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I'm about to run a premade adventure, and reading through it I see a potential situation where one group of monsters could, though PC actions, end up fighting each other.

As a PC, I hate seeing my DM, for lack of better words, play with himself. NPC vs. NPC combats played out blow-for-blow in initiative order, while the party fights other foes or just ends up watching.

Is there a tried and tested better way of doing this?

I want the PCs to be able to take advantage of the situation, but ideally not track HP and roll damage round for round if they decide to sit and wait.

If it's a full "background battle" then I'd assume the highest CR group wins, with losses.

But if this is not the case - both enemy groups are hostile to the party and each other - how do I resolve the NPC-NPC combats quickly and cleanly so my players aren't bored waiting for the results?

Ask the players when they want to intervene, and narrate until that point. Don't get bogged down in mechanics.

The normal flow of the game is for the DM to describe the scene/scenario, the players to declare their actions/intentions, then listen to the DM describe the result (with dice rolling as necessary). If the players choose to sit out of a conflict between NPCs, there isn't much else to do. The players are there to have fun, and if they want to sit out of a fight, they must not think that that fight would be fun, so hit the highlights and move on.

In my experience the best way to handle this is to make your best judgement on either what the outcome would be, or what it should be, either what makes sense or what makes the best story, then start describing how you get there from here. Roll one or two dice if you want help deciding some major turning points, but the story isn't about them, it's about the players, so keep it brief. Focus on the highlights and things of particular interest to the players.

If you're concerned that the players will want to intervene eventually, then describe the conflict in stages, giving the players easy/sensible times to do so. This can be done either explicitly, where you ask "Do any of you want to jump in yet?" or implicitly, where you pause and gauge the players by their expressions, body language and questions. If they still don't want to act, then move on.

In my opinion going through the full combat rules isn't necessary here. As DM you should be familiar enough with both the rules and the story to be able to make a sound judgement call on what the numbers should be if the players decide to surprise you during or after the fight. If the players do decide to jump in, reroll initiative from scratch, as the addition of a new fighting force is enough to throw scrambling.

Remember that experience points should be granted for overcoming obstacles. If the players orchestrated events to cause one band of monsters to eliminate another, then I would reward that success. If this conflict was already in motion and the players sat idly by, then at least their time wasn't wasted while getting to the parts they were interested in.

Give the players something else to do while the sides fight

If you're concerned about them potentially getting bored watching an NPC vs NPC battle, you could give them the chance to do something else while it's ongoing.

Examples:

• Are the NPCs no longer paying attention to the party? - If so, the players could see if there's a camp or something nearby of theirs's to raid while they're all busy fighting.




• Are there any hostages that need freeing or suspicions that there might be?




• Do the players favour one NPC side over the other? Or even a specific NPC? - Give them the chance to try tip the scales. Maybe an archer sends off a potshot or a magic-user casts a ward.




• Perhaps they want to use this opportunity to try sneaking away and so need to try that - while the NPCs are hypothetically too busy fighting to notice each other to notice.




• Have something going on/extra details that an observant PC could potentially spot. Reward observation and such initiative. Perhaps an NPC is holding a MacGuffin or there's another observer watching in the shadows, or someone has a plot relevant insignia on, etc.

This of course depends on your campaigns specific circumstances and is not an exclusive list/all options - the players themselves depending might have ideas too on how they'd want theirs PCs to (or not want to) get involved so definitely make sure they're consulted on that, which also helps them be involved and stay invested - but giving some other options is a good way to prevent the players from twiddling thumbs and keep a feeling of agency.

But

Keeping in mind that as they may well jump in (which you yourself say is an option you want had), you will need to keep a track off the individual HPs and initiatives. Otherwise that could be an issue - if a rogue does let off a potshot at someone, for instance, you need to know if that does some significant, which isn't possible if you don't know the HP of who is hit.

This can be integrated with "give them something else to do" - while the players are busy with that, you could be rolling at the same time/on the side (roll for them, then while they do their bit you roll for the NPCs), so things don't stall up and keep moving faster than they would otherwise.

