So more interesting is a formular that contains the average amount of rounds(n) played in succession as a parameter instead of just assuming the case n->∞.
Sure, now that I don't have to do the algebra by hand, this is pretty simple to do. Correct me if I'm wrong though. I have this nagging suspicion that I've missed something... but here it is as far as I can tell.
n - number of rounds played in one sitting on average
x - win rate
y - average multiplier
When:
n = 1, y = 1
n = 2, y = x + 1
n = 3, y = x^2 + x +1
n = 4, y = x^3 + x^2 + x + 1
n = 5, y = x^4 + x^3 + x^2 + x + 1
n > 5, y = x^4 + x^3 + x^2 + x + 1
As long as you play at least 5 rounds a sitting, and only quit on a x1 (since multipliers don't carry over sittings), the general equation of, y = x^4 + x^3 + x^2 + x + 1, should apply.
I'm not sure where you're getting the numbers you gave in your example. For x = 0.5 (50% win chance),
1 round, y = 1
2 rounds, y = 0.5*1 + 0.5*2 = 1.5
3 rounds, y = 0.5*1 + 0.5*0.5*2 + 0.5*0.5*3 = 1.75
4 rounds, y = 0.5*1 + 0.5*0.5*2 + 0.5*0.5*0.5*3 + 0.5*0.5*0.5*4 = 1.875
5 rounds, y = 1.9375
TL:DR, play at least 5 rounds per sitting, and only exit on a x1.