This is really a poker tournament/payout/bankroll question but I think I can frame it in a dice game for easier understanding. If a player starts with $X$ units, and plays a game where on each throw the cost/payouts are:
1,2 or 3 = -1
4: = 0
5: = +1
6: = +3
If the players wealth is never allowed to rise above $X$, ie, hits a 6 with a current wealth of $(X-1)$ the new wealth is only $X$, the extra two units are ditched. Is there an algorithm/formula to predict/plot the chance of ruin over many throws?
There is a Risk of Ruin formula for when the wealth is unrestricted and this, I think, is
$RoR(X) = e^{-aX}$, where $a$ is a constant related to the relative likelihood and expectations but is there a simple formula for when the max wealth is capped?
Or any clues to how to find a reasonable solution?
In the real results I would like to solve there are more outcomes - perhaps 20 or so +ve outcomes out of 100.