Summary

To this day there is a lack of systematic research to evaluate a cause area with immense potential: aging research. This is the first of a series of posts in which I'll try to begin research to address this gap. The points made in this post are about how to evaluate impact using the concept of Longevity Escape Velocity. Bringing the date of Longevity Escape velocity closer by one year would save 36,500,000 lives of 1000 QALYs, using a conservative estimate. Other sources of impact that arise from the same concept include: increasing the probability of Longevity Escape Velocity, making Longevity Escape Velocity spread faster, and making a new future portion of the population reach Longevity Escape Velocity by increasing its life expectancy. Aging research could also positively impact the cost-effectiveness of other interventions by increasing the probability that Longevity Escape Velocity will be attained in the recipients' lifetimes. I will also discuss why the probability of Longevity Escape Velocity is substantial and why QALYs should be the measure of impact, and I'll give mathematical proofs that the adoption speed of the technologies that arise from research doesn't impact cost-effectiveness analyses.

The need of a theoretical foundation to evaluate aging research

I think one important approach to research in Effective Altruism is to try to lay theoretical foundations and put together tools for helping to evaluate a specific cause area that can be generalised to any intervention inside that cause area. Such work is often not possible because of lack of time and expertise, making it preferable, sometimes, to scout specific promising interventions or refine existing research.

One cause area that absolutely needs this kind of more systematic groundwork is aging research. The current EA research about aging is lacking in number and in what I think are crucial considerations, even though informal discussion with members of the community reveals that many people regard it as potentially promising. The expertise required to make such an analysis possible is rare to find. It requires people with a strong quantitative background who are also interested not only in biology but in the biology of aging in particular, and they must be accustomed to predicting the future of scientific research and making cost-effectiveness evaluations. I observed that the Effective Altruism community seems to have plenty of people with a background in philosophy, economics, social sciences or computer science, but people with a strong background in biology, or at least a strong interest in it, are scarce. This makes it even harder to find people willing to do the work of evaluating the cause area of aging research.

For these reasons, and since I'm very familiar with the topic and I think I have important things to say about it, I am willing to try to lay as much groundwork as possible, at least until I think I'm needed.

My long-term hope is that the groundwork I will lay will be good enough for a more formal discussion about this topic within Effective Altruism, both for evaluating specific interventions inside this cause area and for evaluating the cause area as a whole. I will write about what I think are original points and put together all the existing tools that could help both Effective Altruism organisations and organisations within the cause area of aging to make better decisions.

I have chosen to split the analysis into multiple posts so that I can receive and incorporate feedback during the process and thereby modify my work and its planning along the way. Organising the work in this way will also make the whole thing easier to read.

I'm doing this alone and in my free time, between university and other activities, so the posts will probably come out a few weeks or even months apart.

Although I hope that the bottom line of my arguments is strong, there will probably be many mistakes and many corrections to make. I encourage you to comment, give feedback and to contribute new ideas, especially if you have a consideration about something that I didn't address that would substantially impact the result of a potential cost-effectiveness analysis. Along the way I will probably need to collaborate with other people and coauthor some posts, since my knowledge probably has gaps and needs to be complemented. Nonetheless I will try to learn what I currently don't know along the way.

What will this series of posts be about?

This is the first of a series of posts in which I'll explore different ways of reasoning about the potential cost-effectiveness of aging research. Each post will focus on one or more considerations. In the last post I would like to wrap everything up with a comprehensive framework useful for evaluating the cost-effectiveness of any given avenue of scientific research into the aging processes and how to treat their various facets. The points made will also provide an idea of the potential of the cause area as a whole.

An initial major focus will be on the scope of the problem and on moral considerations that could affect it. Neglectedness and tractability will be given space in later posts, in which I will try to lay out useful methods and heuristics to evaluate them in this cause area.

After this work, I would like to discover the best funding opportunities within this area and compare my conclusions with other past efforts within Effective Altruism that have been made to evaluate aging research.