On experiences as a player:

To give more specific experiences/contrasts to hopefully help:

• What worked: There's been a campaign I was involved in, which involved a big fight with a monster and several guards. The players also got involved and tried to fight the beast - for the NPC vs NPC aspect of that, what the GM did was roll on the side, like what I mentioned before. We would only really hear about the guards when they died and the specific details were quite vague - which helped move the fight much faster along than it would have otherwise (meaning less of a tedious wait).
  And it also meant we as players could still land damage or hypothetically assist the guards if we chose to, with their HP being quietly tracked (opportunity didn't arise, but it was there - so feelings of agency).




• Potential Risk: With the something else to do, there is a definite risk of going a way not expected I will say that and that may need quick thinking (if someone runs away for instance - what then? Especially if there's a party split, which I have seen happen).




• Consideration: But - if you don't ask and give opportunities, there's a risk of feelings of railroading/being ignored: for instance, if a character has a specific way they would react to the situation (flee/get involved/pickpocket) and you don't give them that chance other than to watch, that can lead to upset players who are taken out of the immersion. Even if they choose to just watch, at least they were asked and given that player/PC agency- and that's a happier table there.




• What worked: With the investment - NPC vs NPC fights work best when the players have either investment or reason to get invested in them. If that cult is fighting with that adventurer over there - and you are on a mission about that cult - then you are invested in the outcome.
  If it's people you have never met before and have no reason to care about either way what happens in the fight, or ways to learn about why you should, then it's more of a drain. Another example again being the monster vs guards above - one, good characters may want to help anyway but even aside, players were invested because that monster had also nearly killed one of them once - and was near where they all currently lived. So, the outcome of the battle was important to them.




• What Worked: With the extra details - In several games I've been in, players will try look for this anyway - roll observation. Whether to learn and/or to help decide what to do next, depending on the Player/PC. Thus making this something that could be used and I'd advise be at least considered - as "nothing special/of note" in response to a high roll can be hard to progress from, from the perspective of player trying to reach a decision. Also potential plot hook material here, depending on your campaign.




• What worked: I've also been in a oneshot where a lot of it was background by default. Those NPCs off there are fighting, you hear it and you know it's getting closer and risking effecting you - the players were all still progressing on their own agenda of that mission, but the details of the other stuff also added an urgency to it and feeling of risk: we were involved/invested as a result of that, even though we physical weren't. In that case, the HPs weren't tracked - but there was not a chance to join in, it being too far away and not in player reach.
  This meant, that when the NPC vs NPC finally did catch up too - it felt like a pay off more than anything of all the build up (making build up another positive thing to increase investment): although at this point, the GM did have to start tracking HP as players also got themselves involved in this: deciding to back a side.

So TLDR: Going from experience as a player - in games where agency "things to do/options" are provided the players tend to be much happier as a result rather than twiddling thumbs, and when rolls are done on the side/in-background it's often streamlines and progresses events faster, again keeping the players more engaged.

Simplify the NPC-NPC combat.

Let each side of the conflict be represented by a single dice roll each round. Apply an appropriate modifier to the rolls to represent their relative strengths and compare the totals directly, granting the side with the higher total a "win". Narrate the combat accordingly, giving the fight some flavor and allowing the players to interpret and engage as they wish. The overall outcome can be determined by whichever side is first to 3/5/7 "wins", first to 3 wins in a row, or some other similar solution.

This keeps the combat simple and quick, and with some room for players to respond in between rounds, allows for player involvement as well.

I've used this strategy with my groups both while their characters are busy with other tasks, or when they're just spectating. Both scenarios it works well, because it's fast and simple enough that players can follow along without getting bored.

If players do enter the fray, you can easily roll into a standard combat encounter, assigning some damage and status effects to the NPCs as would fit the narration thus far. There very likely won't be the opportunity for a long drawn-out fight at this point, as both sides should have exhausted some resources (including hit points) and players being involved will help keep their engagement up and their boredom down.