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From AGEWiki

A retreat is checked before each battle round of a given battle. This also include round #0, which is the first round of battle, before any shot is fired. If a retreat succeed at round #0, then we are in 'retreat before battle' and no damage will be impacted on the retreating stack.

A routing is a forced retreat, it works the same as a retreat, but is forced upon a faction because its morale has collapsed. Routing is decided at the faction level, not at the side level (meaning if GBR + IND fight against USA, then the IND faction can rout before the GBR). Routing works from that as a retreat, but with augmented losses during Pursuit.

Retreat Chance formula and modding

Each stack check separately for a retreat, but if some already retreated, then the others get a bonus. Routing is always a success. Finally, there is an auto-retreat threshold, if the losses reach a certain %, then retreat ensue at the beginning of the next round, automatically.

NOTE: As retreat (or routing) is only checked at the start of each round, then a force can still be annihilated or suffers extremely high losses far above the auto-retreat threshold, if the opposing side has a devastating firepower compared to the resistance of the retreating stack.