# What deals more damage on a weapon, +1 to hit or +1 to damage?

Background: I'm introducing masterwork items for 5e as a homebrew rule, both to help players spend surplus gold on something useful when they cannot buy magic items, and to make shopping a bit more interesting -- e.g. when you buy a horse, do you want to spend extra for a well-bred horse that is a bit faster, or economize and get a nag that is slower?

For weapons, I have the option to make the masterwork items deal +1 extra damage, or add +1 to hit. I am currently going with the extra damage, to avoid any risks with bounded accuracy. I suspect that for the levels where such items are making a difference, the extra damage might actually be stronger, and I want the effect to be as small as possible.

Players will get access to magic as in typical D&D adventures, so by the mid-levels they likely will have magic weapons and this will matter less. They also can use feats. I'm most interested in the effect on level 1-10 characters, if that makes a difference or makes the answer easier.

For the campaign world (a rather generic D&D world like Forgotten Realms) assume that for the first 10 levels:

• Characters start with +3 stat bonus on their prime stat (race and point buy) and use their ability score increases to raise it to +5.

• Characters have a to hit chance of 65%: to hit grows from +5 to +9, for simplicity increasing every second level. Monster ACs average from 13 to 17 (see here), and again for simplicity, increase every second level.

• Characters deal an average 9 + level damage per attack (from crits, feats, spells, character features). Those with Great Weapon Master or Sharpshooter deal 19 + level damage instead, at 5 worse to hit (ignoring archery for simpicity).

All things being equal, damage bonuses are more valuable against low-AC targets where you're probably going to hit anyway, while to-hit is more valuable against high-AC targets where misses drag down the average a lot.

If you have a high base damage (that is, your damage total before adding a +1 for the weapon), then the to-hit is much more valuable, because you're depending on actually connecting to get that big ball of base damage, and the +1 is a proportionally smaller part of that damage total. (Taking it to the extreme can be useful for illustration: If you hit for 30 damage, adding 1 to that is nothing, while possibly converting no damage into 30 damage is a big deal!)

At low level, the differences are minimal, but grow as characters gain class features that add to their damage, such as sneak attack, smites, rage, and so on.

Or to put that another way, rogues and greatweapon users are going to want a +1 to hit at all levels, and +1 to hit remains useful across your career. A +1 damage favors characters who use smaller weapons, such as two-weapon specialists, and tends to fade over time as the base damage per hit goes up.

I would suggest having masterwork weapons just be +1 damage. In my opinion, +1 to hit will make magic +1 weapons feel much less special in a way that the damage bonus won't.

• I think this is the most well rounded answer so far. It lacks some of the rigorous numerical treatment of the others, but in exchange considers how this practically will play out with feats, spells and abilities that are to be expected. I do believe one probably can get more numerical in support of your final verdict, even if it is not easily done; at the same time I think it is true, and even if it makes the weapons a bit stronger in levels 1-3, probably the way to go. Apr 17 at 20:32
• There also seems to be a question of how important hitting (e.g., low hit point, higher AC opponents) and damaging (+1 damage appears useless against 1hp opponents). Low hp and low AC opponents might be used in low numbers for narrative reasons or in larger numbers for challenge. The durability of the PCs (which is also relative to the attack capabilities of opponents) would also seem to be a factor both for the kinds of challenges the DM is likely to present and the strategies used by the PCs. Apr 18 at 13:33
• Well, I was hoping my explanations of why various things work the way they do would obviate the need for exact numeric support. I explain why a high base damage makes to-hit more valuable in lieu of throwing up a spreadsheet, and give rules of thumb instead of an incomprehensible equation. Apr 18 at 13:33

### It depends.

As explained in greater detail below, +1 to hit is better than +1 to damage when the following inequality is true:

$$21+H-D

Where:

• $$\D\$$ = Average damage on a hit (before accounting for the possible +1)
• $$\H\$$ = Bonus to hit (before accounting for a possible +1)
• $$\AC\$$ = Target Armor Class

