>>6905790It's not exactly sanity, it's more stability. It represents a character's psyche and self-image. High Integrity has 2 general benefits for CofD, and then a couple more in HtV2e. It's first benefit is that high Integrity grants a bonus to Breaking Point rolls, theses are things that make you lose Integrity. So the more you have the easier it is to keep, and inversely low Integrity imposes penalties. Succeed or failing a Breaking Point effects you the same, out side of failure meaning you lose a dot of Integrity. You gain one of three annoying but not awful Conditions, but it's the dramatic failures that are the really big deal for this. When your dice pool is lowered to below 1 dice, you roll a "chance die" instead. This has a 10% chance of a success, and 80% chance of a failure, and a 10% chance of a dramatic failure. A particularly traumatic event for a low Integrity character can be hitting -6 or -7 dice, which is typically enough to make you roll a chance die. Which means a 10% chance you'll gain a really debilitating Persistent Condition that could potentially get you killed as it can outright cripple you. So, high Integrity means less bad stuff to worry about.
The other one is that it grants bonuses to Abjuration as well as Warding and Binding attempts. Both of these things are vital tools in dealing with Ephemeral Entities, if you're going up against a spirit with low integrity you're not going to have a fun time. It can be a +3 bonus for 10 Integrity, down to a -5 for 1 Integrity.
For Hunter it's more of a big deal. Firstly there is "The Code" a set of beliefs a Hunter places value in that determines their Breaking Points. When they have high Integrity they have more Breaking Points to deal with, as they lose Integrity they lose some of them. They're split into high/mid/low tiers. So a high Integrity Hunter would face a Breaking Point for causing significant harm to a person, at mid they'd have to kill a person, and low they just don't care and have to torture a person before it effects them. Breaking Points at lower tiers are harder to resist though.
Next up as you grow more callous though Integrity loss you gain Vigilant as a Persistent Condition when you hit the mid tier. It means you have to spend Willpower to avoid looking into potential supernatural dangers, it makes you jumpy and easily Spooked (a Condition) as you're always looking over your shoulder, but it lets Virtues (your best trait) grant Willpower in excess of your total and it grants a Beat when you alienate a normal person because of the Vigil. If it drops further and you become more withdrawn you instead become Merciless. Works basically the same except now its your Vices (your worst) that let you exceed maximum Willpower, and you can get a Beat from alienating other Hunters too. If your Integrity drops so low that it hits 0, you become a Slasher. A deranged killer that's a monster all of their own, at this point you hand your sheet to the GM and make a new character. Slasher's like this can be redeemed but it's hard work
Integrity is also tied into Touchstones, these are people/objects/places that keep your character grounded. The Vigil is dangerous work that requires separation from who you protect, Touchstones help you keep your feet on the ground. Touchstones get "attached" to specific dots of Integrity, if you are currently attached to a Touchstone it provides a bonus against Breaking Points. If you lose the dot of Integrity that Touchstone is attached to, you become detached form them and if you have no attached Touchstone you gain a penalty instead. there is a little more too them as well, but that's how it works for Integrity.
Then you've got some little bits here and there, like one of the special Hunter power sets is modified by it, and a few other bits. So it's not a massive deal in CofD, is important in Hunter, but it's not likely to come up at all because that's 2e rules they almost certainly haven't read.