Is 5e character advancement commutative across multiclassing?

Question

Is 5e character development commutative across multiclassing? I'm talking about the mathematical property where given two operands and an operation, \$x \circ y = y \circ x\$. That is, regardless of the order of the operands, the result of a commutative operation remain the same.

We can rule out character creation, because that is not commutative. A fighter 1 / wizard 1 will have different game statistics than a wizard 1 / fighter 1 (saving throws and other proficiencies, for example).

Also, let's disregard the randomness of hit point rolls. Assume all characters take the average of the dice.

So, for two characters with the same 1st level class and the same features, who reach the same levels in the same classes but different advancement paths (i.e., the order in which these features were gained was different), do they have the same game statistics?

Exempli Gratia:

Assume two 1st level Rogues, Alice and Beatrice.

Alice gained 3 levels of Rogue and on her 4th level of Rogue, instead of an ability score increase she picked up the Martial Adept feat. She later gained 3 levels in Fighter, and chose the Battle Master archetype.

Beatrice gained 3 levels of Fighter, and chose the Battle Master archetype. She then gained 3 levels of Rogue, and instead of an ability score increase she picked up the Martial Adept feat.

Alice and Beatrice each now total four Rogue levels and three Fighter levels, obtained via different paths. Is the superiority dice gained from the Martial Adept the same dice for both Alice and Beatrice?

Meta: I know the issue above could sit in its own post, but then it would weaken the broader question.

Answer 1

In general, yes character leveling is commutative

To understand why, we need only consider the factors that change on a level up:

1. Hit Points: From PHB 163

   You gain the hit points from your new class as described for levels after 1st. You gain the 1st level hit points for a class only when you are a 1st-level character.

   This indicates that, regardless of when the levels are gained (aside from 1st level) each level gains (class hit die + constitution modifier) hit points. The net result will be the same.

2. Class features: These always occur based on class level, not character level.

3. Feats: As long as the pre-requisites are met, it doesn't matter when a feat is taken with regards to effect.

I think that is everything. In general, assuming identical starting builds, the pathway taken shouldn't matter assuming I haven't overlooked anything.

Answer 2

The career of a 6 wizard/6 fighter vs. 6 fighter/6 wizard

while the answer is yes, they will have the same game statistics abstractly, HOWEVER, in play, getting to those levels they will not, and there will be many level ratios where their statistics will be very different. This is because unlike 3e, in 5e, ability score increases (ASIs) are tied to your class level, and each class has its own ASI table, independent of the total character level, just like multi-attacks. Also, because things like extra/bonus attacks don't overlap, access to a spell or effect that grants extra attacks will clash with the fighter's existing multi attacks. This happens again and again, and because a wizard's innate abilities or spells can duplicate a fighter ability, the non stacking rules will create inconsistencies DURING PLAY where the commutative property doesn't hold true until after the levels reach parity.

The fighter gets an ability score bonus at 6th level that the wizard does not get, so while you were rising in levels, if you had 16 total levels, 8/8, at 16th level, you would have +3 (6 total) from fighter and +2 (4 total) ability score modifiers from wizard, and that would look commutative.

But if you took the levels back and forth, like 2/2, then 3/3, you would have no ability score bonus at all for the first 6 levels, until 7th then 8th level at 4/4. Again, 9th, 10th, 11th level, 4/5, 5/5, and 5/6 would create an experience where the order you took the classes changed the level you experienced the stat increase from 6th level fighter. At 12th level or 6/6, it would again no longer matter, abstractly what order your 12 levels came in, but during those 11 levels of actual play, it could make a considerable difference.

Since fighter gets a lot of ASIs compared with wizard, if you took fighter levels first, your wizard would have considerably higher stats, and if you were sneaky, you could use your fighter levels to get to a 20 intelligence faster, such a level 6 fighter going from a 16 intelligence to a 20 intelligence. Then swapping to Wizard, you have the modified Saving throw difficulty for a 20 intelligence. Additionally, the proficiency bonus is level neutral and totally commutative, so when you become the 1st level wizard/6th level fighter, your proficiency bonus is +3, so your Save DC would be 8+5+3, or 16! When you got to a 3rd level wizard/6th level fighter (9th total), your save would be 8+5+4, or 17.

Compare that with the first level wizard, 16 intelligence, 8+3+2 = 13 Save, and at 6th level, you only got one intelligence bonus, 18 total, 8+4+3 = DC 15. Then it would be 10th level, or 6th wizard/4th fighter before you got your intelligence bonus, 8+5+4 = 17 again.

So at 10th level, they reach parity again, but as you can see, their stats and saves are very different during actual play, depending on the order they take their classes, even if they have the end goal of reaching exactly the same class levels and take exactly the same options and start with the same ability scores.

Strangely, the Wizard who became the fighter on their 7th level still has their +3 proficiency bonus to attack rolls, and the 6th wizard/3rd fighter has a +4 proficiency to attack, with 1 ability score. Which could be strength, going from, say, 16 to 18 (+4), so the Wizard-Fighter has a +7 to hit at 7th level total (6/1) and a +8 at 9th level (6/3) and a +9 when their strength goes to 20 at fighter/4th, 10th level total.

Swapping that around, a Fighter-Wizard could by 6th level have a 20 strength, +8 total to hit, and +9 at 9th level, with no difference at 10th, 11th, or 12th level.

In both cases, taking the fighter levels first advantages you for spell saving throw difficulties, for attack rolls, and ranged/melee touch attacks, where taking the wizard levels first only provides higher level spells early on, but those spells will be weaker. This can radically change how the characters are played - spells for the Fighter/Wizard could be low level and reliable, while spells for the Wizard/Fighter could be higher level, such as 3rd level spells, but they would be less reliable and you would never want to use a save to Negate spell, instead relying on "always works/flat damage/utility" spells. The differences are NOT trivial.