Relation to the economics and actuarial literature

Draft section for the evaluation manager’s commentary on “The Value of Existential Risk Mitigation” · 28 October 2026 · David Reinstein

WarningDraft, under review

AI-generated draft, not yet checked by an evaluation manager. See “Status of this draft” below. Comments are welcome as a Hypothes.is annotation or by email.

Status of this draft

AI-generated. Claude (Anthropic) drafted this section on 28 October 2026 at the request of an evaluation manager. It is not part of the anonymous evaluation, and the evaluator has not seen it. We checked the bibliographic details and abstracts of the works cited, but did not reread every paper in full. An evaluation manager should verify each claim against its source before this is published.

What it is for. Suggested extensions and connections for the author and readers, not errata. Nothing here says the report’s results are wrong. The point is that older literatures in economics, actuarial science and health economics have worked through closely related problems, and some of their results would sharpen or change the report’s conclusions.

Summary

Much of the report’s mathematics was worked out decades ago, in actuarial science (life annuities), in the economics of uncertain lifetimes, and in health economics (valuing reductions in mortality risk). The report cites almost none of it. Re-deriving known results costs the report little. What it misses is that some of these literatures go further, and the further results matter even for an impartial, risk-neutral approach like the report’s.

Three extensions stand out:

  1. Uncertainty about the level of risk. When long-run risk is uncertain, expected value is driven by the most optimistic risk scenario taken seriously, not by the average risk. Weitzman (1998) and Sozou (1998) showed this for discount rates and hazard rates.
  2. Several risks at once. The value of reducing one extinction risk depends on the others, so values don’t simply add up across risks or interventions (Dow, Philipson and Sala-i-Martin 1999; Martin and Pindyck 2015).
  3. Making risk depend on choices. Growth models in which the “time of perils” emerges from choices about technology and safety (Aschenbrenner 2020; Jones 2024), as opposed to being drawn by hand.

The first two also bear on Rethink Priorities’ Cross-Cause model, whose catastrophic-risk module cites this report as an influence (last section).

Scope and key concepts

The report takes an impartial, risk-neutral view: a life or a year of value counts the same whenever it occurs, and outcomes are ranked by expected total value. We keep to that view here. We set aside economists’ reasons for weighting the present more heavily, and flag where a result depends on departing from risk neutrality.

Economists use several terms that sound alike but mean different things. The distinctions matter for which results carry over.

Term What it means Compatible with an impartial, risk-neutral view?
Pure time preference Valuing a given amount of wellbeing less because it comes later, or to people born later. No. This is the discounting the report, Broome and most effective altruism writing reject.
Hazard rate The probability that an event (death, extinction) happens in a period, given that it hasn’t happened yet. The report’s r_t is a hazard rate. Yes. It’s a probability, not a value judgment.
Survival function The probability of lasting to time t. It’s the product of (1 − hazard) over all earlier periods. This is the chain rule of probability, so it does not assume the hazards are independent. Yes.
Discounting for extinction risk Weighting later value by the probability it occurs at all. Mathematically identical to discounting at the hazard rate. The Stern Review (2006) set its whole 0.1% annual discount rate this way. Yes. This is the only discounting the report uses.
Life annuity A payment stream that stops at death. Its expected present value under a constant hazard is the same sum as the report’s expected value of the future. Yes. Pure arithmetic.
Diminishing marginal utility of consumption An extra dollar adds less wellbeing to a rich person than a poor one. Economists usually summarize it with a curvature parameter (often written η). Yes. It’s a claim about how wellbeing depends on resources, not about time or people. It matters for non-extinction catastrophes that make future people poorer.
Risk aversion over total value Preferring a sure outcome to a gamble with the same expected total. No. A risk-neutral expected-value view rejects it by definition. Related to, but not the same as, Rethink Priorities’ risk profiles (see below).
Risk aversion over time (also called intertemporal or correlation aversion) Caring how outcomes are spread across time and across possible histories. For example, preferring a world where every generation has a 50% chance of a good life over one where there’s a 50% chance all generations do well and 50% that none do. No, not under strict risk neutrality, although it’s compatible with treating all generations equally. It is not a preference for present over future people.
Competing risks Several hazards acting at once; you can only die (or go extinct) once. Yes.

