When estimating the amount of good that can be done by working on a given cause, a good first approximation might be the asymptotic behavior of the amount of good done at each point in time (the trajectory change).
Other important factors are the magnitude of the trajectory change (how much good is done at each point in time) and its duration (how long the trajectory change lasts).
For example, changing the rate of economic growth (population growth * GDP/​capita growth) has an O(t2) trajectory change in the short run, as long as humanity doesn’t expand into space. We break it down into a population growth component, which grows linearly, and a component for the change in average welfare due to GDP per capita growth[1]. GDP per capita typically grows exponentially, and welfare is a logarithmic function of GDP per capita. These two trends cancel out, resulting in a linear trend, or O(t).
If humanity expands into space, then economic growth becomes a quartic trend. Population growth becomes cubic, since humanity can fill an O(t3) amount of space in t time.[2] The average welfare due to GDP per capita growth is still linear. Multiplying the cubic and linear trends yields a quartic trend. So increasing the probability that space colonization goes well looks more important than changing the rate of economic growth on Earth.
Surprisingly, the trajectory change caused by reducing existential risk is not an exponential trend. Existential risk reduces expected welfare at each point in time by a factor of (1−r)t, where r is the probability of catastrophe per unit of time. This exponential trend causes all trajectory changes to wash out as t becomes very large.
The amount of good done by reducing x-risk from ri to rf is
∞∑t=1y(t)((1−rf)t−(1−ri)t),
where y(t) is the welfare trajectory conditional on no existential catastrophe. The factor ((1−rf)t−(1−ri)t) increases to a maximum value and then decays to 0 as t approaches infinity, so its asymptotic behavior is O(1). So the trajectory change caused by reducing x-risk is O(y(t)), or whatever the asymptotic behavior of y(t) is.
On the other hand, the amount of good done by changing the trajectory from yi(t) to yf(t) is
∞∑t=1(yf(t)−yi(t))(1−r)t.
So if you can change the trajectory to a yf(t) that will grow asymptotically faster than yi(t), e.g. by colonizing space, then this may be more important than reducing x-risk while holding the trajectory constant.
Change in GDP per capita may underestimate the amount of welfare created by technological progress. I’m not sure if this makes the change in average welfare a super-linear growth trend, though.
Big O as a cause prioritization heuristic
When estimating the amount of good that can be done by working on a given cause, a good first approximation might be the asymptotic behavior of the amount of good done at each point in time (the trajectory change).
Other important factors are the magnitude of the trajectory change (how much good is done at each point in time) and its duration (how long the trajectory change lasts).
For example, changing the rate of economic growth (population growth * GDP/​capita growth) has an O(t2) trajectory change in the short run, as long as humanity doesn’t expand into space. We break it down into a population growth component, which grows linearly, and a component for the change in average welfare due to GDP per capita growth[1]. GDP per capita typically grows exponentially, and welfare is a logarithmic function of GDP per capita. These two trends cancel out, resulting in a linear trend, or O(t).
If humanity expands into space, then economic growth becomes a quartic trend. Population growth becomes cubic, since humanity can fill an O(t3) amount of space in t time.[2] The average welfare due to GDP per capita growth is still linear. Multiplying the cubic and linear trends yields a quartic trend. So increasing the probability that space colonization goes well looks more important than changing the rate of economic growth on Earth.
Surprisingly, the trajectory change caused by reducing existential risk is not an exponential trend. Existential risk reduces expected welfare at each point in time by a factor of (1−r)t, where r is the probability of catastrophe per unit of time. This exponential trend causes all trajectory changes to wash out as t becomes very large.
The amount of good done by reducing x-risk from ri to rf is
∞∑t=1y(t)((1−rf)t−(1−ri)t),
where y(t) is the welfare trajectory conditional on no existential catastrophe. The factor ((1−rf)t−(1−ri)t) increases to a maximum value and then decays to 0 as t approaches infinity, so its asymptotic behavior is O(1). So the trajectory change caused by reducing x-risk is O(y(t)), or whatever the asymptotic behavior of y(t) is.
On the other hand, the amount of good done by changing the trajectory from yi(t) to yf(t) is
∞∑t=1(yf(t)−yi(t))(1−r)t.
So if you can change the trajectory to a yf(t) that will grow asymptotically faster than yi(t), e.g. by colonizing space, then this may be more important than reducing x-risk while holding the trajectory constant.
Change in GDP per capita may underestimate the amount of welfare created by technological progress. I’m not sure if this makes the change in average welfare a super-linear growth trend, though.
The Epistemic Challenge to Longtermism (Tarsney, 2020)