Formulas and parameters

Every equation the tool computes, in the report’s notation, with defaults and sources

1. The report’s model

Expected value of the world (report equation 1)

A world $w$ faces per-period extinction risk $r_t$ and, if it survives period $t$, realizes value $v_t$. With periods $t = 1, \ldots, T$:

\[ \mathbb{E}(w) = \sum_{t=1}^{T} \left[ \left( \prod_{j=1}^{t} (1 - r_j) \right) v_t \right] = \sum_{t=1}^{T} S_t v_t, \qquad S_t \equiv \prod_{j \le t} (1 - r_j). \]

Engine: survival(r), expectedValue(r, v). The function also returns the truncated sums $\mathbb{E}_H(w) = \sum_{t \le H} S_t v_t$ for every horizon $H \le T$, which the horizon charts plot.

Mitigation (paper section 2, “persistence”)

An action $M$ with efficacy $f$ and persistence $P$ changes the risk path to \[ r'_t = (1 - f)\, r_t \quad \text{for } t_0 \le t < t_0 + P, \qquad r'_t = r_t \text{ otherwise}, \] with $t_0 = 1$ in the paper (no delay). Its value is the counterfactual difference \[ \mathbb{E}(M) = \mathbb{E}(w') - \mathbb{E}(w). \] The paper allows $f \le 0$ in principle (p. 11) but analyses only $f \in (0, 1)$. Engine: mitigateRisk(r, {f, start, P}).

A useful identity (Thorstad; verified in the tests): with constant risk and value and $P = 1$, $\mathbb{E}(M) = f v_c$ exactly, whatever the risk level.

Value cases (paper Table 1 and section 11)

Case	Paper (centuries)	Tool (years, $\alpha = 1/100$)
V0 constant	$v_c$	$1$
V1 linear	$i\,v_c$	$(\alpha t + 1 - \alpha)$
V2 quadratic	$i^2 v_c$	$(\alpha t + 1 - \alpha)^2$
V3 cubic	$i^3 v_c$	$(\alpha t + 1 - \alpha)^3$
V4 logistic	$c / (1 + \frac{c-s}{s} e^{-\gamma t})$	same, with $s$ chosen so that $v_1 = 1$

$v_c = 1$ throughout. The adjustment $v(t) = (\alpha t + (1-\alpha)) v_c$ (paper section 11) makes the year-indexed sequence pass through the century-indexed one at $t = 1, 101, 201, \ldots$. Logistic defaults from the notebook: $c = 10^6$ (“what obtains after 10,000 years of cubic growth”), $\gamma = 0.03$ per year. The paper’s equation 6 gives a different, Weibull-like logistic form $v_t = v_c(c(1 - e^{-(t/a)^b}) + 1)$; it is used only for Figure 2 and is not implemented here.

Risk cases (paper sections 3 and 4)

Case	Definition	Defaults
Constant	$r_t = r_c$	$r_c = 0.0022289477$
Time of Perils	$r_t = r_{high}$ for $t \le L$, else $r_{low}$	$L = 100$, $r_{high} = 0.0022289477$, $r_{low} = 0.0001$
Great Filters	$F$ eras of $r_{high}$ (length $\ell_d$) alternating with eras of $r_{low}$ (length $\ell_s$), last era open-ended	$F = 2$, $\ell = (100, 500, 100, T - 700)$
Exponential decay	$r_t = r_0 e^{-\lambda (t-1)} + r_\infty$	$r_0 = 0.0022289477$, $\lambda = 10^{-4}$, $r_\infty = 10^{-9}$

The per-year constants come from per-century probabilities: $1 - 0.8^{1/100}$ (20% per century) and, for the low value, $0.0001$ taken directly (paper footnote 17 notes $(1 - 0.0001)^{100} \approx 0.990$).

Convergence (paper section 5)

Lemma 1: if $\lim r_t$ exists and is positive, $\mathbb{E}(w)$ converges whenever $\lim |v_{t+1}/v_t| \le 1$, which holds for all polynomial and logistic value paths. The tool does not need the infinite-horizon result; it always sums to a finite $T$ and reports what is left out (next section). The ratio-test quantity is shown as a note when uncapped exponential value is selected: the series converges as $T \to \infty$ only if $(1 + g)(1 - r_\infty) < 1$.

