This sort of “many gods”-style response is precisely what I was referring to with my parenthetical: “unless one inverts the high stakes in a way that cancels out the other high-stakes possibility.”
I think you are making some unstated assumptions that it would be helpful to make explicit. You say your argument is basically just explaining how expected values work, but it doesn’t seem like that is true to me, I think you need to make some assumptions unrelated to how expected values work for your argument to go through.
If I were to cast your argument in the language of “how expected values work” it would go like this:
An expected value is the the sum of a bunch of terms that involve multiplying an outcome by its probability, so of the form x * p, where x is the outcome (usually represented by some some number) and p is the probability associated with that outcome. To get the EV we take terms like that representing every possible outcome and add them up.
Because these terms have two parts, the term as a whole can be large even if the probability is small. So, the overall EV can be driven primarily by a small probability of a large positive outcome because it is dominated by this one large term, which is large even when the probability is small. We rule high stakes in, not out.
The problem is that this argument doesn’t work without further assumptions. In my version I said “can be driven”. I think your conclusion requires “is driven”, which doesn’t follow. Because there are other terms in the EV calculation their sum could be negative and of sufficient magnitude that the overall EV is low or negative even if one term is large and positive. This doesn’t require that any particular term in the sum has any particular relationship to the large positive term such that it is “inverting” that term, although that would be sufficient, it isn’t the only way for the overall EV to be small/negative. Their could be a mix of moderate negative terms that adds up to enough to reduce the overall EV. Nothing about this seems weird or controversial to me. For example, a standard normal distribution has large positive values with small probabilities but has an expectation of zero.
I think you need to be more explicit about the assumptions you are making that result in your desired conclusion. In my view, part of the point of Thorstad’s “many gods” response is that it demonstrates that once we start picking apart these assumptions we essentially collapse back to having the model the entire space of possibilities. That is suggested by what you say here:
I don’t think that dystopian “time of carols” scenarios are remotely as credible as the time of perils hypothesis.
The issues isn’t that the “time of carols” is super plausible, its that if your response is to include it as a term in the EV and argue the sum is still positive, then it seems like your original argument kind of collapses. We are no longer “ruling stakes in”. We now also have to actually add in all those other terms as well before we can know the final result.
I could imagine there are assumptions that might make your argument go through, but I think you need to make them explicit and argument for them, rather than claiming your conclusion follows from “how expected value works”.
The responses to my comment have provided a real object lesson to me about how a rough throwaway remark (in this case: my attempt to very briefly indicate what my other post was about) can badly distract readers from one’s actual point! Perhaps I would have done better to entirely leave out any positive attempt to here describe the content of my other post, and merely offer the negative claim that it wasn’t about asserting specific probabilities.
My brief characterization was not especially well optimized for conveying the complex dialectic in the other post. Nor was it asserting that my conclusion was logically unassailable. I keep saying that if anyone wants to engage with my old post, I’d prefer that they did so in the comments to that post—ensuring that they engage with the real post rather than the inadequate summary I gave here. My ultra-brief summary is not an adequate substitute, and was never intended to be engaged with as such.
On the substantive point: Of course, ideally one would like to be able to “model the entire space of possibilities”. But as finite creatures, we need heuristics. If you think my other post was offering a bad heuristic for approximating EV, I’m happy to discuss that more over there.
I think you have be underestimating to what extent the responses you are getting do speak to the core content of your post, but I will leave a comment there to go into it more.
I think you are making some unstated assumptions that it would be helpful to make explicit. You say your argument is basically just explaining how expected values work, but it doesn’t seem like that is true to me, I think you need to make some assumptions unrelated to how expected values work for your argument to go through.
If I were to cast your argument in the language of “how expected values work” it would go like this:
An expected value is the the sum of a bunch of terms that involve multiplying an outcome by its probability, so of the form x * p, where x is the outcome (usually represented by some some number) and p is the probability associated with that outcome. To get the EV we take terms like that representing every possible outcome and add them up.
Because these terms have two parts, the term as a whole can be large even if the probability is small. So, the overall EV can be driven primarily by a small probability of a large positive outcome because it is dominated by this one large term, which is large even when the probability is small. We rule high stakes in, not out.
The problem is that this argument doesn’t work without further assumptions. In my version I said “can be driven”. I think your conclusion requires “is driven”, which doesn’t follow. Because there are other terms in the EV calculation their sum could be negative and of sufficient magnitude that the overall EV is low or negative even if one term is large and positive. This doesn’t require that any particular term in the sum has any particular relationship to the large positive term such that it is “inverting” that term, although that would be sufficient, it isn’t the only way for the overall EV to be small/negative. Their could be a mix of moderate negative terms that adds up to enough to reduce the overall EV. Nothing about this seems weird or controversial to me. For example, a standard normal distribution has large positive values with small probabilities but has an expectation of zero.
I think you need to be more explicit about the assumptions you are making that result in your desired conclusion. In my view, part of the point of Thorstad’s “many gods” response is that it demonstrates that once we start picking apart these assumptions we essentially collapse back to having the model the entire space of possibilities. That is suggested by what you say here:
The issues isn’t that the “time of carols” is super plausible, its that if your response is to include it as a term in the EV and argue the sum is still positive, then it seems like your original argument kind of collapses. We are no longer “ruling stakes in”. We now also have to actually add in all those other terms as well before we can know the final result.
I could imagine there are assumptions that might make your argument go through, but I think you need to make them explicit and argument for them, rather than claiming your conclusion follows from “how expected value works”.
The responses to my comment have provided a real object lesson to me about how a rough throwaway remark (in this case: my attempt to very briefly indicate what my other post was about) can badly distract readers from one’s actual point! Perhaps I would have done better to entirely leave out any positive attempt to here describe the content of my other post, and merely offer the negative claim that it wasn’t about asserting specific probabilities.
My brief characterization was not especially well optimized for conveying the complex dialectic in the other post. Nor was it asserting that my conclusion was logically unassailable. I keep saying that if anyone wants to engage with my old post, I’d prefer that they did so in the comments to that post—ensuring that they engage with the real post rather than the inadequate summary I gave here. My ultra-brief summary is not an adequate substitute, and was never intended to be engaged with as such.
On the substantive point: Of course, ideally one would like to be able to “model the entire space of possibilities”. But as finite creatures, we need heuristics. If you think my other post was offering a bad heuristic for approximating EV, I’m happy to discuss that more over there.
I think you have be underestimating to what extent the responses you are getting do speak to the core content of your post, but I will leave a comment there to go into it more.