Is this a time to choose minimum damage per hit over average DPR?

Question

A lot of gamers consistently optimize characters average damage per round. My intuition tells me, though, there are times when your character should be opting for an action that does the highest minimum damage per hit. In other words, choosing the action with the lowest maximum hits to kill a monster.

Although I trust my intuition here, not everyone agrees with me. I would like to have mathematical evidence to back up my gut feeling.

Wizard vs. Goblin

Consider this example. A 3rd level wizard finds himself alone, facing a goblin at close range. (Beyond the goblin lies certain safety.) Out of spell slots and other supplies, the wizard considers whether to use Firebolt or his dagger.

Our wizard has:

• Intelligence: 16
• Dexterity: 16
• AC: 13
• HP: 17
• Crossbow Expert Feat (so no disadvantage for attacking with Firebolt at close range.)

So, our wizard has a +5 to hit with either the Firebolt or the dagger.

The goblin has AC: 15, 7 Hit Points, attacks at +4, and does 1d6+2 damage. Assume both combatants fight to zero HP.

My gut is telling me that the wizard should reach for his dagger (that does 1d4+3 damage) rather than casting Firebolt (for 1d10+0 damage) even though Firebolt will do slightly higher average damage.

Including calculations for critical hits, Firebolt will do 3.025 average DPR against the goblin, while the dagger does 2.875 average DPR. The D&D 5e tag is there deliberately - the slightly-trickier math around critical hits is part of this..

Counting on bad luck

My thinking is that bad luck on damage rolls won’t be disastrous for the wizard if he attacks with the dagger. Two hits with the dagger will definitely kill the goblin, while the goblin might survive several Firebolts.

Is my intuition correct? Is the wizard more likely to beat the goblin using his dagger than with Firebolt?

I’m hoping someone can provide some math to prove what the wizard should do.

(Note, I think it's obvious that if you have an attack that can guarantee killing a creature in one attack, you'd want to use that attack over one that hits as often but sometimes does less damage. The above example is meant to be more subtle than that, a example where non-mathematically-minded folks might disagree on the best tactic.)

Answer 1

Background Theory

Broadly when choosing to optimise damage in a single round there are five variables you need to take account of:

• Average damage per round (I’m assuming that chance to hit has been rolled up into this variable)
• Damage variance
• Damage immunities/vulnerabilities
• Enemy HP
• Have you Crit’d (This fact significantly increases the math required.)

When you optimise DPR, you are optimising only one/two of these five potential variables (Criting can alter the ADPR).

If you are only rolling one dice, your damage variance on that attack will be relatively high.

To reduce the variance of the attack there are three broad strategies:

1. Increase the static damage for the attack (as in your firebolt vs dagger example) — Larger static variables weight it towards average damage.
2. Reduce the damage die (as in your firebolt vs dagger example) — Smaller dice have a lower variance.
3. Increase the number of dice you use in the attack (the more dice you roll, the more likely you are to get the average damage for a given attack) - This is an application of the law of large numbers.

In the situation where there is a damage type resistance/vulnerability then the choice between the two becomes clearer as you will be halving/doubling the result from the damage dice which will decrease/increase your DPR.

To answer whether or not you are more likely to beat a specific enemy depends on the HP, and variability in that enemies HP (determined by their hit dice). The same two considerations apply here.

If you are fighting an enemy with a large amount of hit dice, then by the same application of the law of large numbers you are more likely to be fighting an average specemin of the enemy.

Similarly if you are fighting an enemy with a large fixed HP component then you are similarly more likely fighting an average enemy.

Application to your example

The Firebolt has a range of damage values when it hits of 1-10 (with a 10% chance of each)

The dagger has a different range of damage values when it hits 4-7 (with a 25% chance of each).

A Goblin’s HP is calculated by rolling 2d6 (the 7 HP in the Monster Manual is simply the average value of this distribution). This gives the Goblin Monster a range of Hit Point values between 2 & 12 HP (obviously the extremes of 2 and 12 are pretty unlikely).

Both attacks have a 45% chance of a non-critical hit and a 5% chance of a critical hit.

The rest of this analysis will assume that we have hit (that makes the math slightly earier, and we can convert it into number of rounds by using this information later).

As a result, when we hit that translates to a 90% of hits are a non-crit, and 10% are a crit.

I’m also going to assume, for simplicities sake, that we are doubling the result of the dice when we crit, instead of doubling the number of dice we roll.

