Divine Favor (from a level 1 spell slot) lasts for up to 10 rounds, and on a hit does 1d4 radiant damage.
On a critical hit it adds 2d4 radiant damage.
Divine Smite (from a level 1 spell slot) is good for 1 hit, but does 2d8 radiant damage.
If the hit was critical it does 4d8 radiant damage.
Assumption 1: Combat lasts from 3-6 rounds
Why?
a. That's how long combat tends to last with this group
b. For a two round fight divine smite is without question the better choice.
Case 1: hit Armor Class 14
Case 2: hit Armor Class 16
While I can see Divine Favor being a decent use of a level 1 spell slot once the Paladin has two attacks per Attack Action at level 5, I am trying to nail down (in tier 1 play) how many rounds the combat needs to last for Divine Favor to be the better choice for a level one spell slot.
Assumption 2: paladin makes concentration save if hit during combat.
That assumption is to keep the question within a tight scope. As I discuss this with my paladin player, a separate risk assessment needs to be made vis a vis loss of concentration.
Assumption 3: single class paladin, level 2, 3 or 4.
My back-of-the-napkin-math says "4 rounds" but I may not be setting the problem up correctly. If every attack hits, 4d4 is slightly more damage than 2d8 on average: 10 versus 9. But every attack does not hit.
How many rounds does combat have to last to make divine favor the better choice?
My instinct is that against one big monster, use divine smite to nova the monster down; against a mob of small creatures use divine favor to whittle down a lot of targets – but that is a separate question/problem.
Note: I am the DM in this case, not the player. The Paladin is Dragonborn, and does not have the Sentinel feat. Strength is 18 (Rolled stats). Paladin is sword and board. An answer for a strength 16 paladin (point buy) is acceptable.
Lastly: if you believe that the answer is "It's never the better choice, and here is why" – demonstrating the "why" answers the question. (For example, if the "why" shows that it's not better until the seventh round, it meets the question/problem constraints).
Best Answer
It varies with enemies' hit points, up to turn 8
The actual damage dealt is usually not the thing we are interested in. A often more useful metric is how many (more) things will this let us
killreduce to 0 hit points. A major point here is that dealing 2d8 extra damage to something that would die to an extra 1d4 is wasteful. The problem is that how quickly something will die depends on how many hit points it has, so our result is going depend on that.So we need to compare how many enemies with a given hit point total and AC we can kill on average over a number of turns using each of two methods: an extra 1d4 on each attack or dealing an extra 2d8 on the first hit. For each of these methods we're ignoring other attacks being made etc., but that would make the modelling completely unfeasible.
Trying to do this analytically makes by brain hurt, so instead I'll be using simulation. The basic idea is to simulate the given scenario, note how many kills we get, and average across many, many runs of the simulation. My Python code here. A couple of notes:
I simulated crits, but not the paladin waiting on a crit to use their Divine Smite. The Smite is always used on the first hit.
The last enemy attacked is counted as a partial kill, with the fraction being their fraction of lost hit points.
I've run this with 1M tries for each data point (yes, this gives quite long run times), effective AC 8 (equivalent to AC 14 with +6 to hit), and using a d8 damage weapon (longsword, warhammer, etc.). Hit points were chosen with a scaling resolution from 5 to 120. (5, 6, 7, 8, 9, 10, 12, 15, 18, 22, 26, 30, 35, 40, 50, 60, 70, 80, 90, 100, 120.)
This can we generate two tables for the average killrate of each method as a function of number of turns and enemy hitpoints. From this we can obtain the relative difference between the two as a function of the two variables. In the below table a positive value (green) means using divine favor has a higher killrate, and a negative (red) means using Divine Smite is better. The plot of the relative rates can be viewed at one's own leisure, it's a bit messy.
There are a few things to observe from the above results. As enemy hit points become very high, the turnover point becomes 6 which agrees with considerations of damage only. This makes sense as little to no damage is lost as a result of the target dying.
At 15 enemy hit points divine favor becomes particularly poor, being more efficient only after 8 rounds. This is presumably because the enemy is just left at very few hit points quite often. Below 15 hit points favor becomes better earlier. This is presumably because the added d4 is often sufficient to kill of enemies where the normal post-Smite attacks do not. At 6 hit points favor is always better (assuming more than one turn) as any hit is guaranteed to kill with favor, but has a chance not to post-Smite. For 5 hit points either method does nothing, because a normal hit is guaranteed to kill the target. Whether the result is positive or negative for 5 hit points is down to noise/chance.
The above method only considers the paladin attacking the enemies. Trying to factor in party members makes the simulation both trickier (more things to calculate) and more specific (need to set attack options for each party member). In general when also considering the enemies potentially being attacked by others, each hit point would have contributions from lower hit points of the paladin only consideration.