Are attack rolls or saving throws more consistent or predictable?

Question

In D&D 5e, spellcasters often have to choose between spells which affect a target on a successful melee attack and spells which affect a target on a failed saving throw. The fire bolt cantrip (1d10 to 4d10 damage depending on character level), for example, requires an attack roll, while the poison spray cantrip (1d12 to 4d12) requires a Constitution saving throw by the target. Obviously, this difference could also apply to class features that require a saving throw to avoid an effect.

Assume two hypothetical spells which are the same level, cause the same damage of the same type, and affect a single target (and can thus be most directly compared with a single attack roll). Assume no resistances or immunities for the target, and assume the character is facing a creature of a CR matching their level (as they often would be as part of a party of 4-6). Also, assume that players are experienced, but are not allowed to consult the Monster Manual or other DM resources.

Possible considerations:

• Many experienced players will know the AC of common monsters, but even when they don't, it's easy to figure out after a few attack rolls.
• Most players won't often have knowledge of full monster ability scores, which determine saving throws.
• Many DMs roll in secret. Even if they roll adversary saving throws in the open, the fact that there are 6 potential saving throws (and only 3 common ones) will make it hard in any given combat for a player to determine a creature's saving throw bonus.
• Monsters have guidelines for ACs that are typical for a CR ("Monster Statistics by Challenge Rating", DMG p. 274).
• Many monsters do not have any saving throw proficiencies. The DMG recommends that saving throw proficiencies primarily be used to counteract saving throw penalties from low ability scores.

Given this, and other considerations that you explain in your answer, from a player's point of view will a character succeed more consistently with an attack roll or a saving throw, or are the results of attack rolls more predictable than saving throws?

Answer 1

I have actually done analysis on this in the past and the answer is:

About 63/37 on AC vs. Saves, and it really depends on what you expect to see

The way I determined this was that I established equivalent levels of AC and Saves which would have the same rate of defense against a spell cast by the same caster.

You as a caster have a spell attack bonus of \$\text{Proficiency} + \text{Casting Stat}\$.
You as a caster have a spell DC of \$8 + \text{Proficiency} + \text{Casting Stat}\$.

So, we can deduct the variables from the situation and reduce the problem across casters.

Assume an attack bonus of \$+0\$ and a DC of \$8\$.

A spell with a DC of \$8\$ will be failed \$7\$ times out of \$20\$ with a save bonus of \$+0\$.
For a spell with an attack bonus of \$+0\$ to hit \$7\$ times out of \$20\$, the defender needs an AC of \$14\$.

This gives us an answer on a per-defender basis:

For defenders with an AC of over \$14\$ more than their save bonus, attacking Saves is easier.

Including most content released at least over a month ago, the splits across all published creatures seems to be as follows. I considered only the following defenses due to their common occurrence in spell effects: AC, CON, DEX, and WIS.

• 503 creatures have AC as their weakest defense
• 22 creatures have CON as their weakest defense
• 96 creatures have DEX as their weakest defense
• 178 creatures have WIS as their weakest defense

In case of ties, preference was decided as follows: AC > DEX > CON > WIS.

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Some weakest defenses by creature type

• Beasts 104/119 are weakest to attacks on AC.
• Constructs 6/36 are weakest to attacks on AC.
  • 21/36 are weakest to WIS saves, but are immune to many spells which use those saves.
• Dragons 39/47 are weakest to attacks on AC.
• Monstrosities 68/96 are weakest to attacks on AC.
• Ooze 8/9 are weakest to attacks on AC, Elder Oblex being weakest to WIS saves by a small margin.
• Plants 21/28 are weakest to attacks on AC.

This section of the answer could do with a better presentation.

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The following sources were used in this analysis:

• Curse of Strahd
• Dungeon Master's Guide
• Hoard of the Dragon Queen
• Monster Manual
• Mordenkainen's Tome of Foes
• Out of the Abyss
• Princes of the Apocalypse
• Rise of Tiamat
• Storm King's Thunder
• Tales from the Yawning Portal
• The Tortle Package
• Tomb of Annihilation
• Volo's Guide to Monsters
• Xanathar's Guide to Everything

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Note: I use the term "weakest defense" fairly loosely here, but to reiterate, this is when considering only AC, DEX, CON, and WIS.

Answer 2

For cantrips, they are equally predictable in damage output

Ignoring secondary effects and concentrating only on damage, the expected damage from a cantrip that uses an attack roll and allowing for critical hits is given in equation \$\eqref{eq1}\$ and for a saving throw cantrip in equation \$\eqref{eq2}\$. Note: these are strictly speaking only correct for 'middle' values where the chance of success is between 0.05 and 0.95 but that covers the vast majority of cases - if you are always (or never) going to succeed then it's not really a fair fight.

