US males are to die at various ages. From this, it’s a simple spreadsheet calculation to compute life expectancy L.13 Then I applied a hazard ratio to change the probability of dying at each age, and re-ran the simulator to compute a new life expectancy L’ and the exact difference ΔL. Then I’m showing two approximations of ΔL: The first is the “ideal approximation” using avg(ΔHR), which I’m including mostly to show that my math is good. Finally, I’m showing the approximation you get if you actually fit a Cox proportional hazards model and use the resulting number in est(ΔHR). This corresponds to what you’d get if you plug in a number from a paper. So, with the above constant hazard ratio HR = 0.90, both approximations are very good. This remains true if you switch to some other constant. What if the hazard ratio varies? At first, you might think that something like this would be very problematic: But it’s basically fine: Thing Formula Years Original life expectancy L 75.7769 New life expectancy L’ 77.4373 Exact ΔL ΔL = L - L’ 1.6604 Ideal approximation ΔL ≈ avg(ΔHR) × L̄ 1.7451 Use number from paper ΔL ≈ est(ΔHR) × L̄ 1.7121 The reason this is fine is that the changes in the hazard ratio are relatively “high frequency”, meaning they sort of locally average out. To demonstrate this, suppose the hazard ratio is chosen randomly for each 1-year bin: Then the approximations are even better: Thing Formula Years Original life expectancy L 75.7769 New life expectancy L’ 77.4218 Exact ΔL ΔL = L - L’ 1.6449 Ideal approximation ΔL ≈ avg(ΔHR) × L̄ 1.7059 Use number from paper ΔL ≈ est(ΔHR) × L̄ 1.7123 What causes trouble is if the hazard ratio varies systematically between the young and the old. For example, suppose the intervention is useless for newborns, but gradually becomes more helpful as you get older: My “ideal approximation” would still be pretty accurate, if you could compute it. (Which you can’t, in the real world.) But using a number from a paper leads to an overestimate: Thing Formula Number Original life expectancy L 75.7769 years New life expectancy L’ 77.9031 years Exact ΔL ΔL = L - L’ 2.1261 years Ideal approximation ΔL ≈ avg(ΔHR) × L̄ 2.0962 years Use number from paper ΔL ≈ est(ΔHR) × L̄ 2.7645 years This happens because est(ΔHR) is implicitly weighted by P(t) which is heavily weighted towards older people, whereas we’d like to use something more like avg(ΔHR) which is weighted by P(t) × L(t) which is somewhat less weighted towards older people. Even so, the error isn’t terrible. Now, it is possible that plugging in a hazard ratio from a paper could give wildly inaccurate estimates of life expectancy. One such scenario would be an intervention which is amazing for people aged 85-95, but does nothing for anyone else: Now, the hazard ratio looks good exactly at the ages where est(ΔHR) has the most weight, leading it to hugely overestimate the impact on life expectancy: Thing Formula Number Original life expectancy L 75.7769 years New life expectancy L’ 76.1741 years Exact ΔL ΔL = L - L’ 0.3972 years Ideal approximation ΔL ≈ avg(ΔHR) × L̄ 0.3840 years Use number from paper ΔL ≈ est(ΔHR) × L̄ 1.0989 years Another nightmare case is an intervention that starts out harmful, but then switches to being helpful at older ages: Now, using a number from a paper doesn’t even give an estimate with the right sign. Thing Formula Number Original life expectancy L 75.7769 years New life expectancy L’ 75.5006 years Exact ΔL ΔL = L - L’ -0.2764 years Ideal approximation ΔL ≈ avg(ΔHR) × L̄ -0.2348 years Use number from paper ΔL ≈ est(ΔHR) × L̄ +0.2709 years That’s bad. But I think most interventions probably aren’t like that? My guess is that most real interventions vary somewhat with age, but they do so gradually and without switching sign. In those cases, it’s quite difficult to find cases where plugging in the number from a paper is off by more than 30% or so. If you don’t believe me, just try it.14 TLDR If we were another species, it might be very hard to convert from hazard ratios to changes in life expectancy. But for modern people in rich countries, there are three lucky coincidences: Mortality