Points made in this first post:

• Longevity Escape Velocity (LEV) is the minimum rate of medical progress such that individual life expectancy is raised by at least one year per year if medical interventions are used.
• Reasoning with Longevity Escape Velocity substantially changes cost-effectiveness analyses.
• A conservative estimate for life expectancy after Longevity Escape Velocity is 1000 years, although it’s still not a lower bound.
• In order to account for making Longevity Escape Velocity arrive more quickly in cost-effectiveness analyses (CEAs) the relevant variables are deaths by aging per year, life expectancy after LEV and Expected number of years LEV is made closer by. This without accounting for moral weights and other possible discounts.
• The probability of Longevity Escape Velocity is substantial.
• Another factor potentially greatly influencing impact is the life expectancy increase resulting from research projects or health interventions. If the project is not likely to be funded in the future or subsumed by other research, the recipients of the intervention who would have died near LEV get saved.
• QALYs should be the measure of impact, as one life saved counts more than 30-80 QALYs in this cause area.
• Mathematical proofs that cost-effectiveness analyses aren't influenced by how Longevity Escape Velocity, or any technology that arises from a given financed research project, spreads to the whole population
• Making LEV spread faster is another impact consideration that is pertinent to projects potentially leading to policy change or public awareness.
• Certain research projects could also have the effect of increasing the probability of Longevity Escape Velocity, potentially influencing cost-effectiveness analyses substantially.
• Aging research can boost the impact of other altruistic interventions by increasing the probability of LEV happening in the recipients' lifetimes.

The next posts in the series will probably be about:

• A better lower bound for the life expectancy after Longevity Escape Velocity, and how this affects the probability of LEV.
• The longevity dividend.
• The value of information (depending on if I need to include considerations specific enough to aging research).
• Moral weights.
• If old people are replaceable in a utilitarian moral framework.
• What is neglected and what is tractable in the cause area of aging research.
• Putting the framework together.

Longevity Escape Velocity: what it is

Longevity Escape Velocity (LEV) is the minimum rate of medical progress such that individual life expectancy is raised by at least one year per year if medical interventions are used. This does not refer to life expectancy at birth; it refers to life expectancy calculated from a person's statistical risk of dying at any given time. This is equivalent to saying that a person's expected future lifetime remains at least constant despite the passing years.

It's possible, given sufficient ongoing improvement of medicine and its democratisation, that nearly everyone on the planet, at a certain date in the future, will benefit from therapies that allow Longevity Escape Velocity to be attained, at least until aging is eradicated completely. Then, other factors will influence risk of death, and expected future lifetime could start falling again each passing year if risk of death flattens or doesn’t continue to fall fast enough.

How likely that Longevity Escape Velocity is to become a reality in the future depends on a number of factors, which will be explored later in this post.

Reasoning with Longevity Escape Velocity substantially changes cost-effectiveness analyses

If a given intervention "saves a life", this usually means that it averts 30 to 80 Disability-Adjusted Life Years (DALYs). In order to evaluate the impact of aging research, one could be tempted to try to estimate how many end-of-life DALYs that a possible intervention resulting from the research could save and adjust the number using the probability of success of the research.

This is the line of reasoning that OpenPhilanthropy's medium investigation on aging uses, although without making any explicitly quantitative argument. This is part of the impact, and it has to be factored in, but it doesn't consider where the largest impact of aging research is: making the date of Longevity Escape Velocity come closer. This would have the effect of saving many lives from death due to age-related decline and disease, but here, "a life" means, more or less, 1000 Quality-Adjusted Life Years (QALYs). This figure is derived as follows:

Life expectancy after LEV:

In actuarial science, the expected future lifetime of an individual at age $x$ is denoted with $e_x$. It can be seen as the expected value of the random variable $K\left(x\right)$, also called "Curtate Future Lifetime", which is defined as $K\left(x\right):=\lfloor T\left(x\right)\rfloor$ , where $T\left(x\right)$ maps to the amount of additional time that an individual of age $x$ is projected to live. For our purposes, we can use the discrete random variable $K\left(x\right)$ instead of directly $T\left(x\right)$. Thus, with ${}_k{p}_x$ being the probability of surviving between age $x$ and age $x+k$:

