Incorporating these two elements into your model of indefinite races might change the conclusions (as you are probably aware):
1. A cost of conflict: unlike in the past, great powers conflict today is plausibly nuclear, or at least risks nuclear escalation. So the gap between nations would need to be sufficiently large before the more powerful ones find it in their interests to initiate conflict.
2. Convergence in growth between nations, like Karan Makkar points out in another comment. Leader-follower models, e.g. Greenhalgh and Rogers (2010), might help to formalise this. Specifically, it would be interesting to endogenize the absorptive capacity parameter, which captures how readily the technological advancements of the 'leader' nation are adopted by the 'follower' nation. Leader countries may strategically try to reduce their competitors' absorptive capacity (e.g., by banning poaching of top researchers, stronger IP protections, etc.). OR maybe there is an equilibrium whereby, given some uncertainty of who will be the leader in future periods (and uncertainty in the *current* AI development level of other countries), all major players find it in their interests to allow mutual high absorptive capacity (e.g., by allowing researchers to move around different countries) in order to avoid the high costs potentially losing in conflict.
Sorry for the slight ramble--I'm studying for finals at the moment so my thoughts are scrambled. It's a great post; thanks for sharing!
Overall, this was excellent. A quibble that may be important:
Implicit in your model that small differences in takeoff time leading to large permanent differences in GDP was constant growth across the two players. I think this can fail in a couple of important ways (as you're of course aware):
1. Catchup growth. As the technology (in this case AI) diffuses, we should expect catchup growth as latecomers to the technology incorporate the already frontier technologies into their economies. Unless we're assuming that the technological secrets are super closely guarded, we should faster growth rates for further behind countries just as we see for developing countries today.
2. Differences in steady state growth rates: Distinct from the issue of catchup growth, we should expect differences in steady state growth rates to dominate differences in timing of takeoff. For your toy model, a 24% growth rate over 8 years gets you the same as 18% over 10. In the medium run it's these difference in steady state growth rates that will dominate the GDP numbers. You could make a (very very low confidence) argument that being careful/slow about alignment and initial deployment could be correlated with higher steady state growth rates.
You partially pre-empt this by pointing to the fact that pre WW1 Germany wanted to maintain parity at all times because they were worried that even temporary gaps could weaken deterrence/were bad for war-winning if the war happened while there was a gap.
And sure, this may be an issue if you expect an imminent invasion/war if you fall behind, but it's a much smaller concern than permanent technological/production inferiority. You could try and make the case that the US is currently in a situation where it should be worried about being invaded conditional on temporary technological/production inferiority, but I'd be highly skeptical.
Incorporating these two elements into your model of indefinite races might change the conclusions (as you are probably aware):
1. A cost of conflict: unlike in the past, great powers conflict today is plausibly nuclear, or at least risks nuclear escalation. So the gap between nations would need to be sufficiently large before the more powerful ones find it in their interests to initiate conflict.
2. Convergence in growth between nations, like Karan Makkar points out in another comment. Leader-follower models, e.g. Greenhalgh and Rogers (2010), might help to formalise this. Specifically, it would be interesting to endogenize the absorptive capacity parameter, which captures how readily the technological advancements of the 'leader' nation are adopted by the 'follower' nation. Leader countries may strategically try to reduce their competitors' absorptive capacity (e.g., by banning poaching of top researchers, stronger IP protections, etc.). OR maybe there is an equilibrium whereby, given some uncertainty of who will be the leader in future periods (and uncertainty in the *current* AI development level of other countries), all major players find it in their interests to allow mutual high absorptive capacity (e.g., by allowing researchers to move around different countries) in order to avoid the high costs potentially losing in conflict.
Sorry for the slight ramble--I'm studying for finals at the moment so my thoughts are scrambled. It's a great post; thanks for sharing!
Overall, this was excellent. A quibble that may be important:
Implicit in your model that small differences in takeoff time leading to large permanent differences in GDP was constant growth across the two players. I think this can fail in a couple of important ways (as you're of course aware):
1. Catchup growth. As the technology (in this case AI) diffuses, we should expect catchup growth as latecomers to the technology incorporate the already frontier technologies into their economies. Unless we're assuming that the technological secrets are super closely guarded, we should faster growth rates for further behind countries just as we see for developing countries today.
2. Differences in steady state growth rates: Distinct from the issue of catchup growth, we should expect differences in steady state growth rates to dominate differences in timing of takeoff. For your toy model, a 24% growth rate over 8 years gets you the same as 18% over 10. In the medium run it's these difference in steady state growth rates that will dominate the GDP numbers. You could make a (very very low confidence) argument that being careful/slow about alignment and initial deployment could be correlated with higher steady state growth rates.
You partially pre-empt this by pointing to the fact that pre WW1 Germany wanted to maintain parity at all times because they were worried that even temporary gaps could weaken deterrence/were bad for war-winning if the war happened while there was a gap.
And sure, this may be an issue if you expect an imminent invasion/war if you fall behind, but it's a much smaller concern than permanent technological/production inferiority. You could try and make the case that the US is currently in a situation where it should be worried about being invaded conditional on temporary technological/production inferiority, but I'd be highly skeptical.