The Basics

Instead of adding a lengthy parenthetical section in the fighter DPS article, it was decided a separate series of articles covering the combat calculations would be written. Any questions, comments, and critiques are welcomed. The essentials will be covered here.

Chance To Hit

Essential in applying damage to a target is an opportunity to hit it. It could be determined, say using fingers and toes, the number of sides of a d20 that represent success or failure of a roll for each target AC and given bonus to hit. Or, with shoes and socks in place, a generalizable equation could be used.

To quickly develop the equation, common arithmetic routinely used in combat should be considered. Given a target AC = 17 and a to hit bonus of +6, what roll do you need to hit? An 11 and above on a d20 will hit. Since there are only 20 possible outcomes in any single roll of a d20 and 1-10 will miss, the chance to miss 50%. The chance to hit is 100% less the chance to miss.

This method can be generalized to an equation:

And the average chance to hit is generalized to the complement of this as:

The decimal value can be multiplied by 100 to represent the % value if desired. Note also that there are boundaries not represented in equation but present in game rules, namely there is without advantage a 5% miss chance and at least a 5% critical hit chance represented by values on the d20. Last, the reason to use chance to miss will become apparent in Part 2.

Damage

Similarly, the basic form of the damage calculation is not unlike that commonly used at the table. First, after an attack is successful, one determines the damage given by weapon die. Then the weapon’s ability modifier is added. Next, any applicable bonuses are added. Damage on a hit is given, with ability modifier represented by M and other bonuses represented by B, by the equation:

Over time, the average weapon rolls of a weapon will approach the sum of its weapon die’s average values. For example, a great sword has 2d6. Each d6 has an average value of halfway between its maximum of 6 and its minimum of 1, or 3.5. These dice are summed to find the average weapon damage of a great sword at 7 damage. Similar reasoning can be used to discover the average weapon damage of any weapon.

Critical Hit Chance and Damage

Unlike the hit chance, the critical hit chance does not depend on AC of opponent. Thus for any given chance to critical and weapon die, the contribution to average damage over time is constant. In other words, for a critical hit chance of 5%, 1 in 20 of all rolls on average over time will be a critical hit whether those swings are at a coat rack or at Her Dark Majesty Tiamat, Queen of Dragons. The average extra damage beyond a guaranteed hit is given by equation:

Average Damage Per Turn of Combat

Bringing it all together, average combat damage for n attacks is represented as:

Keeping in mind there are bounds on chance to hit without advantage due to a roll of 1 on a d20 always being a miss and a roll in the range of a critical always being a hit, given by:

Part 2 will cover advantage and its relation to critical hits or misses.

Part 1, Or How I Learned to Stop Worrying and Love the Axe

When Moose decided to upgrade from guest star to featured act, he wanted to upgrade his randomly generated dragonborn fighter to a more appropriate DPS option. A progression from a human glaive battle master fighter making use of Polearm Mastery feat’s glaive-to-hilt combat and Great Weapon Mastery feat’s power attack ability, with added accuracy from Precision maneuver, was suggested as an optimal damage output solution.

Here the rationale for rejecting that solution is shown.

The Builds

First, the Human Glaive Battle Master (HGBM) will be examined by level up to the suggested maximum in Princes of Apocalypse, CL15.

Select Human, prioritize strength in array. Take Polearm Mastery feat.
1. Fighter, Great Weapon Fighting specialty.
2. Gain Action Surge
3. Battle Master specialization; take precision, riposte maneuvers..
4. Take Great Weapon Mastery feat.
5. Extra Attack.
6. Upgrade Strength score to 18, gain +1 Strength mod.
7. Bonus superiority die.
8. Upgrade Strength score to 20, gain +1 Strength mod.
9. –
10. Superiority dice grow to d10s.
11. Extra Attack (2)
12. Feat
13. –
14. Feat
15. Bonus superiority die.

The strength of the build relies on the hilt attack functioning as another attack on which to apply the power attack. By level 11, three beefy attacks with the pointy glaive end and a bonus attack with the hilt are possible, each adding 15 to the base weapon damage including the strength modifier. Accuracy can be improved with precision up to five times per short rest, and additional flexibility is given by other maneuvers. Finally, using action surge, burst damage can be made with seven attacks in one round.