$e_x=E\left[K\right(x\left)\right]={\sum }_{k=0}^{\infty }k\cdot P\left(K\right(x)=k)={\sum }_{k=0}^{\infty }k\cdot ({}_k{p}_x-{}_{k+1}{p}_x)=={\sum }_{k=0}^{\infty }k\cdot {}_k{p}_x-{\sum }_{k=0}^{\infty }k\cdot {}_{k+1}{p}_x={\sum }_{k=1}^{\infty }{}_k{p}_x$

If $p\left(x\right)$ is the probability of dying between year $x$ and year $x+1$, then:

$e_x={\sum }_{k=1}^{\infty }{}_k{p}_x={\sum }_{k=1}^{\infty }{\prod }_{i=1}^k(1-p(x+i-1\left)\right)$

For now, I'll use a constant risk of death to calculate the life expectancy of individuals after aging is eradicated completely and risk of death has presumably stopped falling. While this method doesn't yield a lower bound, since it leaves out from the calculation the risk of death when it's decreasing, it can be made conservative using a relatively high risk of death. I'll use $p\left(x\right)=1/1000$, which is more or less the current risk of death of someone between 20 and 30 years old. It is conservative because it doesn't account for future improvements in medicine and general safety outside of aging research. I also don't expect the lower bound to be much smaller. Therefore,

$e_x={\sum }_{k=1}^{\infty }{}_k{p}_x={\sum }_{k=1}^{\infty }(1-1/1000{)}^k=999\approx 1000$

Since we are talking about life expectancy in a world without aging, 1000 years of life expectancy should amount more or less to 1000 QALYs.

Accounting for making LEV come closer in CEAs

Any given aging research project, if successful, could have the effect of making the date at which most people will reach Longevity Escape Velocity come closer by a certain amount of time. We can estimate the expected QALYs gained because of such an effect. We have established that the average lifespan of a person who reached LEV will be around 1000 years, mostly without disability, and somewhat less if we use a lower bound. The number of QALYs saved are then calculated by multiplying 1000 by the number of people who would otherwise have died of aging if LEV wasn't moved closer. Currently, around 100,000 people per day (36,500,000 people per year) die due to age-related decline and diseases, although this figure will be larger when LEV arrives due to population growth.

So, in order to calculate an almost lower bound for how many expected QALYs that a certain research project would save by making LEV come closer, you simply multiply these values:

• 1000 QALYs
• 36,500,000 deaths/year
• Expected number of years LEV is made closer by

This is true for a crude estimate, without accounting for moral weights and potential discount rates.

It is important to stress the fact that none of these variables depend on how soon LEV will arrive, so we can totally ignore this kind of discussion, even if it is a highly debated topic outside the setting of cost-effectiveness evaluations.

The first two variables have been already discussed. Then, we need to examine the third one, which depends on many factors, such as:

• How promising the project being examined is, for different meanings of the word "promising". For example, it may have direct translational value into effective therapies targeting aging processes or hallmarks, or it could have an effect of speeding up the field, such as by providing new tools, by enabling political entities to aid in achieving LEV sooner, or by enabling a new line of research to start sooner or become widespread more rapidly.
• The neglectedness of the project would make the figure larger.
• If there are other projects that could subsume the effect of the examined project, some of which would universally subsume all potential projects. These include technologies outside the field of aging research that are potentially very disrupting and sudden, such as artificial general intelligence.
• The number of years necessary for another group to step in and do the same project.
• The probability of catastrophic events: existential risks or events catastrophic enough to make the information acquired by the project lost or useless.
• Lastly, the probability that LEV will happen in the first place also has a role in estimating this variable. This is because we can model the number of years that LEV is brought closer as the expected value of the random variable that maps to various numbers of years, among which is zero. The probability that the years LEV is brought closer by is zero, in turn, depends not only on the specifically examined project but also on the probability that LEV will not happen. That's why it's useful to outline how to reason about the probability of LEV.

