Can someone tell me a reason, as to why my slaves just escape spontaneously?
I get a new slave, everything goes well, but at the 4th or 5th day the bitch just runs away! Even if my assistant supervises her, the slave can still just escape.
in order for her not to get away, she needs to have a collar and a brand (or, if she has a magical brand, you can track her down without a collar)
lore-wise, i think it's that she needs a collar so people can tell she's an escaped slave, then upon examination they find your brand and know it's your slave (bringing it to you for free). or they find the contract brand and give it to the guild, who then gives it back to you (at the cost of some guild reputation, because then the brass find out you let a slave escape). or if she has a magical brand, the slaver can use it as a homing beacon to track down personally, so other people don't need to figure out she's an escaped slave (and so she doesn't need a collar). i think it's implied that if she just has a collar, they don't have a way to figure out she's yours so she just goes to the guild or farid or something.
the actual trigger for an escape attempt is the slave having mood less than -1 (ie melancholic or worse) and having a temperament greater than her obedience (ie a slave with -1 obedience might make an escape attempt regardless of temperament, while a slave with 3 obedience would only make an escape attempt with A+ or S+ temperament). if a slave's devotion is 2 or greater, they won't ever try to escape regardless of their obedience.
the supervise rule does make escape attempts less likely, but it does not prevent them. the escape attempt code looks like this (i've highlighted the variable 'n' since the assistant impacts it)
the weird formula is basically just "pseudo-random integer between 0 and n (uninclusive of n)", so you have a 1 in n chance for the slave to make an escape (the formula is used to prevent savescumming, since the number is based on the current day).
in summary, to abate escape attempts, you want a fully devoted assistant with high intelligence, high nature, and high gladiatrix skill, while being a slaver with a high personality stat. but in the early game when you don't even have an assistant, it's much simpler to just put a collar and brand on her (which you should really be doing either way, because brands give slaves passive habit over time and collars give slaves passive taming over time). with how cheap the leather collar is (3 sparks) and how cheap branding is (free if you have a dungeon, 10 sparks at uncle bo's otherwise) you should really be doing both no matter what stage of the game you're at.