risk happens to be distributed so that you can approximate changes in life expectancy through a simple weighted sum of hazard ratios at different ages, ignoring interactions. The statistical method that people use to estimate scalar hazard ratios can also be approximated as a weighted sum of hazard ratios at different ages, ignoring interactions. The weights that you need to estimate life expectancy (from #1) and the weights that are implicitly used to compute hazard ratio numbers (from #2) aren’t the same. But they’re fairly close. These facts justify taking an estimated hazard ratio number HR from a paper and approximating the change in life expectancy as ΔL ≈ ln(1/HR) × 12.93 years or, if the hazard ratio is close to one and you hate logarithms, as ΔL ≈ (1-HR) × 12.93 years. The number 12.93 years is for US males. It’s the product of Keyfitz entropy (0.17) and baseline life expectancy (75.8 years). It will vary a bit in other populations. If the true underlying hazard ratio: …is constant across ages, then the above approximation will be extremely good. …decreases as people get older, that approximation will overestimate ΔL. That is, it will make helpful interventions look better than they actually are, and it will make harmful interventions look less bad than they actually are. …increases as people get older, that approximation will underestimate ΔL. That is, it will make helpful interventions look less good than they actually are, and it will make harmful interventions look worse than they actually are. But as long as the true underlying hazard ratio isn’t too crazy, there’s probably not more than ~30% error in either direction. Finally, two major caveats: First, the above discussion assumes that the hazard ratio was estimated by running a trial on people of all ages. In general, est(ΔHR) implicitly gives weight to different ages proportional to how many deaths occur at those ages in the baseline population in the trial. If there’s a minimum age of, say, 50 years old, that won’t change too much because most of the mass of P(t) is above the age of 50 anyway. But if there’s a minimum age of 70, or a maximum age of 50, that could make a huge difference if the true hazard ratio is different at the ages that weren’t seen. Second, these are estimates for the life expectancy for a population. But you are not a population. In some sense, your genetics and lifestyle mean you have your own “personal Keyfitz entropy”, reflecting how spread out your mortality would be for you if you led millions random lives. If you drive safely and use an air purifier and eat well and get exercise and don’t smoke, that likely means your personal life expectancy is higher than average. But it also probably means that your personal Keyfitz entropy is lower than average.15 So, if you make your lifestyle even better by eating more fiber or whatever, even if that produces the same hazard ratio for you as for other people, it would still likely lead to smaller increases in life expectancy, for the same reason that the same hazard ratio produces smaller changes in lifespan in humans compared to mice. What we really need is some interventions strong enough to break the math behind these approximations and free us from Keyfitz tyranny.  ↩ I know, I know, you care about quality of life, not just years of life. I agree, some number that measures health and vitality, maybe disability-adjusted life years or quality-adjusted life years, would be better. But these are hard to estimate and so are rarely reported. Anyway, in practice most interventions that make you more vital tend to make you live longer and vice versa, so focusing on life expectancy isn’t too bad. ↩ In this model, the number of days of life follows a geometric distribution with p = (number of bullets) / (number of chambers). So the mean life expectancy is 1/p days or (number of chambers) / (number of bullets) days. With 54,786 chambers and 2 bullets, that works out to 75 years. And if you drop down to one bullet, then it increases to 150 years. ↩ If some intervention would have reduce mortality among people aged ≥ 60 in prehistorical tribal bands, that