Vanilla Attacks

1 Hit to Kill

Taking your specific example of a Goblin (7 HP). When our attacks hit

• The Firebolt has a 40% chance of killing the Goblin in 1 non-critical hit (7–10)
• The Firebolt also has a 70% chance of killing the Goblin in 1 critical hit (4–10) x 2
• The dagger has a 25% chance of killing the Goblin in 1 non-critical hit (4 + 3)
• The dagger has a 75% chance of killing the Goblin in 1 critical hit ( 2 x (2–4) + 3)

Taking all of this into account, the 1 hit kill percentages are:

• Firebolt: 40% * 90% + 70% * 10% = 43%
• Dagger: 25% * 90% + 75% * 10% = 30%

3 Hits or more to Kill

At the other extreme, the chance of it taking more than two hits to fell the Goblin.

• Dagger: 0% (Even if we roll the minimum non-crit damage two hits will kill it.)
• Firebolt: We have to roll a total of 6 or less between the first two hits. With crits this is a more complicated piece of maths.

Ways we can get this total with Firebolt

2 x Crits:

• 1 x 2 + 1 x 2: ( 10% * 10% * 10% * 10% )
• 1 x 2 + 2 x 2: ( 10% * 10% * 10% * 10% )
• 2 x 2 + 1 x 2: ( 10% * 10% * 10% * 10% )

1 x Crit:

• 1 x 2 + 1: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 2: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 3: ( 10% * 10% * 10% * 90% )
• 1 x 2 + 4: ( 10% * 10% * 10% * 90% )
• 1 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 2 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 3 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 4 + 1 x 2: ( 10% * 90% * 10% * 10% )
• 2 x 2 + 1: ( 10% * 10% * 10% * 90% )
• 2 x 2 + 2: ( 10% * 10% * 10% * 90% )
• 1 + 2 x 2: ( 10% * 90% * 10% * 10% )
• 2 + 2 x 2: ( 10% * 90% * 10% * 10% )

0 x Crits:

• 1 + 1: ( 10% * 90% * 10% * 90% )
• 1 + 2: ( 10% * 90% * 10% * 90% )
• 1 + 3: ( 10% * 90% * 10% * 90% )
• 1 + 4: ( 10% * 90% * 10% * 90% )
• 1 + 5: ( 10% * 90% * 10% * 90% )
• 2 + 1: ( 10% * 90% * 10% * 90% )
• 2 + 2: ( 10% * 90% * 10% * 90% )
• 2 + 3: ( 10% * 90% * 10% * 90% )
• 2 + 4: ( 10% * 90% * 10% * 90% )
• 3 + 1: ( 10% * 90% * 10% * 90% )
• 3 + 2: ( 10% * 90% * 10% * 90% )
• 3 + 3: ( 10% * 90% * 10% * 90% )
• 4 + 1: ( 10% * 90% * 10% * 90% )
• 4 + 2: ( 10% * 90% * 10% * 90% )
• 5 + 1: ( 10% * 90% * 10% * 90% )

All of that is:

• 3 * ( 10% * 10% * 10% * 10% ) + 12 * ( 10% * 10% * 10% * 90% ) + 15 * ( 10% * 90% * 10% * 90% )

Which totals to 13.26%

Final Totals

As a result the probabilities of killing the Goblin with a repeated attack type are:

1 hit to kill:

• Firebolt: 43%
• Dagger: 30%

2 hits or less to kill:

• Firebolt: 86.74%
• Dagger: 100%

3+ hits or more to kill:

• Firebolt: 13.26%
• Dagger: 0%

Mix & Match

This is of course assuming you don’t mix and match attacks.

If we allow mixing and matching your choices change. For example, if you hit with the Firebolt or the Dagger for 6 damage, it doesn’t matter which attack you hit with next.

Similarly if you hit with the firebolt for 3 or more (80% probability for the hit), using the dagger next hit guarantees the kill.

If you hit with the firebolt for 1 on the first hit (10%) then your probabilities are:

• Firebolt: 50% (6 - 10) * 90% + 80% (6 - 20) * 10% = 53%
• Dagger: 50% (6 - 7) * 90% + 75% (7 - 11) * 10% = 52.5%

The Dagger’s smaller variance will probably give it enough to pip the Firebolt (but it’s close).