So the answer depends on the character's accuracy and damage, and the range of armor classes they expect to encounter. I chose this particular arrangement for the inequality because it provides a way to compare multiple character variables to a single campaign variable; it keeps my stats on one side and the stats of what I expect to be fighting on the other. For example, if you have +7 to hit ($$\H=7\$$), and deal 1d8+4 ($$\D=8.5\$$) damage on a hit, then we have:

$$21+7-8.5 = 19.5 < AC$$

With these stats, +1 to hit will be better than +1 to damage when the target armor class is 20 or more. For $$\AC\leq 19\$$, +1 to damage has a higher average damage. Darth Pseudonym's answer outlines some of the general intuition behind the results of this derivation. Finally, here is a table showing the "Armor Class where +1 Hit is better than +1 Damage", with hit bonus ranging from +0 to +13, and base damage ranging from 1d4+0 to 2d6+5:

Hit\Dam 2.5 3.5 4.5 5.5 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12
0 >19 >18 >17 >16 >15 >14 >14 >13 >13 >12 >12 >11 >11 >10 >10 >9
1 >20 >19 >18 >17 >16 >15 >15 >14 >14 >13 >13 >12 >12 >11 >11 >10
2 >21 >20 >19 >18 >17 >16 >16 >15 >15 >14 >14 >13 >13 >12 >12 >11
3 >22 >21 >20 >19 >18 >17 >17 >16 >16 >15 >15 >14 >14 >13 >13 >12
4 >23 >22 >21 >20 >19 >18 >18 >17 >17 >16 >16 >15 >15 >14 >14 >13
5 >24 >23 >22 >21 >20 >19 >19 >18 >18 >17 >17 >16 >16 >15 >15 >14
6 >25 >24 >23 >22 >21 >20 >20 >19 >19 >18 >18 >17 >17 >16 >16 >15
7 >26 >25 >24 >23 >22 >21 >21 >20 >20 >19 >19 >18 >18 >17 >17 >16
8 >27 >26 >25 >24 >23 >22 >22 >21 >21 >20 >20 >19 >19 >18 >18 >17
9 >28 >27 >26 >25 >24 >23 >23 >22 >22 >21 >21 >20 >20 >19 >19 >18
10 >29 >28 >27 >26 >25 >24 >24 >23 >23 >22 >22 >21 >21 >20 >20 >19
11 >30 >29 >28 >27 >26 >25 >25 >24 >24 >23 >23 >22 >22 >21 >21 >20
12 >31 >30 >29 >28 >27 >26 >26 >25 >25 >24 >24 >23 >23 >22 >22 >21
13 >32 >31 >30 >29 >28 >27 >27 >26 >26 >25 >25 >24 >24 >23 >23 >22

And if colors are your thing for visualization, here is the same table, with +1 to hit more favorable the redder it is:

As you can see, on the higher end of the damage scale, the range of armor classes where +1 to hit is better gets bigger. As base damage increases, the value of the damage boost decreases, and as base hit increases, the value of the bonus to hit decreases.

To use this table to make a decision, find the cell corresponding to your stats, and decide if the given armor class range contains ACs you expect to encounter. For example, if you are a greatsword wielder with 20 strength at 9th level, then you have $$\H=9\$$ and $$\D=12\$$, then you would expect +1 to Hit to outperform +1 damage for armor classes of 19 or more. If you expect most of the enemies you encounter to have ACs of 18 or less, choose +1 damage; or if you want to the better option for ACs of 19 or more, choose +1 to hit.

### Derivation

Thankfully, we do not have to account for crits because the expected extra damage from a crit is the same in either case and will cancel out. The formula for expected damage of one attack (without accounting for crit) is:

$$E(Damage)=\frac{D(21-AC+H)}{20}$$

Thus, to figure out which bonus is better, we just need to compare this formula in each situation. We will find the case where +1 to hit is better than +1 to damage, then you need only flip the inequality to find the other situation:

$$\frac{D(21-AC+(H+1))}{20}>\frac{(D+1)(21-AC+H)}{20}$$

With some algebra, we obtain:

$$21+H-D

When this inequality is true, a +1 bonus to hit will, on average, deal more damage than a +1 bonus to damage. To actually make a decision, just fix $$\H\$$ and $$\D\$$ to see the range of ACs where one is better.

Finally, I have discovered a truly remarkable proof for what happens when you have advantage which the margins of this answer are too small to contain.

