# Does rolling AC to oppose an Attack Roll impact game balance?

I really like the concept of rolling d20 against each other during ability checks like grappling, etc. I want the same thing for normal attacks, so the defender would also roll a d20 + (AC-10). Would this modification make the game somehow unbalanced or create side effects on monster challenge levels?

# It doesn't change much

Using the formulas I give further down we can compare the odds of hitting using opposed rolls to normal attack rolls across a wide range of Attack - (AC - 10) values:

For most attack bonus and AC combinations, both methods give very similar results (within 5% of each other). At the extremes, the difference between both methods grows to 6-10%, with one particular combination (e.g. attack +8 vs 10 AC) producing a difference of 11.5%

However, these extreme cases should be rare. Attack bonuses for players usually range from +3 (2 prof bonus + 1 ability mod) to +13 (6 prof bonus + 5 ability mod + magic weapon or fighting style bonuses). For monsters, attack bonuses usually range from +3 at CR 0 to +10 at CR 20 (Dungeon Master's Guide p.275).

Player AC usually ranges from 10 (no armor, 0 Dex mod) to 20 (full plate + shield) and monster AC from 13 to 19 (ibid.) All the creatures in the Monster Manual with an AC higher than 20 also have CR higher than 20, and even the CR 30 Tarrasque maxes out at 25 AC.

By the time characters have attack bonuses as high as +8, they're probably not fighting 10 AC monsters with any regularity. Additionally, players with extremely high attack bonuses will probably invest in Great Weapon Fighting or Sharpshooter for extra damage, which brings their attack bonus back down.

Likewise, it's unlikely that a monster will have much higher attack bonuses than player AC because monsters don't have as many ways of boosting their attack rolls and players tend to invest into AC at least a little.

The red line in the graph can also be shifted up slightly by using AC - 11 instead of AC - 10.

On the topic of randomness...

Since dice rolls are independent events, using two dice instead of one has no effect on the outcome of future attacks. Additionally, since there's only two possible outcomes to an attack roll (hit or miss), it doesn't matter how many dice are used to determine the result as long as the probability of hitting stays the same, or close enough for your purposes.

For example, suppose you were betting on the outcome of a coin toss; there's only two possibilities and each is equally likely, so you have a 50% chance of winning.

Now suppose the game is changed to flipping two coins, and you bet on whether both coins will land same side up. There's 4 possible outcomes, each equally likely: HH, HT, TH, and TT. However, you can win with 2 of those 4 outcomes, so your chance of winning is still 50%, just like when you were flipping one coin.

If you only knew the result (win or lose), someone could change from using 1 coin to 2 coins and you wouldn't be able to notice, because your chances of winning stayed the same.

Every method of resolving attack rolls, regardless of how many dice you use, will boil down to producing hits with certain a probability. Any two methods that produce hits with the same (or close enough) probability are interchangeable. If the differences are small, it would take meticulous tracking of results across multiple game sessions for players to notice a difference.

## Probability for single rolls:

The odds of a d20 + Bonus being greater than or equal to AC is

$$P = \frac{21 + Bonus - AC}{20}$$

Proof:

Since every number on the die is equally likely, we can get the probability of hitting by counting the number of rolls that can hit and dividing that by the number of possible rolls (20).

Since the highest total we can roll is 20 + Bonus, subtracting the highest total that can miss (AC - 1) will leave us with the number of rolls that can hit: 20 + Bonus - (AC - 1) = 21 + Bonus - AC

Example: With an attack bonus of +2 and an AC of 20, there's only three rolls that result in a total of 20 or higher: 18, 19, or 20. The formula correctly gives (21 + 2 - 20)/20 = 3/20 = 0.15, or 15%.

