I did many things to study including make a demonstration than this formula led to a Nash equilibrium with k equal-size pool, as I was asked by IOHK devs.
I did not draw curves, because it sounds to me very straight-forward how it behaves. And very easy to imagine, and because none asked yet. So I could if you really need to. Even why not 3D curves with s, and σ variables.
I don’t think there would be a “steep gradient” problem. But it is possible That I missed something. Adding an exponential factor as you propose may be advantageous,I don’t know, I need to think about it.
Take the time to read this long thread, I’m sure there are many things that was written by me and others, that should interest you a lot, given your reflections:
We may discuss that again in a few days, if you wish.