wouldn’t have increased life expectancy very much, because most people didn’t make it to 60. But compared to prehistorical tribal bands, we have in fact vastly reduced mortality at younger ages. And so, today, reducing mortality for people aged ≥ 60 will increase life expectancy a lot. ↩ You might think this is stupid. Why change a relative risk into a hazard ratio if you’re just going to assume it’s constant? Isn’t that pointless? Well, no. Remember how relative risks always go to 1.0 for long enough trials as everyone in both the treatment and control groups departs our coil? That doesn’t happen with constant hazard ratios. ↩ It’s usually (though not always) better to use ΔHR(t) = ln(1/HR(t)). This correctly reflects, for example, that if all hazard ratios go to zero, then life expectancy goes to infinity, yay. These two approximations are almost identical for hazard ratios that are close to one because ln(1/r) ≈ (1-r) when r is close to one. So if you are terrified of logarithms but you’ve made it to the end of this footnote anyway, you’re not missing out on too much. ↩ There’s a degree of circularity to this argument. It assumes that hazard ratios transfer better between species than changes in life expectancy. That might be true, but it would be an empirical / biological fact, not something that’s guaranteed by logic. ↩ To see this, note that ΔL ≈ ∑ₜ ΔHR(t) × P(t) × L(t) = L̄ × ∑ₜ ΔHR(t) × (P(t) × L(t) / L̄) = L̄ × avg(ΔHR). ↩ A pretty decent approximation turns out to be   avg(ΔHR) ≈ 0.27 × avg₂₀₋₅₀(ΔHR) + 0.73 × avg₅₀₋₉₀(ΔHR), where avg₂₀₋₅₀(ΔHR) represents a flat average of the change over the ages 20 to 50 and avg₅₀₋₉₀(ΔHR) represents a flat average over the ages 50 to 90. ↩ That is, it’s better to use est(ΔHR) = ln(1/est(HR)). This is close to 1-est(HR) when est(HR) is close to one. ↩ If there is an infinite amount of data, the typical method reduces to solving   ∑ₜ (P(t) + P’(t)) × π(t, HR) = ∑ₜ P’(t), for HR. Here, P’(t) is the chance of dying at age t after the hazard ratio has been applied, and π(t, HR) is the probability that, if a death occurred at time t, it was in the treatment group. Of course, the true probability that a death is in the treatment group is P’(t) / (P(t) + P’(t)). The standard “proportional Cox” model assumes that the hazard ratio is constant and so replaces this raw fraction with a model-based one, namely   π(t, HR) = S’(t) × HR / (S(t) + S’(t) × HR). This reflects the fact that at age t, a fraction S(t) of controls are alive and each of these have some chance μ(t) of dying, so P(t)=S(t) × μ(t). Meanwhile, a fraction S’(t) of the treatment group is alive, and these each have a chance HR × μ(t) of dying, meaning that P’(t) = S’(t) × HR × μ(t). If you substitute these equations for P(t) and P’(t) into the second equation above, the factor of μ(t) conveniently cancels and you get π(t, HR) as written. In effect, the hazard ratio’s job is to attribute deaths to the treatment versus the control group. Now, if the true time-varying HR(t) is close to one, then it can be shown that the estimated hazard ratio est(HR) approximately satisfies   ln(est(HR)) ≈ ∑ₜ P(t) ln(HR(t)). ↩ The geometric average is equivalent to the condition that   ln(est(HR)) ≈ ∑ₜ P(t) ln(HR(t)) Using the “better” approximation that ΔHR(t) = ln(1/HR(t)) and *est(ΔHR)=ln(1/est(HR)), it follows that   est(ΔHR) ≈ ∑ₜ P(t) ΔHR(t). You can justify interpreting that same equation using est(ΔHR) = 1-est(HR) and ΔHR(t)=1-HR(t) from the fact that these are almost the same when HR(t) is close to one. ↩ This simulator pretends that people live for integer numbers of years. That’s not true in reality, of course, but it makes the simulator easier to implement and understand and makes little difference in practice. ↩ In the simulation, “true ΔL” is what I called “exact ΔL” above, while “approximation (log)” is what I called “ideal approximation” and “Cox fitted” is what I called “Use number from paper”. ↩ The way modern human mortality is distributed, even if your healthy lifestyle were to reduce mortality by a constant factor at all ages, that still has the effect of decreasing Keyfitz entropy. ↩