Does “risk aversion over time” mean favoring present people? No. Bommier and Zuber (2008) show that a society can behave as if it discounts the future, with no pure time preference at all, if it is averse to catastrophic risk to the whole sequence of generations. That is a departure from risk neutrality, not from impartiality.

Is it the same as valuing population size less? Not quite. Bommier and Zuber evaluate an uncertain future roughly as the expected value of φ(total welfare across all generations), where φ is concave: each extra unit of total counts for a bit less.

  • When the only uncertainty is the extinction date, total welfare is proportional to the number of generations, so this amounts to risk aversion over how long humanity lasts and how many people ever live.
  • Number-dampened or critical-level population ethics, as in Méjean and coauthors (extension 4), is different. It changes how certain populations are ranked; for example, ten billion people might count for less than twice five billion.
  • Applying an increasing φ to a sure total leaves the ranking of sure outcomes unchanged. It only changes how gambles are weighed.
  • In practice both views reduce the pull of small chances of astronomically large futures.

Is that a principled position? It’s contested, and hard to justify from an impartial utilitarian starting point:

  • Harsanyi’s aggregation theorem: if social choices follow expected-utility theory and respect every individual’s own prospects (ex ante Pareto), social welfare is the expected sum of individual welfare, which is risk-neutral in the total. A concave φ has to give up one of those premises.
  • It makes value non-separable. Whether adding a generation is worth much depends on how many other generations exist, including people long dead. This is a version of the “Egyptology” objection to average utilitarianism.
  • Nothing pins down how curved φ should be, or what exactly it is applied to, so results rest on a parameter with no empirical anchor.
  • Its defenders appeal to intuitions about catastrophe avoidance (preferring separate risks to each generation over one correlated risk that ends everyone) and to avoiding fanaticism.

We include it for readers who give some weight to risk aversion. The report’s framework doesn’t need it.

Not the same as Rethink Priorities’ risk profiles. The Cross-Cause model’s risk profiles apply to the distribution of each fund’s impact, that is, aversion to risk in the difference a donation makes. Bommier and Zuber’s aversion is about how the world as a whole turns out. We come back to this distinction in the last section.

What the report re-derives

Each of the report’s formal results has a close, older counterpart. Citing them would let readers see which results are standard and which are new.

Report Older counterpart Comment
Expected value of the future = value per period ÷ per-period risk (constant case) Value of a life annuity at a constant force of mortality. In economics: Yaari (1965) on uncertain lifetimes; Blanchard (1985), where a constant probability of death acts exactly like an extra discount rate. Same formula.
Closed forms for linear, quadratic and cubic value growth (section 6 and the polylogarithm derivations) Increasing annuities in actuarial mathematics, e.g. the sum of t × v^t equals v ÷ (1 − v)². Correct, but standard.
Convergence (Lemma 1, Proposition 1): finite if risk stays above a positive floor and value grows more slowly than exponentially Finiteness conditions throughout economics. The Gordon growth formula (value ÷ (discount rate − growth rate)) requires growth below the discount rate. Catastrophe models such as Martin and Pindyck (2015) state the same kind of condition. With constant risk and exponential value growth, the report’s sum is the Gordon formula.
Persistence: risk cut by a fraction f for P years A treatment that lowers the hazard rate by a fixed proportion for a limited time. Health technology assessment calls the end of such an effect “treatment effect waning”, and has methods and evidence on how long effects last. Offers a template for calibrating P from evidence.
The simple rule E(M) ≈ f × cumulative risk over the window × expected continuation value The standard valuation of a reduction in mortality risk: the change in survival probability times the value of the life that continues. See Rosen (1988) and Murphy and Topel (2006). Florian Habermacher raised this rule independently. It reproduces the report’s worked example (28.5 against 28.6).
Higher background risk lowers the value of reducing any one risk (Thorstad’s “pessimism” result, which the report builds on) Competing risks. Dow, Philipson and Sala-i-Martin (1999) show that reducing one cause of death raises the value of reducing the others. The same mechanism, seen from the other side.