If you hit with the firebolt for 2 damage on the first hit (10%*90% + 10%*10% = 10%) then your probabilities are:

• Firebolt: 60% (5 - 10) * 90% + 80% (6 - 20) * 10% = 62%
• Dagger: 75% (5 - 7) * 90% + 100% (4 - 11) * 10% = 77.5%

You can see where this is trending…

Consolidated Percentages

1 Hit to Kill

• Firebolt: 43%
• Dagger: 30%

2 Hits or less to Kill

• Firebolt, then Dagger: (80%*90% + 90%*10%) * 100% + 10% * 52.5% + 10% * 77.5% = 94%
• 2 x Firebolt: 86.74%
• 2 x Dagger: 100%
• Dagger then Firebolt (we don't care as doing this would be stupid)

3 Hits or less to Kill

• Firebolt, Dagger, Dagger: 100%

3 Hits or more to Kill

• 3 x Firebolt: 13.26%

Strategy against a 7 HP Goblin

Hit with Firebolt first, as it gives you a better chance of one-shot killing the Goblin.

With two hits the probabilities of killing the Goblin are: Firebolt + Dagger = 94% Dagger + Dagger = 100%

If you hit with the Firebolt on your first hit, and it doesn't kill the Goblin (and the Goblin has 7HP) then you are better off switching your second attack to be the dagger, as you have a higher (and less variable) probability of killing the Goblin with the Dagger on the second hit.

Given that the increase of kill % with 1 hit on a Firebolt (13%) is greater than the decrease (6%) in % we get over two rounds from not doing Dagger, Dagger, Optimal play is to do Firebolt + Dagger.

On average how many rounds does it take for you to hit the Goblin?

We effectively have repeated trials to success of a iid Bernoulli random variables with p=0.5. The expected waiting time until a success (a hit) for this type of process is given by the expected value of a Geometric distribution, with p=0.5.

E[Rounds to Hit] = 1/0.5 = 2 rounds.

Thus the expected rounds to get 2 hits is 4.

Conclusion

This sort of spread will change depending on the actual HP of the Goblin.

• Lower than 7 HP will lean towards you using the Dagger for both attacks (6HP is the point where the one shot probabilities for both Firebolt and Dagger are close enough (53% vs 52.5%) that the consistency of the Dagger leans in its favour).
• Higher than 7HP Firebolt first (and potentially 2nd) Dagger later will be the better combo.

Answer 2

I ran some simulation using a tweaked version of DnD-battler by matteoferla and for the above example, here is the distribution of final health for the wizard.

In summary, the Firebolt is more likely to get you killed by about 1.7% but about 4.4% more likely to let you get away with no health loss.

Full stats

Stat               | Dagger | Firebolt
--------------------------------------
Wins               | 85.83% | 84.65%
Perfects           | 27.84% | 31.65%
Deaths             | 14.17% | 15.35%
Goblin Perfects    | 3.83%  | 3.81%
Average Final HP   | 9.38   | 9.49
Standard Dev. HP   | 6.70   | 6.96
Average Damage     | 8.76   | 8.96 
Goblin Avg Hp      | -1.76  | -1.96
Goblin Std Dev Hp  | 2.98   | 3.755 
Goblin Avg Damage  | 7.6198 | 7.502

Answer 3

Let's break it down a bit ...

Building a chart to help visualize (using AnyDice to help out :) )

       --------- 1d4+3 ----------    --------- 1d10 -----------
  HP   % kill 1 % kill 2 % kill 3    % kill 1 % kill 2 % kill 3
  1     100.0%                        100.0%      
  2     100.0%                         90.0%   100.0%   
  3     100.0%                         80.0%    99.0%   100.0%
  4     100.0%                         70.0%    97.0%    99.9%
  5      75.0%   100.0%                60.0%    94.0%    99.6%
  6      50.0%   100.0%                50.0%    90.0%    99.0%
  7      25.0%   100.0%                40.0%    85.0%    98.0%
  8       0.0%   100.0%                30.0%    79.0%    96.5%
  9               93.8%                20.0%    72.0%    94.4%
  10              81.3%                10.0%    64.0%    91.6%
  11              62.5%                0.0%     55.0%    88.0%
  12              37.5%   100.0%                45.0%    83.5%
  13              18.8%    98.4%                36.0%    78.0%
  14               6.3%    93.8%                28.0%    71.7%
  15               0.0%    84.4%                21.0%    64.8%
  16                       68.8%                15.0%    57.5%
  17                       50.0%                10.0%    50.0%
  18                       31.3%                 6.0%    42.5%
  19                       15.6%                 3.0%    35.2%
  20                        6.3%                 1.0%    28.3%
  21                        1.6%                 0.0%    22.0%
  22                        0.0%                         16.5%
  23                                                     12.0%
  24                                                      8.4%
  25                                                      5.6%
  26                                                      3.5%
  27                                                      2.0%
  28                                                      1.0%
  29                                                      0.4%
  30                                                      0.1%
  31                                                      0.0%

This gives us a more complete picture of what's happening. As you can see, the answer very much depends on how many HP the target has.