## Probability for opposed rolls:

If the attacker has a roll modifier of X and the defender has a roll modifier of Y, the odds of the attacker rolling higher is:

$$P = \begin{cases} (20 + X - Y)(21 + X - Y)/800 & \text{if X \leq Y} \\ 1 - (19 + Y - X)(20 + Y - X)/800 & \text{otherwise} \end{cases}$$

Proof:

Since each number on one die is equally likely to come up, every combination of numbers is equally likely when rolling two dice. Figuring out the number of combinations that result in a hit is tricky, but looking at the table of possible outcomes (taken from this answer in Mathematics Stack Exchange) gives us some insight:

\begin{array} {c|cccccc} &1&2&3&4&5&6 \\ \hline \\ 1&=&<&<&<&<&< \\ 2&>&=&<&<&<&< \\ 3&>&>&=&<&<&< \\ 4&>&>&>&=&<&< \\ 5&>&>&>&>&=&< \\ 6&>&>&>&>&>&= \\ \end{array}

(This table is for a d6, but it's not hard to see that the pattern would be the same for a d20.) When both sides have the same modifiers, the combinations that result in a hit (the > and = signs) form a triangle whose sides are as long as the table (20 rows and columns).

If the attacker's modifier is x points lower than the defender's modifier, the triangle shrinks by x units:

\begin{array} {c|cccccc} &2&3&4&5&6&7 \\ \hline \\ 0&<&<&<&<&<&< \\ 1&<&<&<&<&<&< \\ 2&=&<&<&<&<&< \\ 3&>&=&<&<&<&< \\ 4&>&>&=&<&<&< \\ 5&>&>&>&=&<&< \\ \end{array}

So when X ≤ Y the sides of the triangle have length 20 + X - Y. Fortunately there's a handy formula for counting the number of characters in this triangle: n(n + 1)/2, where n is the length of the triangle's sides. Replacing n with 20 + X - Y gives (20 + X - Y)(21 + X - Y)/2. Dividing that the total number of possibilities (20 * 20 = 400) gives us the first part of our function.

If the attacker's modifier becomes positive, we get a similar table:

\begin{array} {c|cccccc} &0&1&2&3&4&5 \\ \hline \\ 2&>&>&=&<&<&< \\ 3&>&>&>&=&<&< \\ 4&>&>&>&>&=&< \\ 5&>&>&>&>&>&= \\ 6&>&>&>&>&>&> \\ 7&>&>&>&>&>&> \\ \end{array}

The > and = signs no longer form a triangle, but we can count the number of < signs (i.e. the roll combinations that miss) and subtract the result from 100% instead. The formula is the same except the triangle length starts at 19 instead of 20 since we're no longer including the = signs.

• Comments are not for extended discussion; this conversation has been moved to chat. Jun 15 '17 at 17:24

# I would strongly recommend against it. The game is balanced for the current amount of randomness in combat. Introducing more swing will hurt the players over time.

If you do, you might want to curtail the number of encounters per day. Or mix in more non-combat encounters.

How to balance encounters is discussed in depth in Chapter 3 of the DMG, and specifically on pages 81 - 85 under the heading CREATING ENCOUNTERS - read closely THE ADVENTURING DAY section.

Two reasons:

• It is going to slow play down for very little gain in simulation or fun.

And my main reason:

• Any increase in exposure to randomness with detrimental effects hurts the players in the long run. Why? because the monsters don't get a long run. They are there to be mowed down. The DM effectively supplies infinite monsters. But eventually the odds are really strong you are going to run into a couple of what would have been balanced encounters that are going to wipe the floor with the party under this new system.

I won't go into the statistical theory behind this because it's super mathy, but at a high level - randomness is clumpy. That is, a string of truly random numbers will have clumps of closely grouped numbers. Blackjack gamblers, who know how / when / where one can beat the house odds, know this "clumpiness" can still wipe them out if too much "bad luck" happens before the "good luck" does.

They account for this by having a large enough bankroll to carry them through the clumps of bad luck. You see, the house basically has infinite resources. So, even when the odds may favor the gambler in the long run, if the odds are small enough and there is enough swing, sometimes the house can simply wait until a string of bad luck drains all the gamblers resources.

