Dungeons and Dragons 5E is an elegant system that has far simplified the math of combat over previous systems. With this simplification, my previous articles on using expected damage are feeling a bit dated so I wanted to revisit Mass Combat math to better match with 5E.
This table for expected damage has values varying from 0.1 to 1. The low end .1 value is the chance we'll hit with a crit at a 5% chance with double damage. The high end value is the guarantee will hit on every dice roll except a 1, with double damage on the 20 crit. The red boxes are the AC/Attack combinations at the min where only crits hit. The green boxes are the AC/Attack combinations where only 1's miss. In between, the values vary in 0.05 increments.
To use the table, plug in the AC of the defender, the attack modifier of the attacker, and get an expected damage factor. Take the average damage of the attacker (max+min)/2 times the factor times the number of attackers and you have the damage for that round. Since average damage never changes, you only calculate it once. If the same attacks happen again and again, you only need to adjust for numbers dying attackers.
If you have minor effects in addition to AC and attack bonus, feel free to add / subtract 0.05 or .1 from the damage factor. This can include things like the champion fighter's improved criticals or advantage/disadvantage mechanics.
5E has a nice feature that average damage is already calculated for monsters. For heroes, small damage numbers give a mean damage pretty easily: (die max - die min)/2. What isn't easy is calculating how much of this damage is expected to hit. Enter the expected damage formula to allow us to figure this out.
The formula isn't as important as what is represents. An attacker can always miss with a 1 and always hit with a 20 for double damage. In between the attacker gets a bonus on their d20 to try to hit the AC of the defender. For every hit, the attacker gets to apply normal damage.
So imagine that we look at a collection of lots and lots of attacks. Based on this we can calculated the expected damage that the attacker will deal in one round. That value is the expected damage. Expected damage is the average damage times the factor that represents how often the attacker hits and a bit extra to account for crit double damage. This factor is the expected damage factor.
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| Expected Damage Factors for Various Attack Modifiers and Armor Classes (Click to expand) |
To use the table, plug in the AC of the defender, the attack modifier of the attacker, and get an expected damage factor. Take the average damage of the attacker (max+min)/2 times the factor times the number of attackers and you have the damage for that round. Since average damage never changes, you only calculate it once. If the same attacks happen again and again, you only need to adjust for numbers dying attackers.
If you have minor effects in addition to AC and attack bonus, feel free to add / subtract 0.05 or .1 from the damage factor. This can include things like the champion fighter's improved criticals or advantage/disadvantage mechanics.
In mass combat, with troops organized in groups, each group attacks another group or individual. Calculate the damage per the above formula. This damage hits, taking out the defenders one at a time until all of the damage is used up, or until you run out of defenders. Using the table may seem slow, but it is easy to boil it down to just the numbers you need for the groups and characters you have in play. Using a slightly more complex formula for Pathfinder, I have successfully run an over 100 combatant battle easily in a 4 hour session.
This method has a number of advantage over other mass combat methods. First, all of the normal values have an impact on combat. There are no new values or stats made up or added in. Second, PCs can still have a significant effect on the combat. Because PCs can do lots of damage with fireballs and multiple attacks and special abilities, those can still be used. Just roll their damage normally and apply it. Third, you can easily mix groups of foes with large foes and battlefield implements like cannons, warships, airships, etc. The combat of 12 city guards in a group with the PCs versus an ancient dragon still makes sense.
To add to the narrative, put your players and their PCs in charge of groups. Match the PC to the group, so your ranger is controlling and leading the archers, and you have the start of a great story. Much like Aragorn, Gimli, and Legolas lead the troops at Helm's Deep, your PCs can lead great battles on their own, all without the DM pulling his hair out.
This method has a number of advantage over other mass combat methods. First, all of the normal values have an impact on combat. There are no new values or stats made up or added in. Second, PCs can still have a significant effect on the combat. Because PCs can do lots of damage with fireballs and multiple attacks and special abilities, those can still be used. Just roll their damage normally and apply it. Third, you can easily mix groups of foes with large foes and battlefield implements like cannons, warships, airships, etc. The combat of 12 city guards in a group with the PCs versus an ancient dragon still makes sense.
To add to the narrative, put your players and their PCs in charge of groups. Match the PC to the group, so your ranger is controlling and leading the archers, and you have the start of a great story. Much like Aragorn, Gimli, and Legolas lead the troops at Helm's Deep, your PCs can lead great battles on their own, all without the DM pulling his hair out.
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