Closed forms under constant risk (paper Table 2, appendix 12)

With $q = 1 - r_c$ and $L_n = \sum_{t \ge 1} t^n q^t$ (polylogarithms $\mathrm{Li}_{-n}(q)$): $L_0 = q/r_c$, $L_1 = q/r_c^2$, $L_2 = q(2 - r_c)/r_c^3$, $L_3 = q(1 + 4q + q^2)/r_c^4$. Then, for $v_t = (\alpha t + b)^n$ with $b = 1 - \alpha$, \[ \mathbb{E}(w) = v_c \sum_{k=0}^{n} \binom{n}{k} \alpha^k b^{\,n-k} L_k . \] Engine: closedFormConstantRisk(type, r_c, alpha). Checked against the numeric sum to $10^{-9}$ in the tests.

2. Beyond the paper

Risk floor and truncation error (switch 1)

With the floor on, $r_t \leftarrow \max(r_t, c)$. Since $S_t \le (1 - c)^t$, the survival probability is below $\varepsilon$ after $H_\varepsilon = \ln \varepsilon / \ln(1 - c)$ years (approximateSupport(c, eps)). For any horizon $H$ the value left out satisfies \[ \sum_{t > H} S_t v_t \;\le\; \sum_{t > H} (1 - c)^t v_t , \] which has a closed form for polynomial value (tailBound): writing $q = 1 - c$ and $A_0 = 1/(1 - q)$, $A_j = \mathrm{Li}_{-j}(q)$, \[ \sum_{t > H} q^t t^k = q^{H+1} \sum_{j=0}^{k} \binom{k}{j} (H + 1)^{k-j} A_j , \] and the polynomial $(\alpha t + 1 - \alpha)^n$ is expanded binomially. For logistic and hard-capped value the bound uses the cap times the geometric tail; for uncapped exponential value it is $\sum_{t>H} (q(1+g))^t / (1+g)$ when $q(1+g) < 1$ and infinite otherwise. When the floor is off, the tool uses the smallest risk in the (mitigated) path as $c$, which is still a valid bound. The horizon table reports, for $H \in \{100, 500, 1000, 10^4, 10^5\}$, the share of $\mathbb{E}(w)$ and of $\mathbb{E}(M)$ realized within $H$ and beyond $H$ (relative to the total at $T$), and the bound on what lies beyond $T$ itself.

Sign-uncertain mitigation: Monte Carlo (switch 2)

Draws $i = 1..n$ with a seeded Mulberry32 generator:

$f_i \sim \mathrm{Normal}(\mu_f, \sigma_f)$, clipped at 1. Negative draws raise risk during the window.
With probability $p_b$, a delayed backfire: $r_t \leftarrow (1 + b_i) r_t$ for $t \in [t_b, t_b + P_b)$, with $\ln b_i \sim \mathrm{Normal}(\ln b_{med}, \sigma_b)$.
Optional rebound multiplier $m_i = 1 + \beta f_i P / D$ (or $1 + \beta f_i$ if permanent) on the $D$ years after the window.

Each draw’s $\mathbb{E}(M)$ is exact. Because $S'_t / S_t = \prod_{j \le t} (1 - r'_j)/(1 - r_j)$ only changes inside the modified windows, $\mathbb{E}(w')$ is a sum of window terms (computed per draw) plus constants times precomputed tail sums. With rebound on, the post-window sums are precomputed on 65 evenly spaced levels of $m$ and interpolated geometrically; the tests check agreement with brute force to $10^{-4}$ (observed about $4 \times 10^{-6}$). Reported: mean, median, p5, p25, p75, p95, $\Pr[\mathbb{E}(M) < 0]$, $\Pr[f < 0]$. With fluctuating value, 16 seeded value paths are cycled through the draws.