\$\begin{align} E(A)&={{D_a + (D_a+d_a)(21-AC+b+b_a)}\over 20}\tag{1}\label{eq1}\\ E(S)&={{(D_s+d_s)(7-s+b+b_s)}\over 20}\tag{2}\label{eq2}\\ \text{where}\\ D_a &= \text{Random variable representing dice from attack cantrip}\\ D_s &= \text{Random variable representing dice from from saving throw cantrip}\\ d_a &= \text{fixed damage bonus from attack cantrip}\\ d_s &= \text{fixed damage bonus from from saving throw cantrip}\\ b &= \text{spellcasting attack modifier = proficiency + ability modifier}\\ b_a &= \text{any bonus to hit with attack cantrip (e.g. *Wand of the War Mage*)}\\ b_s &= \text{any bonus to DC with save cantrip (AFAIK nothing does this but I don't know everything)}\\ AC &= \text{target's Armour Class}\\ s &= \text{target's saving throw bonus}\\ \end{align}\$

It follows that the saving throw cantrip is better when:

\$\begin{align} {{(D_s+d_s)(7-s+b+b_s)}\over 20}&>{{D_a + (D_a+d_a)(21-AC+b+b_a)}\over 20}\tag{3}\label{eq3}\\ {D_s(7-s+b+b_s)+d_s(7-s+b+b_s)}&>{D_a(22-AC+b+b_a) + d_a(21-AC+b+b_a)}\tag{4}\label{eq4}\\ \end{align}\$

Simple, right?

Well, let's see if we can make some assumptions.

• \$D_a=\$ nd8 or nd10 for all Player's Handbook attack cantrips.
• \$D_s=\$ nd12 for Poison Spray, nd8 for Sacred Flame, nd6 for Thorn Whip and nd4 for Vicious Mockery. Acid Splash we'll ignore because it can have multiple targets.
• The \$n\$ in \$D_a=\$ or \$D_s=\$ is the same and in equation \$\eqref{eq5}\$ every term is a multiple of at least one of these so the level at which the cantrip is cast doesn't influence which is better, although it does influence how much better.
• \$d_a=0\$ for everybody except a Warlock with the Agonizing Blast invocation using Eldritch Blast. We'll ignore them.
• AFAIK there is no feature or magic item that makes \$d_s\ne 0\$ so we can let \$d_s= 0\$. Similarly for \$b_s\$.
• There are things that can set \$b_a\ne 0\$ but they are pretty niche so we'll ignore them.
• \$b\$ is in the range of 5 to 11 for most spellcasters.

So adopting these assumptions we get:

\$\begin{align} {D_s(7-s+b)}&>{D_a(22-AC+b)}\tag{5}\label{eq5}\\ \end{align}\$

This is more like it! However, it's still a function of 5 variables, two of them random variables. However, given the ranges of the random variables we know that there will not be a situation where one is strictly better than the other so we can substitute the expected values and get some sensible insight. \$E(D_a)\$ is either 5.5 or 4.5 and \$E(D_a)\$ can be 6.5, 4.5, 3.5 or 2.5.

I'm not going to comprehensively compare cantrips but I will look at the one you start with Fire Bolt v Poison Spray and I'll assume a caster with \$b=5\$ (typical for a 1st level). The saving throw cantrip is better when \$s\$ is less than or equal to:

\$\begin{array}{c|c} AC & s\\ \hline 8 & -5\\ 9 & -4\\ 10 & -3\\ 11 & -2\\ 12 & -1\\ 13 & 0\\ 14 & 1\\ 15 & 1\\ 16 & 2\\ 17 & 3\\ 18 & 4\\ 19 & 5\\ 20 & 6\\ \end{array}\$

So, it's almost a 1 for 1 exchange - high AC monstres are more likely to be better targeted by a saving throw spell. Which is not really a deep insight for all that maths but now you know that the "normal" range of AC and "normal" range of save bonuses overlap, more or less so there really isn't much to choose between these cantrips.

At the other end of the scale Vicious Mockery is never better than Fire Bolt but then, you shouldn't be using it for it's direct damage output anyway.

Answer 3

The question wrongly assumes landing an attack is just as important as failing a saving throw

In reality this is almost never the case as many saving throw spells can deal half damage even on a successful save whereas attack rolls will entirely miss. Likewise, many saving throw spells will have more lasting effects, often involving even more saving throws which is very rarely the case with attack rolls. On top of this nearly ever attack roll will be single target and many saving throws will be multi-target. There is no reason to claim these methods of spell delivery have an even playing field. They have different effects, benefit from different features and abilities, and target very different aspects of a given enemy.

Even taking the assumption for granted, it requires nothing more than a basic statistical analysis of every single monster

For the reasons already explained, comparing the two is a pointless endeavor, but let's humor the hypothetical. It would suffice to check every single monster of each CR and see which options are best to use and worst to use and by how much. This is a rather horribly lengthy endeavor but requires little more than calculating the average AC, Str, Dex, Con, Wis, Int, and Cha save bonuses of every monster of each CR. And so, it would be immediately apparent which option is "best" on average against the CR-appropriate monster.

Any such analysis will be unusable/irrelevant because the assumption is false

You cannot simply learn that, for example, AC is on average the best thing to target, and then go about choosing only spells that involve attack rolls, or even mostly spells that involve attack rolls.

So many options and effects will be completely missed out on simply because of what was statistically "best". Even if you looked at the average scores are chose a number of spells of each method to match the percent of creatures where that is their weakest defense you would still have problems.

This entire analysis would depend directly on the premise that all attack rolls spells and all saving throw spells are entirely and exactly equal in their damage and effects. This simply cannot be the case or we would not even have the massive variety of spells that we do. No meaningful information or data can be extrapolated from this assumption.

Additionally, as MarkWells pointed out, such an analysis would assume an equal likelihood of encountering every single monster (including NPCs existent in a single module) as well, which is never the case.