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Darkest Dungeon has a stress system. It is cruel, unrelenting, disturbing and I love it.
First lets quickly run down of how the system works. Darkest Dungeon is a turn based dungeon crawler. You get a missions and you complete them by exploring a small-medium sized randomly generated area with 4 characters. As your characters go through the dungeon they get stressed and can become traumatized by the terrible things that happen on your adventures. This is expressed via a stress meter each character has which will fill while exploring dungeons either from seeing terrible things or from watching their comrades die (perma-death is a very real and constant threat in this game).
Too much stress can cause them to pick up new ticks and peculiarities (Kleptomaniac, Gambler, Known Cheat...). They might even have a full on mental break down and become irrational, depressed, abusive towards their party members, or pick up any of the many many things that can go wrong with your characters.
These "Afflictions" can cause them to disobey commands, refuse healing (because they don't trust the healer), or just increase everyone else's stress by being a jerk (causing everyone else to go crazy). Stress doesn't go away when they get out of the dungeon but you can lower it back in town, even get rid of problems they already have accumulated. Things like giving them some time drinking or having them pray can help them sort themselves out, or you could have them committed to a sanitarium.
The system is complex, interesting, subtle and engaging, but if we really want to talk about what makes it so great we need to talk some more about perma-death.
Perma-death (short for "Permanent Death") is when a character is for realz dead. The irreversible kind of dead. In some games perma-death means you have to start the whole game over again because you only control one character and they are now irrevocably dead. Others though you control multiple characters and so the death of a couple characters isn't the end of the road for the player, it is more like a speed bump. Games like Final Fantasy Tactics, XCOM, and Darkest Dungeon fall in the latter category.
I like perma-death as a mechanic, but that isn't to say that it works in every game. Some games use it to make the game artificially harder or more stressful, but it can be used well and deepen a game. No mechanic is good or bad but on their own (fine some are bad), but as I've said many times the same mechanic in different context can be either amazing or terrible. How well a mechanic works depends on how the rest of the game enables that mechanic, so with that in mind lets talk about how Final Fantasy Tactics, XCOM, and Darkest Dungeon try to support the mechanic of perma-death.
First lets talk about Final Fantasy Tactics. Now most of the characters the player gets (the ones that no longer matter to the story) can die and the game just keeps on going. Their death may be a tragedy, but it is also a footnote. The reality of one of your preferred characters dying is that you have to replace them, or rather you must already have a replacement for them ready. Otherwise any character death is followed by a grind fest in order to get a new character into a workable shape. We want the loss of a character to be a speed bump, but we don't want it to cause everything to shut down for a while. This means that we have to figure out a way to encourage the player to have extra characters sitting on the bench that are decently usable as a replacements for any character that might die. Not as good as, but one that can be used in a fight (by which I mean they aren't going to die when the enemy looks at them).
Final Fantasy Tactics does several things to encourage the player to have people ready to step up. First there are heavy mechanical differences between units which means that some units will do better on certain terrain or against certain foes. So the player is incentivised to have a diverse group of people to pull from because their needs for every fight can change which means they are naturally going to have a deeper bench. Secondly, the game changes how many units can be used in each fight. Sometimes you field 4 units, sometimes 8. This means the player will have at least a couple characters that they use only on a semi-regular basis. A B-team that they can pull from if a slot opens up on the A-team.
Even with these mechanics though the game still focuses on having a strong core team that you rely on and not on having a deep bench that is continuously rotated through. There might be mild variations in who is used depending on terrain or team size, but even with the terrain and enemy changing having a couple extra levels (and as such greater stats) is so strong that someone who has 2 levels on another unit is almost always the better choice even if that weaker unit is perfectly suited to the terrain of the level. You can't beat numbers. That means that your party gets completely wiped out, even it it isn't technically the end of the game, is going to be the end of the line for the player. A lot of players will simply stop playing or simply reload a safer save file. That is why XCOM games force the player to bench characters after nearly every mission, because they want the player to have a lot of units so that they can kill and entire squad without stranding the player without any usable units.
In XCOM characters who get heavily injured on missions have to heal for a while, meaning you can't use them. Someone can't get their head cracked open and barely survive only to go out on another mission the next day. This means that you will have to keep changing up your group as often many of your favorites will be in the medbay healing off injuries they took on the last mission. There are still soft reasons for the player to want to switch up their party like in Final Fantasy Tactics, but XCOM also forces the matter and forces the player to have a diverse cast of people to send out on missions. Some of them won't be coming back and the player needs to managing a team and not just a small group if they are to make it though the game.
Not having injuries in the game would actually make the game harder in the long run for newer players because it would make the deaths of their units significantly more devastating. A player might not be able to recover from having an entire team die because that was all the units who knew how to hold a gun that they had. But the game forces you to diversify and so losing an entire squad is a set back but not the end of the game.
They also do things to support switching up your characters like de-emphasizing levels. Sure, levels to matter a great deal. More levels means more abilities you can use. But a low level character can still be equipped with all the equipment that person sitting in the sick bay was using and be decently helpful.
The system isn't perfect though. One problem is that because it is tied to damage not all units are going to be getting benched regularly. For instance you don't need a large number of snipers you can call on, only 1 or 2 max, because they won't be getting hit. Sniper's roll is to be as far away from the action as humanly possible so you can just use the same one in fight after fight. Some classes though will be getting the shit continually kicked out of them, they will need someone to take their place on follow up missions.
A much more basic problem with the XCOM system though is that it is binary, either someone can go on missions or they can't. There is no wiggle room, no "They are a bit injured but I need them to do this anyways...". Here is where Darkest Dungeon's system shines like a beacon, because it allows for so many gradients and complexities. In Darkest Dungeon you often have to try and weigh the costs/benefits of bringing someone who is strong but stressed along on a mission versus someone who is a less experienced but calm and relaxed. You are worried about the veteran developing even more psychological problems and getting everyone killed, in fact they already have so many that you should probably commit them to the sanitarium. But the only other person you have available that has it even vaguely together is green and you aren't sure they will be able to make it through the mission alive even if they aren't going to be a danger to their teammates.
And everyone on the mission gets stressed, not just those in the front who get hit the most. Things that can stress a character out are things like not having enough food, getting hit by a trap, random interactions, walking through a dungeon with low torch light, and so on and so on. Half my roster is either crazy or getting drunk to deal with the stress of dungeon exploration, my favorite healer (seen above) has became a coward after a couple of her teammates died on a failed mission.
Final Fantasy Tactics has a bunch of soft ways to encourage the player to have a usable bench, and XCOM's system forces you to use more characters. Darkest Dungeon's system though doesn't just get you to have a large and diverse cast of characters to pull on, it forces you to make more choices in places most games don't even think to look. What makes this system so great isn't that it makes things harder, it is that it creates more choices with unclear answers. Questions that are interesting and meaningful to the people in the game world. That is what makes it great, choices, not difficulty.