This put into well-written words a lot of thoughts I've had in the past but never been able to properly articulate. Thank you for writing this.

This sounds rather like the competing political economic theories of classical liberalism and Marxism to me. Both of these intellectual traditions carry a lot of complicated baggage that can be hard to disentangle from the underlying principles, but you seem to have a done a pretty good job of distilling the relevant ideas in a relatively apolitical manner.

That being said, I don't think it's necessary for these two explanations for wealth inequality to be mutually exclusive. Some wealth could be accumulated through "the means of production" as you call it, or (as I'd rather describe it to avoid confusing it with the classical economic and Marxist meaning) "making useful things for others and getting fair value in exchange".

Other wealth could also, at the same time, be accumulated through exploitation, such as taking advantage of differing degrees of bargaining power to extract value from the worker for less than it should be worth if we were being fair and maybe paying people with something like labour vouchers or a similar time-based accounting. Or stealing through fraudulent financial transactions, or charging rents for things that you just happen to own because your ancestors conquered the land centuries ago with swords.

Both of these things can be true at the same time within an economy. For that matter, the same individual could be doing both in various ways, like they could be ostensibly investing and building companies that make valuable things for people, while at the same time exploiting their workers and taking advantage of their historical position as the descendent of landed aristocracy. They could, at the same time, also be scamming their venture capitalists by wildly exaggerating what their company can do. All while still providing goods and services that meet many people's needs and ways that are more efficient than most possible alternatives, and perhaps the best way possible given the incentives that currently exist.

Things like this tend to be multifaceted and complex. People in general can have competing motivations within themselves, so it would not be strange to expect that in something as convoluted as a society's economy, there could be many reasons for many things. Trying to decide between two possible theories of why, misses the possibility that both theories contain their own grain of truth, and are each, by themselves, incomplete understandings and world models. The world is not just black or white. It's many shades of grey, and also, to push the metaphor further, a myriad of colours that can't accurately be described in greyscale.

Another thought I just had was, could it be that ChatGPT, because it's trained to be such a people pleaser, is losing intentionally to make the user happy?

Have you tried telling it to actually try to win? Probably won't make a difference, but it seems like a really easy thing to rule out.

Also, quickly looking into how LLM token sampling works nowadays, you may also need to set the parameters top_p to 0, and top_k to 1 to get it to actually function like argmax. Looks like these can only be set through the API if you're using ChatGPT or similar proprietary LLMs. Maybe I'll try experimenting with this when I find the time, if nothing else to rule out the possibility of such a seemingly obvious thing being missed.

I've always wondered with these kinds of weird apparent trivial flaws in LLM behaviour if it doesn't have something to do with the way the next token is usually randomly sampled from the softmax multinomial distribution rather than taking the argmax (most likely) of the probabilities. Does anyone know if reducing the temperature parameter to zero so that it's effectively the argmax changes things like this at all?

p = (n^c * (c + 1)) / (2^c * n)

As far as I know, this is unpublished in the literature. It's a pretty obscure use case, so that's not surprising. I have doubts I'll ever get around to publishing the paper I wanted to write that uses this in an activation function to replace softmax in neural nets, so it probably doesn't matter much if I show it here.

So, my main idea is that the principle of maximum entropy aka the principle of indifference suggests a prior of 1/n where n is the number of possibilities or classes. P x 2 - 1 leads to p = 0.5 for c = 0. What I want is for c = 0 to lead to p = 1/n rather than 0.5, so that it works in the multiclass cases where n is greater than 2.

Correlation space is between -1 and 1, with 1 being the same (definitely true), -1 being the opposite (definitely false), and 0 being orthogonal (very uncertain). I had the idea that you could assume maximum uncertainty to be 0 in correlation space, and 1/n (the uniform distribution) in probability space.

I tried asking ChatGPT, Gemini, and Claude to come up with a formula that converts between correlation space to probability space while preserving the relationship 0 = 1/n. I came up with such a formula a while back, so I figure it shouldn't be hard. They all offered formulas, all of which were shown to be very much wrong when I actually graphed them to check.

I was not aware of these. Thanks!