• It might be worth plugging in a few more examples and comparing against expected ACs to determine if there is a good rule of thumb that either one or the other is usually better. (My money says that the damage bonus will usually be better.) Apr 15 at 18:51
• @Carcer Consider the form H < D - (21 - AC). If most ACs are in the range 12~20, say, then that's H < D - (1~9). So if H < D - 9, +1 to hit is broadly better. If H > D, +1 to damage is broadly better. And the middle is a range between those two points. Apr 15 at 22:45
• Your equations don’t account for the fact that a +1 to hit makes no difference if you need a 21 or mor (only crits hit) or a 1or less (since 1s always miss). Admittedly, this happens at the edges but in those extreme cases, the +1 to damage is always better. Apr 16 at 0:56
• I think this is very elegant overall, and well written. To me it would be desireable to include reasonable expectations for the chance to hit (from expected H, AC), and reasonably expected D (from compentent ASI/feats). I understand that this is less safe, as you have to make some assumptions about the monsters, but otherwise, this does not answer the question beyond "It depends", and thus does not help make the call what bonus to use. (PS: I like the Fermat riff at the end). Apr 17 at 20:21
• @GroodytheHobgoblin A person trying to make this decision should be able to use the formula just fine because they know what their numbers are. Apr 17 at 20:28

### The bonus to damage is more beneficial when the probability to hit is greater than the expected damage -1 divided by 20.

When the chance to hit is greater than the average damage diminished by 1 and divided by 20, then a +1 on the damage roll is better: $$p > \frac{\text{E}[d]-1}{20}$$

It then depends on the character, the average damage and on the average probability to hit. Consider a 1st level fighter with a longsword, as an example: the expected value for their damage is 7.5 (1d8+3, assuming an optimal strength of 16), then a +1 bonus on damage is convenient if they hit with a probability greater than 32.5%. Since the average hitting probability is 65%, such bonus is more convenient rather than a bonus to hit.

##### There is always a chance to miss.

Suppose that a fighter (5th level, STR 18) has a bonus to hit equal to +7, they are under the effect of Bless and they are fighting a group of Ogre Zombie. The probability to hit is 95% and not 100% since a natural 1 is always a miss.

##### Critical hits.

With a straight roll, if the only way to hit an enemy is a 20 on the attack roll then the expected damage is $$\frac{1}{20} 2\,\text{E}[d] = \frac{\text{E}[d]}{10}.$$ A bonus to damage does not change the probability to hit, it increases the final damage: $$\frac{1}{20} \left(2\,\text{E}[d]+1\right) = \frac{\text{E}[d]}{10}+\frac{1}{20}.$$

If even with a +1 on the attack roll one still needs a 20 to hit, then the expected damage is the same of a straight roll: $$\frac{1}{20} 2\,\text{E}[d] = \frac{\text{E}[d]}{10}.$$

##### Target's AC is 20.

When the AC of the target is 20, with a straight roll or with a flat +1 damage bonus one needs a critical hit: then the damage is $$\10\% \text{E}[d]\$$. On the other hand, a +1 on attack rolls allow to hit both with a critical hit and with a 19 on the roll: then the expected damage is $$\15\%\text{E}[d]\$$.

The Mathematical explanation is reported below, for the interested readers.

The expected value $$\\text{E}[D]\$$ for the damage is given by the sum of two terms:

• the probability $$\p_{nc}\$$ to hit without a crit multiplied by the expected value $$\\text{E}[d]\$$ of the damage roll.
• the probability $$\p_{c}\$$ to hit with a crit multiplied by the 2 times the expected value $$\2\text{E}[d]\$$ of the damage roll.

The final formula reads $$\text{E}[D] = p_{nc}\,\text{E}[d] + p_c\,2\text{E}[d].$$

In order to clarify this, let's take an example. Suppose that, considering all the bonus, one needs a 14 or more on the d20 to hit an enemy: then, the probability to hit is (21-14)/20=7/20=35%. But we have to discern non critical hits and critical hits for the expected value. Indeed, for rolls between 14 and 19 (extrema included) the expected damage is $$\d\$$, but with a natural 20 the expected damage is doubled, i.e $$\2\,d\$$. Then, the formula for the expected damage is $$\begin{split} &P(14\text{-}19 \text{ on d20 })d + P(\text{critical hit})d=\\\\ &\frac{6}{20}d + \frac{1}{20}2\,d =\\\\ &\frac{3}{10}d + \frac{1}{10}d =\\ \\ &\frac25 d. \end{split}$$

If the target’s AC is such that the d20 roll needed to hit is $$\x\$$, then $$p_{nc} = 1-F(x)+\frac{1}{20} - \frac{1}{20} = 1-F(x)$$ where $$\F(x)\$$ is the cumulative distribution function, $$\1-F(x)\$$ is the probability to roll a number greater than $$\x\$$, the first 1/20 term takes into account the probability to roll a $$\x\$$ and the last 1/20 excludes the prob to get a nat 20. Consider the above example for an explanation: if one needs a 14 on the roll to hit, then the probability to hit without crit is 6/20, which is exactly $$\1-F(14)\$$, where $$\F(14)= 14/20\$$.