As discussed in the DMG in aforementioned THE ADVENTURING DAY section, D & D is balanced for about six encounters per day, and adventurers can't rollover these daily resources to bankroll them for a bad patch. You can think of the players as the gamblers with constrained resources and the DM as the house with basically infinite resources.

Thus - introducing more randomness beyond the established balance almost always means less fun for the players in the long run.

# MAJOR EDIT:

There is a great deal of confusion around what I am stating here. There is a tendency to treat each event as independent of each other. In terms of probability this is true. But if we look at a combat as a whole, it becomes obvious that a bad roll early on can affect the results of the entire combat; even though all the die rolls made in combat are still independent.

This effect can be summed up as Risk is the ally of the underdog. One of the major ideas in Risk Analysis is exposure. The idea is that any amount of risk can be tolerated if there is no exposure, and even a tiny bit of failure chance can be intolerable if your exposure is overwhelming (say dozens of times every combat). It gets complicated, but it boils down to the idea that the monsters (the underdogs) are there to be defeated. They are there to lose. Anything you do that increase exposure to risk for the PCs increases the overall risks the PCs face, and gives the underdogs that many more chances to kill a PC.

There are a great deal of dynamics in combat, so let me pick a simpler example.

When the DM calls for a perception roll from a player, and they fail, normally the other player instinctively ask if they can try too. They know that additional die rolls increase their chances of seeing the thing in question. Each die roll the player makes may be independant, but systematically it only takes one player to see the item and the whole party benefits. Once the object is found, there is no going back to it being unfound. Adding die rolls changed the outcome in the aggregate, even if it didn't change it for that individual.

Another example: In 5e they stopped calling on Stealth rolls every round. The explicit reason for this is that it made it really hard for characters to be stealthy. Even though a character could have world class stealth, if they had to make 5 stealth rolls a combat, they are failing every fourth combat on average. And once they are seen, there is no going back to unseen. Once again, when there is a resource that can not be easily recovered (in this case being hidden) adding die rolls changed the overall outcome even though the individual probabilities didn't change.

So how does this tie in to the proposed system? Well you are explicitly adding a second die roll for every single attack. That is a second bite at the apple for a player's hit to become a miss - thus wasting a spell or magic item charge or an action in combat - or a monster's miss into a hit, burning hit points, and other side conditions. And at least per day once those HP, spells, magic item charges, whatever, are gone you don't get them back until tomorrow (for the most part).

Thus the bankroll analogy. The extra die roll is another drain on the player's resources. It is most obvious in things like the School of Divination Wizard. The extra die roll halves their portent ability. Sure, it is also a drain on the DM's resources, but the DM is supposed to put up monsters for the players to kill, right?

Slightly related - this is why detailed critical hit tables tend to suck for players. Sure it's cool to lop off monster's body parts. But the PCs only have so many limbs, and eventually some monster is going to get lucky and take your sword arm. Limbs are a limited resource, and once they are gone, they are not easy to get back. And typically that sucks for the player.

• the concept I think you're looking for is referred to as gambler's ruin - the principle that even a perfectly fair game based on luck will, eventually, always be won by the side with infinite resources (assuming the gambler doesn't adjust their play dynamically) Jun 8 '17 at 15:57
• – JWT
Jun 8 '17 at 21:23

# It doesn't make a big difference

To answer this question, I wrote up an AnyDice program to simulate it. I calculated average damage per round on a target with both static Armor Class and Rolled Armor Class.

The end result is this: Rolled AC will actually result in marginally less average damage being dealt per round, and marginally higher deviation in your results, than in dealing with Static Armor Class.

Consider a PC with a +6 to hit using a d6 weapon, opposed by an Armor Class ranging from 10 to 20 (note: this is AC as listed in the PHB. For rolling AC, I subtracted 10 as per the question. Thus the 'rolled' AC was treated as a d20 + 0..10). Rolling your AC results in an average damage that is 0.14 points of damage per round less than if you used static AC.

This should not unbalance anything when considered in the long run.

