Modelling People Smarter Than Me
One thing I find interesting is thinking about the brains of people smarter than me. There's some sense in which I probably can't model the brains of people smarter than me, because if I could model what they think then I would be as smart as they are. But I suspect that it's like trying to visualise the 4th/5th/6th dimensions: in some sense it's impossible, but there's benefits to trying anyway.[^1]
A claim I've seen is that more intelligent people are able to do more levels of recursion when modelling other people's minds (e.g. from "I believe X" to "I believe that Anna believes that Brian believes that I believe...")
To be clear, I've never seen this claim verified or "proven" in any sense, but it does seem intuitively plausible to me: modelling other minds is cognitively expensive, and I'm willing to believe that every additional level of modelling requires more and more cognitive capacity.
So, for example: level 1 is only modelling your own thoughts, "I believe X", with no model at all of what anyone else believes.
Level 2 is being able to think: "I believe X, but I believe that Anna believes A and that Brian believes B".
Level 3 is reaching: "I believe X, but I believe that Anna believes A, and that Anna believes I believe AX, and that Anna believes Brian believes AB, and I believe Brian believes...."
I'm going to come clean here and say I couldn't figure out the maths myself for this, but our friendly neighbourhood LLM claims the formula is
where P is the number of people and L is the number of levels. Said LLM also generated this table claiming the number operations required for each level and group size is:
| 3 | 5 | 10 | 20 | |
|---|---|---|---|---|
| Level 1 | 1 | 1 | 1 | 1 |
| Level 2 | 3 | 5 | 10 | 20 |
| Level 3 | 9 | 25 | 100 | 400 |
| Level 4 | 21 | 105 | 910 | 7.6k |
| Level 5 | 45 | 425 | 8.2k | 140k |
| Level 6 | 93 | 1.7k | 74k | 2.8M |
If you have a better function (and can check the maths here) I'd be super happy to hear from you, but instinctively this pattern seems approximately right: every additional person and every additional level gets more and more expensive in terms of the operations you need to model everyone in the group.
Where does this leave us in terms of modelling people smarter than ourselves? Intuitively, I think there's a few recommendations.
First, I think we should at least try modelling people to an additional level of depth than we currently do. Perhaps it's not practical or even possible to build out the entire tree, but if you're able to ask yourself "what does my manager believe is happening right now? And what does my CEO believe? And what does my CEO believe I believe?" Call this level 2-and-a-half, or something – it's not a full map of beliefs to a level-3 depth, but it incorporates at least one piece of level 3 modelling for whoever seems most important in your current situation.
Second, I think it's worth trying to figure out the implications of someone modelling to a deeper depth than I am, even if I can't do the modelling myself. What would that person's actions and behavior look like? What would I need to be careful of when interacting with them? What would other people react to?
My guess is that usually the implication of anyone being at a level (or more) deeper than you is just "this person is likely to outplay me in any social-strategic game". Which is not at all a fun thing to feel, but at least is better than having it be true without realising it.
[^1]: Geoffrey Hinton says: "To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it."