LW - How do open AI models affect incentive to race? by jessicata

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Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: How do open AI models affect incentive to race?, published by jessicata on May 7, 2024 on LessWrong.
I see it said sometimes that open models contribute to AI race dynamics. My guess is that they don't, and if anything, reduce AI race dynamics.
I will consider a simplified model that only takes into account the cost of training a model, not the cost to deploy it (which tends to be small relative to revenue anyway). Let f(x) map a training expense x to a "value per day per customer" of the trained model, under the assumption that the training makes efficient use of the cost. That is, a customer values using an AI model trained with x compute at $f(x) per day.
I assume there are n identical customers here; of course, there are complexities where some customers value AI more than others, incentivizing price discrimination, but I'm abstracting this consideration out. (In general, variation in how much customers value a product will tend to increase consumer surplus while reducing revenue, as it makes it harder to charge customers just under the maximum amount they're willing to pay.)
I'm also assuming there is only one company that trains closed models for profit. This assumption is flawed because there is competition between different companies that train closed models. However, perfect competition assumptions would tend to reduce the incentive to train models. Suppose two companies have closed models of equivalent expense x. They each want to charge slightly less than the minimum of f(x) and the competitor's price, per customer per day.
If each competitor undercuts the other slightly, the cost will approach 0. See the Traveler's Dilemma for a comparison. The reasons why this doesn't happen have to do with considerations like differences in models' performance on different tasks, e.g. some models are better for programming than others. If models are sufficiently specialized (allowing this sort of niche-monopolization), each specialized type of model can be modeled independently as a monopoly.
So I'll analyze the case of a closed model monopoly, noting that translation to the real world is more complex.
Suppose the best open model has compute x and a company trains a closed model with compute y > x. Each customer will now spend up to f(y) - f(x) per day for the model; I'll assume the company charges f(y) - f(x) and the customers purchase this, noting that they could charge just below this amount to create a positive incentive for customers. So the company's revenue over m days is nm(f(y) - f(x)). Clearly, this is decreasing in x.
So the better the open model is, the less expected revenue there is from training a closed model.
But this is simply comparing doing nothing to training a model of a fixed cost y. So consider instead comparing expected revenue between two different model costs, y and z, both greater than x. The revenue from y is nm(f(y) - f(x)), and from z it is nm(f(z) - f(x)). The difference between the z revenue and the y revenue is nm(f(z) - f(y)). This is unaffected by x.
This can model a case where the company has already trained a model of cost y and is considering upgrading to z. In this case, the open model doesn't affect the expected additional revenue from the upgrade.
Things get more complex when we assume there will be a future improvement to the open model. Suppose that, for k days, the open model has training cost x, and for the remaining m-k days, it has training cost x' > x.
Now suppose that the closed AI company has already trained a model of cost y, where x < y < x'. They are considering upgrading to a model of cost z, where z > x'.
Suppose they do not upgrade. Then they get nk(f(y) - f(x)) revenue from the first k days and nothing thereafter.
Suppose they do upgrade, immediately. Then they get nk(f(z) - f(x)) revenue from the first k days, an...