Probability of LEV

If we had the minimum rate of decrease of risk of death such that LEV would happen, then the probability of LEV happening is the probability that the risk of death would fall at that rate or faster, and so the probability largely depends on that rate and on how fast medical research will be.

For now, we can reason about the problem by dividing the situations in which LEV will not happen in at least two scenarios:

• Very slow research scenario: In this scenario, each new therapy is developed in the span of an entire generation and contributes only a few more years of healthy life. This slow rate of progress is maintained more or less constantly within the span of a few centuries. Example: every 30 years or so, only one new therapy grants 5 more years of healthy life for the general population. If progress is this slow, LEV will never be reached. Negligible senescence will eventually be met after a few centuries, and generations will have progressively longer lifespans without anyone suddenly making very large jumps in life expectancy. New therapies may rely on previous ones for having substantial effects, forcing new treatments to come sequentially and not in parallel. It's also possible that they could theoretically be developed in parallel, but an incredibly inefficient research community develops them sequentially. This scenario seems somewhat unlikely. This tells us that reaching the minimum rate of decrease of risk of death shouldn't be too difficult.
• Dire roadblocks scenario: This is the scenario in which there are roadblocks so dire that aging research is stalled for enough time that the recipients of previous interventions die. This doesn't necessary prevent LEV all the time; these kind of roadblocks must be enough in number to effectively make the average decrease of risk of death the same as the one of the very slow scenario until aging is cured completely.

The scenarios in which LEV will happen, instead, are the ones in which the risk of death falls fast enough, which means that new therapies would be developed sufficiently close together. This would be brought about through steady progress in medicine or relatively large jumps in life expectancy that enable previous recipients of therapies to extend their lives by another large amount of time. We can imagine how such scenarios could unfold:

• Today's therapies or future therapies appear to be somewhat effective on humans or very effective on mice. This increases public focus on translational aging research, which, in turn, results in a multiplication of resources dedicated to it. It's argued that that this first "proof of concept" required to convince the world is Robust Mouse Rejuvenation, which would double the remaining life expectancy of elderly mice, as demonstrated and then replicated in rigorous laboratory studies. A multiplication of resources for the field should result in therapies following the first proofs of concept. After this, the rate of therapy development and improvement will increase exponentially following the initial success of therapies. The history of technology is full of examples of this feedback loop, in which successive improvements are faster than the development of the proof of concept, a prominent one being flight.
• Without invoking a large public interest, LEV could also be caused by combinations of different treatments coming in waves and by the improvement of treatments over time. This would mean sudden jumps in life expectancy that would buy enough time for other treatments to be developed. A sudden future enlightenment about the nature of aging could be also possible, or the first therapies could also have the effect of slowing down the accumulation of other damages, other than doing the job of addressing their specific targets. This would happen by breaking negative feedback loops of damages or processes. LEV could also happen in a sudden way if effective delivery methods are developed after many proof-of-concept therapies have been demonstrated, for example, in vitro.
• The two scenarios above sound somewhat optimistic, but they might not be needed at all. The research could unfold silently but surely and the risk of death could still fall fast enough to ensure LEV. This would happen if the current situation of very slow improvement is overcome and there isn't a large number of new dire roadblocks ahead.

Given these scenarios, can we have a preliminary idea, without knowing how fast the risk of death needs to fall, of how likely LEV is? There are, at least, probably some relevant points to make regarding the current best guesses about aging and the present state of research.

It's difficult to predict major future roadblocks, but at least it seems that the "very slow research" scenario is proving less and less likely. This doesn't mean that we already have effective therapies against aging, or that the pace of science is optimal. But how research is distributed and the theories about what aging is make believable the possibility of therapies being developed closer together, thereby enabling a high-enough rate of decrease of overall risk of death.