The probability $$\p_c\$$ to get a crit instead is simply 1/20.

Gathering everything, one then has $$\text{E}[D] = \left(1.1-F(x) \right)\,\text{E}[d]$$

A flat +1 bonus to hit consists in increasing the probability to hit of a 5%$$\=1/20\$$ more. Then, the two scenarios are given by $$\text{E}_{hit}[D] = (1.15-F(x) )\,\text{E}[d]$$ for the bonus to hit and $$\text{E}_{damage}[D] = (1.1-F(x) )\,(\text{E}[d]+1)$$ for the bonus to damage roll. The breakeven point, i.e. where the two bonuses are equal, is given by $$\begin{eqnarray} (1.1-F(x) )\,(\text{E}[d]+1) &=& (1.15-F(x) )\,\text{E}[d]\\\\ F(x) &=& 1.1 - \frac{\text{E}[d]}{20} \end{eqnarray}$$ and then having a +1 on the damage is more convenient when $$\F(x) < 1.1 - \displaystyle\frac{\text{E}[d]}{20}\$$. Then, since the total probability $$\p\$$ to hit is given by $$\1-F(x)+\frac{1}{20}\$$, one obtains the final result $$p > \frac{\text{E}[d]-1}{20}$$

• convenient or beneficial? Apr 15 at 19:49
• +1, I think this is the most rigorous mathematical treatment of those offered, and it has the benefit that it also considers to hit probability. I feel like Thomas's answer, it would benefit from investigation what expected amounts of average damage could be at various levels. This is of course a very broad question and hard to generalize. I suspect that at tier one the +1 to damage is better, but with feats like GWM or Sharpshooter and more ways to gain advantage, at Tier two this shifts to favor the to hit bonus. But I cannot tell from the answer yet. Apr 17 at 20:28
• @GroodytheHobgoblin I may provide a couple of example, but really it depends on the character and on the monster one is fighting. Given the formula and the fact that the prob to hit is around 65% for any tier, +1 to hit is better when the average damage is 14. I will complete the answer asap, maybe tomorrow. Apr 19 at 21:49
• @Eddymage, thank you. Using my expected damage per level for typcial builds, this would mean that around level five +1 to hit is getting better than +1 to damage (for non GWM/SS). Apr 21 at 18:49
• @GroodytheHobgoblin Yep, it depends on the situation. I will try to adapt my answer with the new information you provided in the question. Apr 22 at 6:34

## In General, +1 to hit is more valuable than +1 to damage.

At most levels of AC present in the game by monster weight (number of monsters with) and common npc choices, factoring in to some degree the more 'popular' types of enemies, and the class abilities and expected ability scores of most characters using the common generation rules at the typical levels of play (1-3 and 3-11, with most time spent at 3-7).

The degree to which it is more valuable (in terms of Average Damage Output) depends heavily on whether Feats are in the game, and to what degree players have managed to optimize their characters to gain regular Advantage. In general, at higher levels of optimization (which is quite common in 5e - the simple nature of the game means that optimizations for character success are well known and publicized) the +hit will be worth more in terms of overall damage output, as attacks will do more damage and/or take advantage of Great Weapon Master/Sharpshooter to directly transfer to-hit into damage, but overall the value of the +hit greater than +1 damage in most (nearly all) cases.

At first level, or with very unoptimized parties, the reverse can be true, but is not guaranteed to be.

Therefore we can fairly safely say that if you want to make degrees of quality with weapons, adding +damage will be a safer/weaker bonus than adding +hit. I would say you could add up to +3 damage at most tables with it still being weaker than +1 to hit, although again, that will change depending on level of optimization etc.

• Feats would be in the game, I can update the question to clarify that. I think this gives a helpful perspective on the matter and am not sure why some downvoted it, maybe because of the claim in the last paragraph, which is a pretty bold and might benefit from backup calculation? I think you are right, with GWM/SS, +1 to hit translates to +2 to damage without penalty and should be stronger. Based on Eddymage's answer, at 1st level +1 damage is generally clearly stronger, maybe not be for the corner case of a variant human with GWM/SS. Apr 17 at 20:44