Value growth alternatives

Polynomial of order $n$ (any real $n \ge 0$): $(\alpha t + 1 - \alpha)^n$.
Exponential: $v_t = (1 + g)^{t-1}$; may fail to converge.
Hard cap: $v_t = \min((1 + g)^{t-1}, \bar v)$.
Fluctuating: $v_t = \bar v_t \exp(x_t - \tfrac{1}{2}\mathrm{Var}\,x)$ with $x_t = \rho x_{t-1} + \sigma \epsilon_t$, $x_0$ drawn from the stationary distribution, so $\mathbb{E}[v_t] = \bar v_t$ for the chosen base path $\bar v$.

Action-dependent risk: rebound (switch 3)

After the window (years $P + 1$ to $P + D$), $r_t \leftarrow r_t (1 + \beta f P / D)$; if permanent, $r_t \leftarrow r_t (1 + \beta f)$ for every later year. $\beta = 1, D = P$ adds back exactly the proportional risk-years removed; $\beta < 0$ is a virtuous cycle. Engine: reboundMultiplier, mitigateRisk(r, m, rebound).

Time-bucket discounting (switch 4)

Let $\Delta_t = S'_t v_t - S_t v_t$ be the mitigation’s contribution in year $t$ (so $\sum_t \Delta_t = \mathbb{E}(M)$). Buckets $[0,5], (5,10], (10,20], (20,100], (100,500], (500, T]$ with factors $d_k$; the discounted value is $\sum_k d_k \sum_{t \in k} \Delta_t$. Defaults $d = (1.00, 0.95, 0.85, 0.70, 0.50, 0.10)$ from the Donor Compass walkthrough. Engine: bucketValue(diff, edges, factors).

Sensitivity

tornado(params) halves and doubles each active input one at a time and reports $\log_{10}(\mathbb{E}(M)_{\text{changed}} / \mathbb{E}(M)_{\text{now}})$. Horizon is varied as $T/10$ and $2T$. The logistic cap is varied through its headroom above today’s value, $c - 1$, divided or multiplied by 10, so that small caps such as 1.6 stay above 1.

Presets

The links at the top of the model page set these values; everything else stays at the report’s defaults.

Preset	Settings
Report default	Time of Perils, constant value, $f = 0.0001$, $P = 5$, $T = 120{,}000$
Report, cubic value	as above, cubic value
Report figures	as above, $f = 0.5$
RP 2026 central values	Time of Perils at 20% per century for 100 years; low risk 0.0833% per century ($8.3 \times 10^{-6}$ a year, an expected 120,000 more years); logistic value with cap 1.6 and growth 1% a year; $f = 0.0001$; effect starting in year 4 and lasting $P = 11$ years; $T = 400{,}000$

The RP preset approximates the central values in Rethink Priorities’ GCR fund write-up (working draft, read 12 October 2026). RP’s risk peaks in about year 17 and stays elevated for about 90 years; the preset uses the report’s 100-year step instead. It leaves out RP’s space-settlement branch, its parameter uncertainty, its split of risk across causes and its counterfactual adjustments. See Departures.

3. Parameter table

Symbol	Meaning	Default (paper)	Where in the paper / notebook	UI
$T$	horizon (years)	120,000 on the model page (the PDF’s stated cap); per cell on the replication page	footnote 27; notebook `Ts_experiment`	select
$r_{high}$	high per-year risk	$1 - 0.8^{1/100} = 0.0022289477$	footnote 16	slider, % per century
$r_{low}$	low per-year risk	0.0001	footnote 17	slider, % per century
$L$	length of Time of Perils	100	notebook `top`	slider
$F, \ell_d, \ell_s$	filters, filter length, safe length	2, 100, 500	section 3 example	sliders
$\lambda, r_\infty$	decay rate and floor	$10^{-4}$, $10^{-9}$	notebook `generate_risk_vector`	log sliders
$\alpha$	value time scale	0.01	section 11	select
$c, \gamma$	logistic cap and rate	$10^6$, 0.03	notebook `generate_value_vector`	sliders
$f$	efficacy	0.0001 (tables), 0.5 (figures)	p. 29	select
$P$	persistence	5	Figure 6; Table 3 uses 1, 5, 50, 500, 2000	select
$t_0$	window start	1	section 2.2 (“without delay”)	slider
$c$ (floor)	minimum per-year risk	off; $10^{-4}$ when on	evaluation	toggle + slider
$\mu_f, \sigma_f$	distribution of $f$	off; mean $f$, sd $= $ mean	evaluation	toggle + sliders
$p_b, b_{med}, \sigma_b, t_b, P_b$	backfire	0.1, $\mu_f$, 0.5, 6, 5	evaluation	sliders
$\beta, D$	rebound	off; 1, 5	evaluation	toggle + sliders
$d_k$	bucket factors	off; 1, 0.95, 0.85, 0.70, 0.50, 0.10	Donor Compass walkthrough	toggle + text