Extensions the older literature suggests

These are ordered by how much they matter within the report’s own impartial, risk-neutral framework.

1. Uncertainty about the level of risk

The idea. The report treats each risk path as known. Suppose instead we’re unsure which path we’re on. Expected value is then the average of the values across paths, not the value at the average risk. Because value is a convex function of risk (roughly 1 ÷ risk), the low-risk paths dominate the average.

This is not a risk attitude. A risk-neutral decision-maker maximizes expected value, and when risk is uncertain the expected value is the average across paths. So this is what risk neutrality itself requires. Using a single best-guess risk instead is the error, and it matters most for exactly the risk-neutral reader.

A toy example. Value is constant, and long-run annual risk is either 1% or 0.01%, with equal probability.

  • Expected years of survival: 99 under 1% risk and 9,999 under 0.01% risk, so the average is about 5,050.
  • Plugging in the average risk (0.505%) instead gives about 197.
  • The correct answer is about 26 times larger, and almost all of it comes from the optimistic scenario.

Try it. Enter long-run annual risk levels and their probabilities. The page computes expected value at each level (value from year 1 on, as in the report, so constant value gives (1 − r) ÷ r survival-years), the average of those values, and the value at the average risk.

The older literature. Weitzman (1998) showed that when the discount rate is uncertain, the effective long-run rate falls toward the lowest plausible value, with the debate continuing in Gollier and Weitzman (2010). Sozou (1998) showed the same for an uncertain hazard of death, and that it produces hyperbolic discounting: discount rates that fall with the horizon.

Does the report make this error? Not directly. It never averages values at an average risk, because it doesn’t model uncertainty about risk at all: each scenario fixes a known path, and the report names the absence of uncertainty analysis as its main caveat. The gap shows in how the results get used:

  • A reader who picks one best-guess path is implicitly plugging in a point estimate. For example, the worked example uses 0.22% annual risk, calibrated to a one-in-five chance of catastrophe this century.
  • The pessimism result (high risk means low value of mitigation) assumes we’re confident risk stays high. With uncertainty, the low-risk scenarios dominate the expectation, and the pessimistic conclusion weakens considerably.
  • The evaluator’s “fixed and independent” point is really this one: the binding restriction is that risk is known, not that it’s independent.

Rethink Priorities’ Cross-Cause model, by contrast, does draw long-run risk from a distribution, so it computes the average correctly (see the last section).

Because this is pure probability, with no time preference and no risk aversion, it applies fully within the report’s approach.

Suggestion. Add a version of the scenario grid that averages over a distribution of long-run risk, and report how sensitive results are to the lower end of that distribution.

2. Several risks at once: competing risks and non-additivity

The idea. Humanity can only go extinct once. If two independent risks each have a 10% chance of ending things this century, survival is 0.9 × 0.9 = 81%.

What is eliminated Survival Gain in survival
Nothing 81% —
Risk A only 90% 9 points
Risk B only 90% 9 points
Both 100% 19 points, not 9 + 9 = 18

Eliminating risk A is worth more once B is gone (10 points instead of 9). For extinction risks, interventions are complements: each makes the others more valuable. This is the same mechanism as the pessimism result, run in reverse.

Try it. Set each risk’s chance of ending humanity this century (independent risks).