For example: 4 hp? Dagger. No question.

6 hp? Again, I'd go dagger. Both have 50% to 1 shot, but dagger is 100% to kill in 2nd shot only, firebolt has 10% to miss 2nd, and need a 3rd.

8 hp? Harder to call, firebolt looks better to 1 shot at 30% . but when you realize it also has a 70% chance NOT to .. and on top of that, a good chance to require a 3rd shot .. and even off chance to require a 4th!!! but Dagger: 2 shots .. guaranteed .. every time .. every day .. no matter what? Hmm ... possibly personal preference at this time, but I'd probably go Dagger myself.

17 hp appears to be the breaking point to me ... 50% both to 3 shot .. but a 10% firebolt takes him in 2 .. I'm not 100% sure how to work out the math on this (or for larger numbers and such), but hoping this chart helps people visualize the problem :) it's not really as simply as saying "oh, we might 1 shot him, firebolt for the win" ..

Answer 4

I'm taking this as a general gaming question with a D&D-specific example; my answer addresses the general case and not the D&D 5E specifics.

What you usually want out of a fight is "kill your opponent before they kill you". Average damage is one way of trying to choose strategies for this, and in a long fight where random stuff averages out, it's pretty good. But in a short fight, as your intuition suggests, it's not so great.

A better way to choose strategies is to look at the probabilistic distribution of the time it will take you to down your opponent; Ditto has given an example of this. But this still isn't perfect.

Let's suppose I am a wizard with 20 HP, fighting my evil twin who also has 20 HP. We each have the same three spells in our repertoire, each of which takes one round to cast:

Rory's Reliable Roasting does 4 HP damage with no miss chance, and we can cast it every round.

Otto's Overpowered Orb does 12 points of damage, but it only has a 40% hit chance. Each of us can only cast it once per combat.

Amelia's Amazing Aim gives +20% to hit on my next spell. Each of us can only cast it once per combat.

Three possible tactics are:

• A: pew pew pew with Rory's Reliable Roasting. Guaranteed to down my opponent in 5 rounds.
• B: Overpowered Orb, then pew pew pew. If the orb hits (40% chance), total time is 3 rounds (12 HP from the orb, then two hits from Reliable Roasting). If it misses (60% chance), then it takes me 6 rounds.
• C: Amazing Aim, then Overpowered Orb, then pew pew pew. If the orb hits (60% chance), total time is 4 rounds; if it misses (40% chance), 7 rounds.

So, if my twin is choosing from the same three options, which is the best?

It's easy to see that Option A beats B 60% of the time (i.e. when the orb misses) and C beats A 60% of the time (when the orb hits).

But if you work the numbers, B beats C 64% of the time. (60% that B takes only 3 rounds, and is guaranteed to win; 40% x 60% = 24% that B takes 6 and C takes 7.)

So, in general, choice of strategy is non-transitive, meaning that the "best" option depends on your opponent's attack mode.

It may also be advantageous to switch tactics partway through the fight depending on how your luck has been so far; in general, the more of a lead you have over your opponent, the less variance you want.

Answer 5

Depends on your estimations and priorities, and of course on the specific circumstances.

The DPR is similar, but for your example the Firebolt has a better chance of ending the conflict in a shot, and to me that's the most relevant criterion. A hit with the dagger will only drop the goblin to 0hp on maximum damage roll of 4, so about 25% chance, whereas the Firebolt will drop the goblin to zero on a 7 or better out of a D10, so about 40% chance.

If your opponent has 6hp, then a hit from either will drop it on a 50% chance. A 5hp (or less) opponent favors the dagger, with 75% chance to drop it, rather than 60%. An opponent with 8 to 10 hp very much favors the Firebolt, as it could end the conflict with one hit, whereas the dagger has no chance of that. More than 10hp for the opponent slightly favors the Firebolt due to the small advantage in average damage.

So, as usual, it's highly situation-dependent. But I'd be inclined to use the Firebolt against anything bigger than a Kobold (unless it had fire resistance).