Thanks for posting this! Your linkpost actually got me to watch the talk for the first time, even though I was aware of this paper for a while.
I think some variant of the cubic growth model could be useful for figuring out whether trying to reduce x-risk is better than trying to make durable changes to the long-term “trajectory” of the social welfare curve. I spent some time a few months ago trying to address this by modeling the trajectory of humanity, so I appreciate this paper for proposing even a simpler toy model.
I have rough thoughts about how the utility from economic growth could be incorporated: Assume that each star system has a growth rate g that the residents of that star system can influence (e.g. through policy). The economy of each star system tends to grow exponentially, but GDP per capita has logarithmic utility, so the utility of the star system us(t) grows roughly linearly.
If the economy of each star system starts at a steady state, then grows exponentially at g starting at time t0, the time at which humanity arrives at the star system, we get us(t)=u0+max(0,g(t−t0)). If the star system’s GDP is capped at exp(umax), then we get us(t)=u0+min(max(0,g(t−t0)),umax).
To incorporate economic growth into the trajectory model used in the paper, we can replace n(s⋅(t−tℓ)) with the cross-correlation of us(t) and n(s⋅(t−tℓ)) (this assumes that all star systems have the same growth rate). Since us(t) is piecewise linear and n(s⋅(t−tℓ)) is cubic, the cross-correlation is piecewise quintic (it’s the integral of a cubic function times a linear function). My gut tells me that having a piecewise quintic term in the trajectory function instead of a cubic term isn’t going to change much about the implications of the model.
Note: I realize that by using GDP per capita, I’m leaving out the population of each star system. This would result in multiplying us(t) by a function that models the population over time, starting at time t0.
Thanks for posting this! Your linkpost actually got me to watch the talk for the first time, even though I was aware of this paper for a while.
I think some variant of the cubic growth model could be useful for figuring out whether trying to reduce x-risk is better than trying to make durable changes to the long-term “trajectory” of the social welfare curve. I spent some time a few months ago trying to address this by modeling the trajectory of humanity, so I appreciate this paper for proposing even a simpler toy model.
I have rough thoughts about how the utility from economic growth could be incorporated: Assume that each star system has a growth rate g that the residents of that star system can influence (e.g. through policy). The economy of each star system tends to grow exponentially, but GDP per capita has logarithmic utility, so the utility of the star system us(t) grows roughly linearly.
If the economy of each star system starts at a steady state, then grows exponentially at g starting at time t0, the time at which humanity arrives at the star system, we get us(t)=u0+max(0,g(t−t0)). If the star system’s GDP is capped at exp(umax), then we get us(t)=u0+min(max(0,g(t−t0)),umax).
To incorporate economic growth into the trajectory model used in the paper, we can replace n(s⋅(t−tℓ)) with the cross-correlation of us(t) and n(s⋅(t−tℓ)) (this assumes that all star systems have the same growth rate). Since us(t) is piecewise linear and n(s⋅(t−tℓ)) is cubic, the cross-correlation is piecewise quintic (it’s the integral of a cubic function times a linear function). My gut tells me that having a piecewise quintic term in the trajectory function instead of a cubic term isn’t going to change much about the implications of the model.
Note: I realize that by using GDP per capita, I’m leaving out the population of each star system. This would result in multiplying us(t) by a function that models the population over time, starting at time t0.