Action Economy and the Power of Numbers

Something I've been thinking about writing for a while...

Understanding Action Economy Correctly

It's conventional to declare that groups in D&D are disproportionately dangerous due to 'action economy', but a bit of inspection shows that this is oversimplified: a half dozen bandits have a big action economy over a purple worm, but no-one would think the bandits are more dangerous. On the other hand, they might well be more dangerous than a bandit captain. The proper way to look at this is:

The threat an enemy poses is equal to the number of actions they get over the course of an encounter, multiplied by the value of those actions.

How Numbers Influence Threat

Let's say our party of PCs can drop one monster per round on average. Against one monster, that means (depending on initiative order and luck) the monster goes zero or one times; against multiple, while the first monster goes zero or one times, the second monster goes one or two times, the third goes two or three times, and so on. The expected number of monster actions, and thus the total threat, winds up being (number of monsters) * (number of monsters) * 1/2. This may be familiar as Lanchester's Square Law.

The above assumes that all the monsters can take effective actions. If this is not the case the advantage of numbers is reduced; for example, if you're fighting in a choke point where only one monster can attack at a time, the threat of a group of monsters is linear in the number of monsters (Lanchester's linear law). The threat of a groups can also be reduced by the use of area effects. You can make those laws more general by introducing a Lanchester exponent, indicating the true power of numbers, which is 1 for the linear law, 2 for the square law, but can have other values for specialized situations; 1.5 (N*sqrt(N)) is a fairly common approximation for modern warfare.

Understanding 2014 Monster and Encounter Building

The 2014 rules appear to have been designed by someone familiar with the scaling I discuss above, because it turns out the xp value of a monster can be approximated as (dpr) * (hp) * 2 / 3. If we assume that HP are a proxy for mean survival time, which seems like a fair assumption, the xp value of a monster is just equal to its expected damage against a standardized party. The 'group size' modifier is roughly equal to the square root of the group size, so it turns out that adjusted encounter XP is equivalent to a lanchester exponent of 1.5.

While this model is mathematically justified, it has two problems: it works very poorly when monsters have different xp values, and it's a big pain to actually use.

2024 Monster and Encounter Building

The 2024 rules do not appear to have been written by someone who understands Lanchester's Laws, because there are really only three significant differences from 2014: monsters do about 20% more damage at the same xp value, actual budget values got tweaked, and the group size modifier got killed. The last is a serious problem, because it means groups of monsters are dramatically more dangerous than single creatures at the same xp value, unless you happen to be fighting in a situation where the Lanchester exponent is 1.

How the power of numbers varies with encounter design

While 2014 used a Lanchester exponent of 1.5, and 2024 uses 1.0, the actual appropriate value can very quite a bit with encounter design. The big factors to consider are

1. Are the monsters all able to effectively target the PCs? This is most often a concern for melee, but can be an issue for ranged.
2. Are the monsters likely to bunch up (making them easy targets for area damage)? Again, most often a concern for melee.
3. Do the PCs have access to area damage with no friendly fire concerns?
4. Do the monsters all arrive at the same time, or over the course of multiple rounds?
5. Do the monsters have high initial effectiveness that drops off rapidly?

If none of those apply, the exponent winds up at 2; a large number of ranged enemies in an open field battle is a big problem. Also, note that this same list of factors determines how effective the PCs are against the monsters.

When Action Economy Really Matters: Status Effects and Non-Damage Actions

In a simple slugging match, it doesn't actually matter whether an enemy gets 1 action for 20 damage or 4 actions for 5 damage -- either one winds up being 20 damage. However, this dramatically changes when dealing with status effects, because the value of a status effect is equal to the change in value of the actions it affects. Stun a monster that does 5 damage per round and you've prevented 5 damage; stun a monster that does 20 damage per round and you've prevented 20 damage. This also applies to buffs -- cast bless on someone who attacks once per round for 8 damage and it's worth an average of 1 damage per round, cast it on someone who attacks twice for 16 and it's worth 4 damage per round. This is why monsters intended as solo bosses need methods (legendary resistance, immunities) of dealing with status effects, and conversely, weak monsters with incapacitating status effects can be a threat entirely out of proportion to their nominal power. 

This is also an issue for non-damaging effects. For example, let's say you're up against a guard captain. Under normal conditions, if he has a pair of guard with him, they're barely relevant... but if there's an alarm bell that takes an action to ring, it's suddenly very important that there's a guard available to run off and ring the bell while the captain fights.