In my master's thesis, which you can find here, I looked at the dynamic problem of an infinitely-lived philanthropist aiming to maximise the expected lifetime of humanity. 

In my model, the philanthropist has a given starting endowment and can reduce the extinction rate at every instant by spending on risk mitigation and alternatively invest in a riskless asset to grow her funds. Based on the Hamilton-Jacobi-Bellman equation of the problem, I characterize the optimal spending rate as a function of funds and of time (respectively). I show that optimal spending is zero for low levels of philanthropic funds so that funds grow at the interest rate until reaching a certain threshold. Once funds exceed a certain threshold, the spending rate as a function of time grows at an asymptotically constant rate, i.e. the time trajectory of the spending rate is asymptotically linear. (I do not prove but reasonably  conjecture that this is also the case for the spending rate as a function of funds and for funds as a function of time.)

The model is quite limited (it already wasn't easy for me to derive analytical results from this simplistic model). Maybe the most important limitation is the fact that risk mitigation spending only affects the contemporaneous extinction rate but not extinction risk over future periods. This and other limitations probably bias the results toward giving later than would be optimal under a more realistic and more complicated model. 

Nonetheless, I post it here in case someone is interested in a starting point for some explicit mathematical reasonning about optimal dynamic philanthropy in the presence of controllable existential risk.

If you find a mistake, please write a comment.

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Interesting, if as you say a bit unrealistic. If I'm interpreting your graph correctly (although I feel like I am probably not; I'm definitely not an economist), you end up describing an endowment-like structure, where if you're going to live forever, you'll want to end up giving away a constant amount of money each year (your b=0 line in the chart), or maybe an amount of money that represents something like a constant fraction of the growing world economy?? (Your b = (r-p)/r line?) It might be helpful for you to provide a layman-accessible summary, if I'm getting this all wrong.

In your conclusion, you talk about a couple ways that your model could be extended:

  • adding financial details like income, multiple investment assets
  • adding multiple charity types (like x-risk vs global health and development) -adding factor models of x-risk-reduction influence (via building up institutions, etc)
  • changes in x-risk over time, as dangers wax and wane
  • changes in whether spending should be thought of as reducing the current level of X-risk, versus permanently reducing the future rate of risk.

Of these, personally I'd be most interested in hearing about the last two, as they seem perhaps the most answerable with a pure mathematical approach. It would be interesting to know what ideal investment strategies look like under different combinations of assumptions:

  • X-risk increasing up until a "time of perils", then decreasing afterwards
  • X-risk greatest right now, steadily declining
  • X-risk just steadily getting worse and worse as dangerous new tech is developed
  • spending to temporarily suppress immediate risk levels, versus spending on permanently lowering the rate of risk
  • being able to flip back and forth between both kinds of spending

And other things like that.

In fact, the optimal spending rate is not constant but starts growing (approximately and asymptotically precisely) linearly once funds exceed a certain threshold. The solid lines in the hand-drawn phase diagram you are refering to are the points where the growth rate of funds (b_dot) is the same. The optimal spending policy is the one starting at the threshold b_hat approaching the line where the growth rate of the budget is constant (at (r-rho)/r). Although I do not prove that this is the optimal policy, what I do prove is that the time trajectory of the spending rate is asymptotically linear. I edited the post to make this more clear.

In case this led to confusion: By spending rate, I am not referring to a proportion of available funds but to the rate of change of how much money has been spent at a given point in time. Since the model is in continuous rather than discrete time, I talk about a spending rate at a certain point in time rather than spending in a certain time period.

Maybe you are interested in the following paper, which deals with similar questions as yours:

Existential risk and growth - Leopold Aschenbrenner (Columbia University) - Global Priorities Institute

Thanks. Actually, I know the paper, but maybe I could have referenced it in my thesis...

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