The only thing you'll run into is that you're making more work for yourself, as the DM. Instead of just memorizing the AC of a monster and instantly being able to relay back to your players whether they hit or missed, you have to roll and add their AC bonus. And this requires the same of your players. If you don't mind this...have fun!

To explain my rationale for the values I selected, consider the Armor Class range in D&D 5E. Your low end for monsters is 6 (Gelatinous Cube) and the max is 25 (Tarrasque). However, most creatures lie between 10 and 20. Furthermore, between 10 and 20 is the typical AC range for player characters (not counting magic items).

The +6 to hit was chosen rather arbitrarily. However, that's a fair estimate of a typical melee character's to-hit bonus around level 4-5 (depending on ASIs)

• Comments are not for extended discussion; this conversation has been moved to chat. Jun 15 '17 at 17:24

# The Odds

Since this is something that affects the number of the game, we can turn to our good friends, statistics and anydice.com. Since the attacker must meet or exceed the AC of the target, we should consider this in any analysis we attempt.

Looking at this anydice script, you'll note that I put a person with a total attack bonus of +3 against two targets with AC 13 and AC 16. To summarize and compare these chances to normal rules:

• the AC 13 is hit 55% of the time under normal rules, and this drops to 52.5% under the new ones. That's a 2.5% reduction in the odds to hit.
• the AC 16 40% of the time under normal rules, and this drops to 38.25% of the time under the new ones. That's a 1.75% reduction in the odds to hit.

Fooling with the script a little, you can see a +2 or +4 attack roll changes the odds by about 5% under the new rule.

# The Balance

Does it make it more challenging for the attackers? Yes. Is it game breaking? In my opinion, no, not really. It can introduce some upsets, such as a fighter failing to hit a run-of-the-mill goblin for a few rounds or rolling a "16+3=19" and then seeing the defender roll a "19+1 = 20". On the whole, however, it should not dramatically change the outcome of most battles.

This form of combat could take longer, but could also be more entertaining. Situations will eventually occur in which the underdogs of the fight win, but not much more than normal. It will also be, according to some tastes, more "realistic" - "people don't just stand still and wait to be hit in real life" (in spite of the dex bonus to AC). (1)

I recommend only using this if (1) the table (that's the DM and players) does not mind the extra die roll and (2) if the table is OK with the upsets.

(1) As a practitioner of HEMA, I doubt any combination of attack bonus / AC mechanics will ever model reality enough. Fighting is just too complicated.

• Comments are not for extended discussion; this conversation has been moved to chat. Jun 15 '17 at 17:17

## It has a large impact on very easy/hard to hit enemies.

As many other answers showed, the chance to hit is mostly the same when you need to roll around 10 to hit. However, when you need much more, or much less, to hit, the new system has a much different probability curve.

Consider this simple anydice program. Rolling for AC as you describe is mathematically (mostly) the same as rolling 2d20-11 for your attack (the simplification breaks down for critical hits and misses). The "at least" view is the most useful, as it shows what the chances are to roll at least a certain target number; for example, if you roll at +5 and need to hit AC 15, you need to roll at least 10 on the dice. For AC 20, you need to hit at least 15.

As you can see in the program, the probabilities are mostly the same around 10. However, the further you get from 10, the bigger the difference from a straight D20 you go. If you normally need to roll a 2 to hit, you go from a 95% chance to hit to a 83.50% chance to hit; if you normally need a 20 to hit, you go from 5% from 13.75%.

In D&D5, since it uses bounded accuracy, the number you need to roll tends to be around 5-15, where this influence is relatively small, so it will not greatly impact balance. In general, however, it greatly favors the party that would normally be disadvantaged.

• ...and penalizes the party that would normally have the advantage. Jun 15 '17 at 17:33

The biggest thing is that while it does not significantly affect game balance (a lot of the other posts prove that), it does effect game play at your table: the precious resource of time is at issue.

The biggest impact on the game is that combat will take longer, especially in more extensive combats. Now if you and your group don't mind that, go for it.