The current best guess about how to tackle aging rests on a milestone paper from 2013: The Hallmarks of Aging. The paper has citations in the thousands and counting, and researchers are using it as a framework to orient and justify their own research. It proposes various categories of dysfunction. Every category, or almost every category, should be addressed periodically in order to maintain a youthful state of health. Reversing one hallmark would mean restoring an internal state of the body that is typical of a youthful body. It could also prove true that it will not be necessary to address every hallmark, due to the possible cause-effect relationships between each of them.

What does this say about how close together therapies will come? It says a lot: a paper like The Hallmarks of Aging means that the field already has an idea of what combination of foreseeable therapies will bring major gains in health and, in turn, life expectancy. This is because this theoretical categorisation constitutes what needs to be addressed.

It also implies that it enables thinking about rejuvenation, not only "slowing down" aging. This is because the dysfunctions described are exactly what is "wrong" with an old body, and not how those dysfunctions arise, so getting rid of those kind of dysfunctions means rejuvenation.

It's a "downstream" view of aging that decomposes the problem and leaves out what is unnecessary to know in order to intervene, increasing the tractability of the problem. We don't need to know how the Hallmarks arise in order to develop therapies that address them. One added benefit is that the hallmarks influence each other in negative feedback loops; reversing one slows down the progress of many others.

Theoretically, interventions aiming at reversing all of the hallmarks of aging could be developed in parallel, and, in fact, they currently are (although not optimally so). Interventions to ameliorate each one of the Hallmarks, at least in specific parts of the body, are underway. You can follow the progress of each research targeting each hallmark by using the Rejuvenation Roadmap made by the Life Extension Advocacy Foundation. This map tracks the progress of research projects that ameliorate each hallmark and provides links with explanations of each project; it also contains citations to the relevant papers.

As you can see, there are some hallmarks, such as mitochondrial dysfunction and loss of proteostasis, which are in the very early stages of research: the furthest they have reached, so far, is the preclinical stage. Research on how to ameliorate mitochondrial dysfunction, in particular, is in such an early stage of research that it is only pursued by nonprofits and academia, but it needs to be addressed in the wider scheme of therapies that will be needed in order to address all of the dysfunction arising from aging.

There are other hallmarks, such as cellular senescence and stem cell exhaustion, which are in fairly advanced stages of research (phase 1 and phase 2 trials), and research on them is pursued by well-funded, for-profit companies, such as Unity Biotechnology.

The fact that all of these lines of research are pursued in parallel is important. It means that at an unspecified time in the future, near or far, lines of research could come together in a relatively short period of time. The fact that right now, many interventions are being researched on specific diseases (e.g. Unity Biotechnology's trial is for arthritis) does not negate the previous point: treatments that are being researched using the Hallmarks framework, even though they are being tested for specific conditions, are relevant for therapies that treat a wide range of diseases. Parallel development makes it more likely that therapies will come in waves, with each therapy being released shortly after another.

There are also other approaches in aging research, such as targeting aging in a more upstream fashion, with less ambitious interventions that target metabolic pathways. One example is metformin, although I don't think that, right now, science is advanced enough for research on specific medical interventions using this approach to substantially make the date of LEV come closer or substantially impact its probability. These kinds of research projects, nonetheless, could have the effect of buying some time for an additional slice of the population to reach LEV. This brings us to another way of accounting impact in this cause area.

Accounting for making an additional slice of the population reach LEV

Another factor potentially greatly influencing impact is the life expectancy increase resulting from research projects or health interventions. If the project is not likely to be funded in the future or subsumed by other research, the recipients of the intervention who would have died near LEV get saved. I think the health interventions or projects for which this factor is relevant are very few or maybe even non-existent. This consideration influenced the impact measure I used in my previous analysis on the TAME trial, but in retrospect I think I overestimated the probability that the health benefit of metformin will not be subsumed by other research.

