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"running away was futile" - I don't believe that for a second. Of course most casualties happened during pursuit, but that's not the way to look at it if you are a soldier on the losing side of a battle. In fact most soldiers who rout could survive and that would not contradict the statement "most casualties happen during rout"...

From the perspective of a common soldier, once the battle goes badly and the enemy starts to get into your line's rear, he has to consider where he is more likely to die - in the failing battle line, with his officers (who unlike him will can surrender to the enemy and expect to be treated well), or on the road / in the hills / in the forests away from the enemy sabers and lances. The decision to bail and run would in most cases be the better decision.

"Often quite futile" and comparable to the chance of escape when running from mechanized troops in modern combat was the point I was trying to make.

As for running being the better decision it could certainly be true in many situations, especially if there is danger of being encircled. However, in the slow shoving matches of ancient battles a soldier would be more likely to survive a losing battle if his unit stayed in formation and fought on until nightfall rather than if everyone made a run for it. Once the army started to rout it was of course wise to join in .
 
"Often quite futile" and comparable to the chance of escape when running from mechanized troops in modern combat was the point I was trying to make.

As for running being the better decision it could certainly be true in many situations, especially if there is danger of being encircled. However, in the slow shoving matches of ancient battles a soldier would be more likely to survive a losing battle if his unit stayed in formation and fought on until nightfall rather than if everyone made a run for it. Once the army started to rout it was of course wise to join in .
I think the problem here isn't that it is the rational decision. It's a panic, with people just wanting to get as far away from carnage as possible. I think even if the soldiers knew that running away was 100% futile (which it definitely wasn't), as soon as a unit breaks, it breaks and all its soldiers just run, as fast as they can.
 
In that regard I find the question of what made people rush out of their trenches into machine gun fire more interesting than the question of what made them run away when a unit breaks. Although both probably aren't exactly rational decisions, I find one decisively easier to understand ^^;
 
Actually, in the former case, where rushing out of a trench into machinegun fire would tend to result in the first to run being cut down by a hail of bullets, the later routers would stand a marginally better chance of the machineguns either overheating or being in the process of reloading. In the case of a line breaking, the first to rout would be the most likely to escape (unless their own side posted guards to stop them), and those who hung on the longest would be most likely to be the ones run down by the opposing cavalry, well after the first fleeing troops had left the immediate vicinity.
 
On the pre-modern battlefield a formed unit that held until the end of a battle tended to take fairly small casualties, often a few percent, even in hotly contested battles. However, if it broke then casualties could be much higher, with those men who fled last the most likely to die.

This puts the individual soldier in an interesting conundrum, a bit like the prisoner's dilemma, if he holds and the unit around him holds then he has an excellent chance of survival. If he holds but the units runs he has a very high chance of death. If he runs first he has a higher chance of death than if the unit holds but a lower chance than if it runs. Hence, as soon as someone starts to run everybody else has to join them as quickly as possible or risk being the last one to run, and hence the most likely to die. In this situation the degree of trust each soldier has in his comrades is critical.

This appears to be part of the reason why the Ancient Greeks and Macedonians put a row of officers at the back and front of their formations - file leaders to lead the unit and file closers to keep the men at the back from running.
 
A little bit of trivia: the term "trophy" comes from the Greek "tropaium" or "turning point". So critical was the moment when the first enemy soldier turned to start the rout that, in the aftermath of the battle, a totem would be erected on that exact spot on the field - the "turning point".

The tropaium was usually a wooden pole or tree trunk, slung with defeated enemy's armor, helmet, etc. It marked "ownership" of the battlefield, and sanctified it. It was raised almost immediately, so there was no rallying & returning to the field once it is trophy-marked. Dead bodies could not be recovered for burial on a trophy-marked field without the victor's permission, i.e. a treaty.

The times when different opposing wings routed sometimes led to two competing trophies being raised on the same field by opposing armies.