The older literature. Dow, Philipson and Sala-i-Martin (1999) establish this complementarity for causes of death and test it with immunization data. Martin and Pindyck (2015) analyze several catastrophes that permanently cut consumption, rather than ending everything. There, diminishing marginal utility of consumption can reverse the sign: averting one catastrophe can make averting another less valuable. Willingness to pay to avert several catastrophes then doesn’t add up, and the best set of projects has to be chosen jointly, not one at a time.

Why it matters here. Within the report’s framework, the interaction terms are products of two risk reductions. They are small for marginal interventions, but not for large ones or across a portfolio. The consumption-loss case matters for impartial welfare whenever catastrophes leave survivors poorer. Diminishing marginal utility is a fact about wellbeing, not a time preference.

Suggestion. Note explicitly that E(M) for one intervention is conditional on the other risks being held fixed, and that values can’t be summed across interventions that target different risks.

3. Risk that depends on choices

The idea. The report’s Time of Perils profile is drawn by hand. Aschenbrenner (2020; updated 2024) gets a time of perils from a growth model: risk rises as technology advances, then falls as societies grow rich enough to spend on safety. Jones (2016, 2024) models the trade-off between faster growth and safety, including from AI.

Why it matters here. If risk depends on choices, an intervention can change the whole future risk path, not just cut risk by a fraction f for P years. That is the endogenous, uncertain-sign case the evaluator says the model doesn’t address. The report cites both works but doesn’t use them. A worked example with a path derived from one of these models would connect the persistence analysis to a mechanism.

4. Population ethics within welfare economics

The idea. The report takes the value sequence v_t as given. Welfare economists have formal frameworks for valuing populations of different sizes: Blackorby, Bossert and Donaldson (2005) for the theory; Méjean, Pottier, Fleurbaey and Zuber (2020) for an application. The latter put extinction risk into a climate-economy model and rank policies under a family of population-ethics views that includes total utilitarianism as a special case.

Why it matters here. Méjean and coauthors find that the preferred policy depends on how much weight goes to population size and to inequality, and the ranking can reverse. Citing this would show readers which of the report’s conclusions survive outside total utilitarianism.

5. Evidence on persistence

The idea. The report hypothesizes that interventions rarely keep working beyond about 50 years, and presents this as a hypothesis. Health technology assessment faces the same question for drugs (how long does a treatment effect last after the trial?) and has methods for modeling and testing it.

Suggestion. Borrow that practice: state the persistence assumption as a waning curve, show results under several curves, and look for analogues (policy and institutional persistence studies) to calibrate P.

6. Beyond risk neutrality

For readers who give some weight to risk aversion over total value, Bommier and Zuber (2008) and Bommier, Lanz and Zuber (2015) show that the choice of preference model can change the value of reducing collapse risk by a large factor, even with no pure time preference. This lies outside the report’s framework, but it’s the formal backing for the risk-aversion options in Rethink Priorities’ Cross-Cause model.

What the report adds

The report’s contribution is applied, not mathematical:

  • It turns scenarios from the longtermist philosophy literature (Time of Perils, Great Filters) into explicit hazard profiles.
  • It compares persistence systematically across those profiles and several value-growth paths.
  • It provides a notebook so readers can run their own scenarios.

Its intended readers are people working in effective altruism and global priorities, and the papers it extends (Ord, Adamczewski, Thorstad) were answering a debate within philosophy. Some of the re-derivation is translation for that audience. A short related-literature section would let it keep that role while pointing readers to where these questions have been taken further.

Relevance to Rethink Priorities’ Cross-Cause model

Exploratory and preliminary. This section applies the literature above to Rethink Priorities’ publicly linked methodology documents, as they stood on 28 October 2026 (both are living documents). It was largely AI-generated. Several points are inferences from published numbers that would need the model’s code to confirm, and are offered as questions.

Rethink Priorities’ Cross-Cause Fund and Donor Compass use a model whose catastrophic-risk module is based on Tarsney (2023) and cites this report as an influence. Of the extensions above, uncertainty about the level of risk matters most for that model. Non-additivity probably matters little at current donation sizes for risk-neutral worldviews, with two exceptions worth checking.