# It can be.

It is slower to implement, since it requires two rolls per attack instead of one, but it does make combat more dynamic in practice, as the players are often more engaged when they get to roll dice to defend as well as attack.

What I do with my group is allow them to choose to make active defense rolls if they're aware of an attack incoming and are able to move. This allows them to spend Inspiration to gain advantage on important defense rolls, and I automatically give them Advantage on the rolls if the enemy is flanked or otherwise less able to concentrate on hitting them. On the other hand, I'll also occasionally force them to roll and give them disadvantage if they're left open by their combat choices, such as when our Barbarian makes a charge while raging.

Thanks to the answer of user Doval i was able to create an excel sheet diagramm with the given formula.

I added the d20 vs d20, the d20 vs static, but also a d10 vs d10 and a d12 vs d12. I personally realy like the hit curve of d12 vs d12

Summary:

The summarized impact is that it slightly lowers the value of armor, reduces some predictability, and slows down the game.

Detailed:

There are four main effects from doing this.

First is that average AC increases by .5 for everything. This is because default AC is 10, while a d20 will on average roll a 10.5. Since the victory is still given to the attack on a match (to hit you must match or beat AC), this difference is less pronounced. Average case without rolling is a 10 AC vs a 10.5 attack, so the attack wins. Average case with rolling is a 10.5 AC vs a 10.5 attack, and the attack still wins.

Second, it slows down game play. Instead of having AC already calculated, you have to roll and calculate. While this is only adding a few seconds per attack, that can add up over a single adventure. Part of the effect depends upon how your group handles rolls, if you have players who can roll at the same time, make reasonable rolls (so dice aren't likely to go off the table), who are fast at doing the mental addition, and who don't spend much time double checking, the total effect may not be big enough to care about. If your group is the opposite, the slow down could be significant.

Third, it removes some predictability. Think of the difference between taking 10 on a skill instead of rolling. Even though taking 10 is worse than the average roll (10.5), being able to know the total value allows the player to treat the check differently. Imagine picking a lock. If a rogue has +12 to the skill check, knowing they can get a 22 is different than knowing that the range is 13 to 33. For an example, imagine some fighter with +10 to AC. A kobold with a +1 to hit will only be able to hit 20AC on a roll of 19 or 20. But if the fighter has to roll their AC, they have an AC range of 11 to 31 vs the kobolds attack range of 2 to 21. Much harder to determine how well defended one is.

Finally, and almost an extension of the last point, is the impact of nat 1s and 20s. If you go by the rules that a nat 1 is always a miss and a nat 20 is always a hit, consider the following.

Fighter with +10 AC vs zombie with +0 to attack. Without rolling, the zombie can only hit by rolling a nat 20. With rolling, if the fighter rolls higher than 10, he gains no benefits because the zombie still hits with a nat 20. If the fighter rolls lower than 10, the zombie can roll lower than a 20 and still hit. So the zombie goes from having a 5% chance to hit to having a higher than 5% chance to hit (I haven't rant the math to know how big a difference, only that it makes a difference).

The opposite also holds, where if a Wizard had an AC of +0 against a ogre with an attack of +9. If we go without rolling AC, the only way for the ogre to miss is to roll a 1. If we roll for AC, if the wizard rolls less than 10 they aren't any worse off (ogre can only miss on a 1), but if they roll higher than a 10 the ogre could miss on rolls higher than a 1. This means that the wizard goes from having a 95% chance of being hit to a lower than 95% chance of being hit (once again, haven't calculated the exact numbers). The impact of these really depends character by character and how the average PC's AC compares to that of the average encounter's AC.

So the fighter with higher AC receives a small nerf while the wizard with low AC receives a slight buff, indicating an overall slight devaluing of increasing AC.

The largest impact on balance will come from the fact that AC is now susceptible to advantage and disadvantage, and will often mirror advantages and disadvantages in attack. This greatly decreases the swinginess of combat in most situations, making characters with situational superiority stay ahead of their competition.