In order to account for this, the relevant factors to multiply are:

• Life expectancy after LEV.
• Recipients of the interventions who would have died just before LEV if their life expectancy wasn’t extended by the intervention.
• Probability that the project will not be funded by someone else, or is subsumed by other research.

QALYs should be the measure of impact

Due to the possibility of LEV, expected QALYs should be the measure of impact of aging research. Lives saved lose their original meaning, unless 1 life of 1000 QALYs is counted as multiple lives of 30-80QALYs. Exactly how many also depends on how moral weights are chosen. In my previous analysis about the cost-effectiveness of the TAME trial, I made the mistake of measuring impact in lives saved instead of directly in QALYs, without considering the fact that a life saved in that context amounted to 1000 or more QALYs and actually counted as multiple lives saved. In that analysis, I also didn't account for DALYs averted at the end of life and every other factor that influences impact, which I will discuss in future posts.

How LEV spreads will have no impact on CEAs

A concern sometimes comes up when I present LEV-based reasoning: how do we account for the fact that LEV will probably spread to the whole population over a large period of time (e.g. following the sigmoid technology adoption curve)? This consideration has no effect on the final estimate of cost-effectiveness, and making the date of LEV closer by any given period of time prevents exactly the number of deaths by aging occurring during that period of time, regardless of how LEV spreads. Let's first see a simple example where this holds and then prove it mathematically in the general case. I came up with two proofs, each one of which is sufficient alone.

These same arguments and proofs work for any other outcome of a given technology. How a certain technology (health-related or not) will spread doesn't influence the cost-effectiveness of financing the research leading to it. I may include the generalised version, which trivially follows from this one, in a separate post.

Keep in in mind that this does not mean that making LEV spread faster doesn’t impact CEAs. In fact, this is a potential impact factor that I will discuss. This result means that the impact of making the date of LEV come closer isn't influenced by how LEV spreads.

I will use deaths prevented in the example and in the proofs, but a generic measure of impact yields the same result. Using QALYs is not necessary in this case.

Example

Let's consider two specific scenarios as an example: in the first scenario, LEV spreads to the whole population instantly, and in the second, it spreads over four years.

First scenario: A particular piece of research makes LEV come closer by one year. Since LEV spreads instantly over the whole population, it's easy to see that the resulting deaths prevented are exactly the deaths by aging occurring during one year: more or less 100k.

Second scenario: A particular piece of research makes LEV come closer by one year, but LEV spreads over the world during a period of four years. In the first year, 1/4 of the population reaches LEV; in the second year, 1/2; in the third, 4/5; and in the fourth, 5/5. If we shift this gradual transition by one year, then in the first year, we prevent, on the margin (deaths that would have occurred if we didn't move the date), 1/4-0 = 1/4 of the deaths of aging occurring during that year. In the second year, we prevent 1/2-1/4 = 1/4 of the deaths by aging that occur during that year. In the third year, we prevent 4/5 - 1/2 = 3/10 of the deaths by aging occurring during that year. In the fourth year, we prevent 5/5 - 4/5 = 1/5 of the deaths by aging occurring during that year. So, in total, by shifting the date of LEV by one year, we prevented: 1/4 + 1/4 + 3/10 + 1/5 = 1. That is, we prevented the deaths by aging occurring during one year: more or less 100k.

As you can see, the number of deaths prevented in the two scenarios is the same: the number of deaths by aging occurring during one year. LEV is moved closer by one year in both scenarios, but it spreads differently.

Now, I'll prove, more generally, that making LEV closer by any given period of time prevents exactly the number of deaths by aging occurring during that period of time, regardless of how LEV spreads.

Proof 1:

$n=$ the number of years needed for therapies to spread to the whole population.

$y=$ the year in which the therapies leading to LEV begin spreading.

$d=$ number of deaths caused by aging each year ($d$ could be the number of deaths by aging occurring in any arbitrary unit of time; the proof remains the same).