Uncertain long-run risk: matters a lot

What the model does. Unlike the report, the model draws long-run annual existential risk from a distribution in each of its one million simulations: central value 8 × 10^-6, range 10^-8 to 5 × 10^-5. So it averages over risk uncertainty, which is right in principle.

The consequence. As in the toy example in extension 1, the average is dominated by the lowest-risk draws:

  • The model’s mean life-years per $1 million for the catastrophic-risk funds are around 10^19 to 10^20, while their 99th percentiles are around 10^16. So the mean is set by a small fraction of draws, presumably those with very low long-run risk and successful space settlement.
  • In a simple version with cubic value growth, expected value scales with 1 ÷ risk^4. Halving the lowest plausible risk would multiply those draws’ value by 16, unless something else caps it.
  • The lower end of the long-run risk range (10^-8) and the distribution’s shape near it are therefore effectively the most important parameters in the module. That’s a judgment, not an estimate, and worth stating as such.
  • The same tail dominance makes the mean hard to estimate by simulation. Results can shift noticeably between runs unless sample sizes are very large or the tail is sampled deliberately.
  • The documentation describes the risk distribution as implying “an average civilizational lifespan of about 120,000 more years”. That is 1 ÷ average risk, the plug-in calculation the toy example warns against. The expected lifespan is much longer.

Does it matter for allocations? It depends on the aggregation method.

  • Under the moral marketplace, each worldview gets a budget share and spends it on its top fund. Once the catastrophic-risk funds win by orders of magnitude for a worldview, larger numbers change nothing.
  • Under maximizing expected choiceworthiness, scores are averaged across worldviews. There, the size of the catastrophic-risk scores determines how far they swamp everything else, so the lower tail of long-run risk feeds straight into the allocation.
  • Under the risk-averse profiles, the upside trimming above the 97.5th percentile removes exactly these draws. So risk-neutral and risk-averse worldviews will disagree sharply, and the disagreement comes down to a single parameter.

Non-additivity: probably second-order, with two exceptions

What the model does. It scores each fund separately, then allocates in $2 million (or $10 million; the documents differ) steps with fund-specific diminishing returns. Values are treated as additive across funds.

Why that’s mostly fine. For risk-neutral worldviews, the interaction between two interventions is the product of two tiny risk reductions. At donations of millions of dollars, it should be negligible next to each fund’s own effect.

Exception 1: the background risk each fund sees. The value of cutting one risk does depend, to first order, on the level of the other risks: a biosecurity success is worth less if AI risk is high. Does each fund’s model include the other causes’ peak risks in its survival calculation, from the same simulation draw? If each fund’s model only includes its own cause’s risk, values are overstated and not consistent with each other.

Exception 2: correlated outcomes under risk aversion. The three catastrophic-risk funds share uncertain inputs, such as the value of the future, long-run risk and the probability of space settlement, so their outcomes are highly correlated. The risk profiles are applied to each fund’s own distribution. Two possibilities:

  • If risk aversion is meant to be about each donation’s own impact (“difference-making” risk aversion), applying it fund by fund is consistent.
  • If it’s meant to be about how the world as a whole turns out, the relevant distribution is the portfolio’s total. Correlated funds then hedge each other less than separate scoring implies. This is the lesson of Martin and Pindyck (2015): evaluate jointly.

The documentation should say which interpretation is intended.

Non-existential catastrophes. The model also values catastrophes that kill large numbers without ending humanity. For those, Martin and Pindyck’s result applies directly: when catastrophes leave survivors poorer, averting one can lower the value of averting another. This is probably second-order at current scale too.

References

Links were checked on 28 October 2026. Works marked with an asterisk were cited from standard knowledge and not looked up for this draft; verify before publishing.