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Oldschool D&D has rules for morale, such as these from Labyrinth Lord:

Monsters have a listing for morale, which represents how likely they are to fight or flee when in an aencounter. Morale is rated from 2-12, and while a score of 2 indicates that the monster never fights (unless absolutely cornered), a score of 12 indicates that the monster will fight until killed, with no morale roll necessary in either case.

The Labyrinth Lord usually makes a morale check under two conditions, when one side of an encounter has lost a member due to death, or when half the group on one side is either killed or otherwise incapacitated. The Labyrinth Lord will roll 2d6, and if the roll is higher than the morale of the monster, the check is a failure and the monster will either attempt a full retreat or a fighting retreat. If the result is equal to or lower than the morale score, a monster will continue to fight. If this roll is made successfully two times in one encounter, the monster will fight until killed.

Is there an equivalent variant rule in D&D 3e / D20?

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No, nothing official. You can make Intimidate checks but that's a poor substitute. I made a quick cut at a 3e morale system with two factors, aggression and morale, on my blog. There's not a lot of reason not to just use the old school morale system straight, though, it's not like you threw away all your d6's with the d20 system.

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The stumbling block is the lack of a Morale stat for creatures. If a DM has the old books that fills in a lot of them (at the expense of extra page-flipping), but the rest add extra prep or on-the-fly judgement burdens. –  SevenSidedDie Sep 20 '12 at 16:18
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Well sure, and there's plenty of retroclone Monster Manuals out there to pull values from. Best thing to use till Paizo pulls their heads out and adds it back in in PF 2.0! –  mxyzplk Sep 21 '12 at 0:32
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Going to crib your morale system for my own game. :] –  tex Apr 3 '13 at 19:49

Actually, there is something official out there; Page 72 of Heroes of Battle. The rules are specifically for large scale battles, but the book has the following to say about using the rules in any fight;

You can also take these rules beyond the battlefield if you wish, using them in dungeon-based adventures, urban settings, or elsewhere. In most cases, using the morale rules means that monsters run away more often than they do in a typical D&D adventure. That situation is common in organized warfare, but rare in a dungeon because the monsters are found in their lair and have nowhere else to go. In contrast, a smart commander encourages her units to retreat (or at least make a strategic withdrawal) when they’re over matched or needed elsewhere. If you use these morale rules in a traditional D&D adventure, make sure you account for the greater likelihood of NPCs retreating.

It is, at its essence, simply a DC 20 Will save to avoid the effects of fear. The check is modified by certain conditions, like overwhelming force or some such. This is not exactly what you're looking for, but in my experience it is almost always best to make new rules use similar mechanics to those already in the game. Tacking on new rules is troublesome enough, we don't need to redefine the mechanics each time.

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Yeah, but this is a pretty bad system because it means fighters are the most likely to give up in battle (low Will saves). Don't think it gives the results one would expect. –  mxyzplk Oct 5 '12 at 23:18
    
That is true. I guess that's why having a leader who can bolster them against fear effects is so important. –  Robobot Oct 5 '12 at 23:47

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