$f:\{y,...,y+(n-1\left)\right\}\to [0,d]$ expresses how many deaths from aging are prevented in a given year during the time that therapies are spreading. How exactly $f$ is defined depends on how the therapies spread (e.g. exponentially or linearly), but we know that $f(y+n-1)=d$ and that $f\left(y\right)=0$.

If LEV spreads to the whole population all at once, then $n=1$ and $f:\left\{y\right\}\to \left\{d\right\}$. In this case if the date of LEV is moved closer by $1$ year, then the resulting new function $f^{\left(1\right)},$ has $y-1$ as the only member of its domain, also mapping to $d$. So the deaths prevented on the margin by making the date of LEV closer by one year are exactly $d$.

We want to prove for all values of $n$ and $y$ that if the date of LEV is moved closer by one year but the therapies do not spread to the whole population all at once, the number of deaths prevented on the margin still amounts to $d$.

Let $f^{\left(1\right)}$ be the function that expresses deaths by aging prevented each year after making LEV come closer by one year and $f$ the (already defined) function that expresses deaths by aging prevented each year without LEV being moved closer. Therefore $f^{\left(1\right)}$ has the following properties:

$f^{\left(1\right)}:\{y-1,...,y+n-2\}\to [0,d]$

$f^{\left(1\right)}(y-1)=f\left(y\right),f^{\left(1\right)}\left(y\right)=f(y+1),...,f^{\left(1\right)}(y+n-2)=f(y+n-1)$, this holds under the very solid assumption that making the date of LEV closer only shifts $f$: it doesn't change how it is defined, but only subtracts $1$ to all the members of its domain.

Then, the deaths by aging prevented each year on the margin if we make LEV come closer by one year are:

$f^{\left(1\right)}(y-1)+\left[f^{\left(1\right)}\right(y)-f^{\left(1\right)}(y-1\left)\right]+...+f^{\left(1\right)}(y+n-3)+\left[f^{\left(1\right)}\right(y+n-2)-f^{\left(1\right)}(y+n-3\left)\right]=$

$=\left[f^{\left(1\right)}\right(y-1)-f^{\left(1\right)}(y-1\left)\right]+\left[f^{\left(1\right)}\right(y)-f^{\left(1\right)}(y\left)\right]+...+\left[f^{\left(1\right)}\right(y+n-3)-f^{\left(1\right)}(y+n-3\left)\right]+f^{\left(1\right)}(y+n-2)=$

$=f^{\left(1\right)}(y+n-2)=f(y+n-1)=d$

$■$

Note that the same exact proof works if the date of LEV is moved closer by more or less than one year: It is sufficient to let $d$ be the deaths by aging prevented in an arbitrary unit of time. Another proof, with $f$ having a continuous domain, involves manipulating integrals. Here it is:

Proof 2:

Let $f\left(t\right):R\to R$ be the function that associates time with deaths by aging prevented at that time. Then, the total number of deaths prevented in a given time interval $[t_0,t_1]$ is ${\int }_{t_0}^{t_1}f\left(t\right)dt$ . The number of deaths averted on the margin if we make the date of LEV come closer by the time interval $\Delta t$ is:

${\int }_{t_0}^{t_1}f(t+\Delta t)dt-{\int }_{t_0}^{t_1}f\left(t\right)dt$

Let's divide the interval $[t_1,t_{n+1}]$ in $n$ smaller periods of time of length $\Delta t$ (the period of time LEV is moved closer by). Let's call those subintervals $\Delta t_i=[t_i,t_{i+1}]$. Then the above integral can be rewritten as a sum of integrals over the smaller intervals.

