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The Random Encounter System posted:

The way the random encounter system works is that each tile you can step on in the Labyrinth has a certain value assigned to them. Most of the time, this number ranges from 0 to 5, but rarely you'll see a number from 6 to 9 as well. This is known as their Danger Value. This value is then added onto a hidden counter, which is indicated by the random encounter radar turning from blue to red as it increases. When you get out of a battle or enter a new floor, the game sets a random limit on the counter, ranging from 16 to 64 inclusive. When the counter reaches or surpasses that limit, you get into a battle. Skills can increase or decrease the danger values of the tiles. Troublemaker will increase the danger values, Vigilance will decrease them, and Safe Stroll will force all tiles in the Labyrinth to have a danger value of 0, making it impossible to get into battles in that case.

This ties into FOEs in that a significant portion of getting through the Labyrinth involves navigating around FOEs and avoiding them. FOEs help provide tension when you're exploring the Labyrinths since they outclass your party (Or are supposed to, but let's ignore how weak they are in this game for now). What doesn't help matters is that they can join in on random battles, so traversing their territory can be dangerous. But you shouldn't worry too much. Typically the tiles in the territories that FOEs live in have low danger values, so it's possible, but hard to get into an encounter in there most of the time. But you should make sure to reset the counter by getting into a random battle outside their territory before attempting to traverse through it. Tiles near chests also tend to be fairly dangerous, so make sure you're prepared before you go on over to open those.

The encounters you get into also depend on the tile you're on. This is accomplished by assigning each tile a list of possible groups of encounter formations. So a monster that's a rare encounter on one part of the floor might be a common encounter on a different part. In some cases, they can even change based on the time of day, or if you have certain quests active. If you're trying to hunt down a certain enemy and aren't having much luck finding them, try looking on a different part of the floor.

A couple of things to note here. While the encounter system is overall similar to Etrian Odyssey 1 and 2's encounter systems, there have been some changes made to it, and a bug introduced in one case in an attempt to fix a bug from the previous system. First of all, in previous games, you only got into an encounter when your Danger Level surpassed the generated threshold. In this game, you get into an encounter when you reach or surpass the generated threshold. This means that you'll get into encounters slightly more often than in the previous games on paper, but in practice that's not really the case, as encounter rates are a bit lower overall in the maps for this game.

Another change is that in the previous games, your danger level is stored as a 1 byte integer. This means that decimals are not taken into account, and ways to reduce the danger value cause the game to round down to the nearest whole number, making those methods far more effective at reducing encounter frequency compared to what their data would suggest. This is no longer the case in EO3, as the danger level is stored as a 4 byte floating point number. This means that decimals are taken into account, so encounter reduction methods do exactly what they intended.

As for the bug, the previous games had it so that no matter how you started off a floor, whether it was coming from a set of stairs or a Geomagnetic Field, your danger level always started out at 0. In EO3's case, falling through a bottomless pit, going through a set of stairs, or warping to a Stratum through the use of a submagnetic pole, causes it so that the tile you start out on for that floor is added to your danger value, so you don't start off with a value of 0 in those cases. The exception is if you warp back to those points through the use of Tonsou Jutsu or a Pole Stone. This bug was introduced in an attempt to fix a different bug from the previous games which caused it so that going through doors, shortcuts, teleporters, and forced movement tiles did not add the danger value of the destination tile to your danger level. So while all those cases were fixed in this game, it did introduce these odd cases, which proceeded to be fixed in later entries.

One last thing. While you always do start out with a danger level of 0 when leaving a random battle, escaping from an FOE causes you to take a step back from it. That's not just a cosmetic effect, it does count as a turn passing and does add on the danger value of the tile you stepped back to, despite the fact that no other FOEs will move during that time, so be aware of that.