Note that the base build can be transitioned into multiclass of choice as the last 2 feats and bonus superiority dice can be traded for a dip into another class or two without harming the maximum damage efforts. A discussion of this is beyond the scope of the article, but suggestions can be posted in the comments.

Compared to many other builds, HGBM appears consistently more destructive over long adventuring days and resilient with high Armor Class, second wind, evasive maneuvers, and reach weapon. However, several more potent and resilient builds were found in exploring options for our
“Tiamat Takedown” several months ago. Some are pure fighter, others with splashes of lesser classes. What became Jeff, Ender of Dragons began as the mono-class fighter specialization covered next.

Half-Orc Great Axe Champion (HOGAC) is put together as follows.

Select Orc, prioritize strength and constitution. Gain Relentless Endurance, Savage Attacks.
1. Fighter, Great Weapon Fighting specialty.
2. Gain Action Surge
3. Champion specialization; critical range improved to 19-20
4. Take Great Weapon Mastery feat.
5. Extra Attack.
6. Upgrade Strength score to 18, gain +1 Strength mod.
7. –
8. Upgrade Strength score to 20, gain +1 Strength mod.
9. –
10. Gain Defensive Fighting Style; +1 AC
11. Extra Attack (2)
12. Feat
13. –
14. Feat
15. Improve critical range to 18-20.

The use of a great axe instead of a great sword should stand out to the 5^th Edition Veteran. Looking purely at the base weapon damage dice, the average damage of the great sword (2d6, 7 damage average) does indeed exceed that is the great axe (1d12, 6.5 damage average). In general, a great sword, or its blunt equivalent, are the damage dealing weapons of choice.

Yet the choice of great axe is made for precisely two reasons: the rules-as-written wordings of Savage Attacks and Great Weapon Fighting. Each allows you to roll (or reroll) one of the weapon’s damage dice and add it to damage. Yes, only one. And great weapon fighting has two damage die. The math works out in favor of great axe because of these quirks.

Like the HGBM, the HOGAC’s damage output relies on the bonus attack, here generated from Great Weapon Mastery by either critical hits or dropping enemies to 0 hp. It can be shown that as the number of attacks increases per turn and as the critical hit chance improves, bonus attacks generated approach 70% uptime from critical hits alone on any given turn with advantage. Additionally, another 1d12 is dealt to each critical hit from Orc racial perk.

Before jumping into damage numbers, the other advantages will be enumerated:

1. Higher HP total level for level from Orc constitution bonus, up to +15 HP.

2. Staying erect made easier with Relentless Endurance.

3. Delicious Half-Orc in plate with great axe flavor.

4. +1 AC is gained thanks to second fighting style.

5. Bonus feat gained proudly because no multi-classing planned.

6. Fewer combat resources, less thinking, more axe.

To sum, significant non-damage bonuses to resilience and simplicity are given by HOGAC when compared with HGBM.

By the Numbers

First, a comparison by damage milestone for HOGAC and HGBM will be discussed, as shown below in figure, by damage per turn (DPT) on average of an infinitely long fight. All sources of damage are accounted for in equation and assume advantage approaching 100%. Armor class 16 is used, approximating average of all creatures in Monster Manual per Surf Archer. The precise calculation methodology will be elucidated in a series of separate posts.

It is observed that HOGAC outpaces HGBM at every milestone except the first. As the proportion of time that the HOGAC bonus attack is active increases, so does the gap in damage per turn between the builds. This gap is expanded when action surge is used for burst damage.

Second, by cursory inspection it is also clear that the upper limit of damage achieved with HOGAC is greater than HGBM. In fact, 112 damage is calculated as the difference, with 45.7% more damage potential demonstrated by HOGAC.

Last, curiosity may have grown regarding how AC affects damage output comparisons. The answer can be found below in figure. The similar arcs formed demonstrate the similarity of the damage die used by both weapons, but a damage source independent of AC separates them. This constant can be shown to be the damage due to critical strikes. It should also be noted that the HOGAC GWM bonus attack active time is 63% of turns, but only triggers from critical strikes were considered and activations of ability due to killing enemies would increase percentage.

Conclusion

It has been demonstrated that in a sufficiently long combat scenario HOGAC surpasses damage output from HGBM. Many additional perks to survivability and simplicity were also outlined. Further discussion regarding the length of combat and role of HGBM superiority die, as well the effect of multi-classing on these calculations will be made in Part 2, after the ridiculous build that made the amazing average damage arc shown below is discussed.