If I remember correctly, I managed to derive functions for the expected number of special spins per game given winrate. They compensate for the time used up by early losses, which take up fewer games than a loss that happens after you manage to fill up the rings most of the way (curse the RNG!).
(w is the probability of winning a game)
Plat: w^2 / (w+1)
Gold: w^3 / (w^2 + w + 1)
Silver: w^4 / (w^3 + w^2 + w + 1)
Bronze: w^5 / (w^4 + w^3 + w^2 + w + 1)
The average number of games needed per spin can be found by finding the inverse of these.
Can someone make a table for this please?