${\sum }_{i=1}^n\left({\int }_{t_i}^{t_{i+1}}f\right(t+\Delta t)dt-{\int }_{t_i}^{t_{i+1}}f(t\left)dt\right)$

But since it's true that:

${\int }_{t_i}^{t_{i+1}}f(t+\Delta t)dt={\int }_{t_{i+1}}^{t_{i+2}}f\left(t\right)dt$

Then, the terms of the sum simplify with each other and we have:

${\sum }_{i=1}^n\left({\int }_{t_i}^{t_{i+1}}f\right(t+\Delta t)dt-{\int }_{t_i}^{t_{i+1}}f(t\left)dt\right)={\int }_{t_n}^{t_{n+1}}f(t+\Delta t)dt-{\int }_{t_1}^{t_2}f\left(t\right)dt$

Notice that if $t_1$ happens one unit of time before LEV begins spreading and $t_{n+1}$is the time at which LEV has reached the whole population, then ${\int }_{t_1}^{t_2}f\left(t\right)dt=0$ and ${\int }_{t_n}^{t_{n+1}}f(t+\Delta t)dt$ is exactly the number of deaths by aging that would have occurred in the time interval $\Delta t$; this is exactly the number of deaths by aging prevented if LEV was moved closer by $\Delta t$ and the therapies spread instantly. This proves that the number of deaths by aging prevented on the margin by moving the date of LEV closer by $\Delta t$ is always equal to the number of deaths by aging occurring during $\Delta t$, regardless of how the therapies spread over the world.

$■$

Accounting for making LEV spread faster

As anticipated, another potential source of impact to consider is if a certain project, for example an advocacy-related or policy change, can change how fast people get access to treatments. This would make LEV spread faster, and, in turn, save the people who otherwise wouldn’t reach it.

In order to evaluate this, we need to come up with an estimate of how the future adoption curve will look like. This could possibly be achieved by looking at the way that current health treatments spread, and then evaluate how many people, and, in turn, QALYs, get saved by a change in the curve resulting from the project.

This consideration will probably be given more space in a separate post investigating current adoption curves for health-related technologies and the impact of advocacy and inducing policy change in this area.

Accounting for increasing the probability of LEV

Another consideration that may be taken into account to evaluate impact is how much a given research project increases the probability of LEV. This doesn't mean increasing the probability of aging getting eradicated completely; it means increasing the probability of research being fast enough to ensure LEV and not a scenario in which research is so slow, or roadblocks so dire, that aging eventually gets eradicated but no one experiences LEV in the meantime.

This probably depends much on the project in question, but it is also possible that, in general, the impact of this consideration could be small, unless we assume a really inefficient research community, or we are analysing a specific research project that is highly neglected and has the potential effect of removing a major roadblock or unlocking further progress, thus speeding up the general pace of research and making the risk of death for future recipients of interventions fall faster. This could be true for research on new scientific tools or research done in a particularly original way that shows a new approach to problems that wasn't seen before.

In case this consideration has to be taken into account and we need to calculate how many QALYs that increasing this probability would save, we should come up with a distribution of probabilities (with the sum of the probabilities = 1 - the probability of LEV not happening) about how fast the risk of death would fall after LEV. Each outcome yields a different number of QALYs saved by LEV. Then, we should calculate the expected value in QALYs of the distribution with an increased total probability of LEV and the expected value in QALYs of the distribution without an increased total probability of LEV, then determine the difference between the two results.

Aging research boosts the impact of other altruistic interventions

It should be noted that another effect of aging research is to increase the chance for people saved by other interventions to reach LEV. If a life in Africa is saved thanks to insecticidal nets, then the expected QALYs saved will be more or less the person's expected remaining life plus his/her life expectancy after LEV in QALYs (more or less 1000 as we have seen) multiplied by the probability that person has to achieve LEV during the rest of his/her life.

The probability of an individual reaching LEV depends on:

• The probability of dying of causes not related to aging.
• The probability of LEV arriving in the lifetime of the recipient of the intervention.

The first depends on the recipients of the intervention, but even in the worst cases, it should be a pretty high number, considering that even in Africa, the lowest average life expectancy for children born in 2018 is 57 years. The second probability depends on how likely LEV is to appear in any given year. In order to determine this, a very thorough and detailed analysis is needed, and so I will probably tackle this problem in another post.

Crossposted to LessWrong after the publication here.