Settings, links and source tags

Every input has a short key (the one that appears in a share link, for example rl for low risk or P for persistence). The engine holds the list of keys with their defaults, slider ranges and menu entries (SETTING_DEFAULTS, SETTING_RANGES, SETTING_OPTIONS), and the model page builds its controls from it, so the page, share links and shared scenarios can’t disagree. parseSettings checks a link’s values: an unknown key is ignored, a value outside a slider’s range is moved to the nearest end, a value not on a menu moves to the nearest entry, and malformed time-bucket factors fall back to the defaults. Each change is listed at the top of the results, never made silently. If a key’s meaning ever changes, the settings schema version (now 1) goes up and old links are translated.

Each setting that affects the current result is tagged by where its value comes from (settingProvenance):

Tag	Meaning
2023 report	the value the report or its notebook uses (for $T$, any of the notebook’s per-cell horizons)
RP 2026 model	the central value in Rethink Priorities’ 2026 GCR fund write-up, or the Donor Compass walkthrough’s time-window factors, as we read them
illustrative	our default for an extension, chosen to show the mechanism; no empirical basis
your choice	anything else, including values from a link someone sent you

Seeds and the number of Monte Carlo draws are technical settings and aren’t tagged. A tag says where a number came from, not that it’s right: the report’s and RP’s values are their authors’ judgments.

4. Numerical notes

All sums run forward in double precision with $S_t$ updated multiplicatively, exactly as in the notebook. $\mathbb{E}(M)$ is a difference of two sums of similar size; at $f = 10^{-4}$ that costs about $10^{-8}$ relative precision, which is why the tests compare to the notebook at $10^{-8}$ and not tighter.
Beyond roughly $t = 320{,}000$ years of constant risk 0.0022, $S_t$ underflows the smallest normal double; the notebook discusses the same effect. The tool caps $T$ at 400,000.
Plots downsample the $T$-length series to about 400 log-spaced points plus the first 300 years.
The scenario grid, persistence curve, sensitivity chart and Monte Carlo wait until the inputs have been still for 0.3 seconds, so dragging a slider stays smooth. The headline numbers update at once.
Headline numbers are shown to three significant figures; hovering shows ten. The extra digits are for checking against the report, not a claim of precision.

Case	Paper (centuries)	Tool (years, \(\alpha = 1/100\))
V0 constant	\(v_c\)	\(1\)
V1 linear	\(i\,v_c\)	\((\alpha t + 1 - \alpha)\)
V2 quadratic	\(i^2 v_c\)	\((\alpha t + 1 - \alpha)^2\)
V3 cubic	\(i^3 v_c\)	\((\alpha t + 1 - \alpha)^3\)
V4 logistic	\(c / (1 + \frac{c-s}{s} e^{-\gamma t})\)	same, with \(s\) chosen so that \(v_1 = 1\)

Case	Definition	Defaults
Constant	\(r_t = r_c\)	\(r_c = 0.0022289477\)
Time of Perils	\(r_t = r_{high}\) for \(t \le L\), else \(r_{low}\)	\(L = 100\), \(r_{high} = 0.0022289477\), \(r_{low} = 0.0001\)
Great Filters	\(F\) eras of \(r_{high}\) (length \(\ell_d\)) alternating with eras of \(r_{low}\) (length \(\ell_s\)), last era open-ended	\(F = 2\), \(\ell = (100, 500, 100, T - 700)\)
Exponential decay	\(r_t = r_0 e^{-\lambda (t-1)} + r_\infty\)	\(r_0 = 0.0022289477\), \(\lambda = 10^{-4}\), \(r_\infty = 10^{-9}\)