It’s so great to see how the community has advanced its thinking in this area over the last year+.
@kbambridge I think you’ve definitely landed on the key concept around symmetry, in that symmetry in the rebase response curve graph doesn’t translate to symmetry of behavior of supply adjustments. And the historical data definitely supports this.
We’ve explored this idea also but came to a different implementation, actually. You can still see it if you do some code archaeology on the AIP-5 commit history. (See this version for instance)
At the time, the community didn’t seem very receptive to the class of “multiplicative symmetric” functions, because it meant by definition that negative rebases affect the supply “more”. (I.e. it looks steeper than the linear version). So this mirrored property was removed for the straight-up sigmoid version. I’m happy to see that now there’s a greater appetite for both more aggressive contractions around the origin and symmetry.
I like how “natural” the log function solution is… but I really do like the positive and negative asymptotes for security reasons. Can we get both symmetry and asymptotes? We can, if we break the rebase function into two parts then calculate the other part as a transform of the first.
So the positive rebase is the sigmoid, then the negative side is t(F(x)). This gives you the freedom to really have any function you’d like on one side, then simply compute the other based on that. I’d write more here, but the best description is probably in the old AIP draft itself, so I’d recommend referencing that.
Another, simpler way to get this symmetry is to approximate it by configuring that basic sigmoid function the right way. If we don’t need 100% exact symmetry, we can set the ceiling and growth factor how we want, then back into the floor so that it’s still basically close to a mirrored function. So we can still get “basic symmetry” with the simple sigmoid. This gives us symmetry, asymptotes, and simplicity (at the cost of some precision).