BY RICK MOSCATELLO
Dungeons and Dragons 5E has a few “roll two dice, take the highest” mechanics. The most popular, for good reason, is the Advantage mechanic for rolling a D20 to hit, which is fairly close to a +4.
There are also a few ways to get a “roll two dice, take the highest” effect when rolling damage. Before looking at how this mechanic works for damage, let’s first take a quick look at how one figures the average damage for a single die.
When rolling a single die for damage, you take an expectation. For a D4, for example, you take 1/4 times 1 + (1/4) times 2 + (1/4) time 3 + (1/4) times 4…which adds up to an expected damage of 2.5. I could also just factor out the (1/4), and get (1/4) * (1 + 2 + 3 + 4), a slightly easier way to calculate.
Rather than do that kind of calculation, most players, when dealing damage, just add up the lowest and highest number on the die, and divide by two. That works for a single die
So, the average damage for a D6 is 3.5, (1 + 6)/2. This trick works for a single die roll, but not when rolling multiple dice or dealing with a a strange new mechanic, although taking an expectation still works.
Let’s take a look at expected damage with this mechanic for various dice:
Roll 2 D4, take the highest
Let’s consider the four possible values, and the number of ways you can get those values—each way is equally likely amongst the 16 possible outcomes from rolling a D4 twice.
1 damage is achieved in 1 way, rolling (1,1)
2 damage is achieved in 3 ways, rolling (1,2), (2,1), or (2,2)
3 damage is achieved in 5 ways, rolling (1,3), (2,3), (3,2), (3,1) or (3,3)
4 damage is achieved in 7 ways, rolling (1,4), (2,4), (3,4), (4,3), (4,2) (4,1) or (4,4)
Note how the number of ways of rolling a damage number increases by two each time the damage increases by one—this pattern works for larger dice as well.
Note also how when I add up all the ways, I get 16—I’ve accounted for all the possible outcomes of rolling a D4, two times.
So now I take expectation.
(1/16) * ( 1 * 1 + 3 *2 + 5*3 + 4 *7) = 3.125.
Thus the new mechanic isn’t much of an improvement for a D 4—you basically expect to do an extra 0.6125 point of damage per attack. Whoop de do.
Roll D6 twice, take the highest
Again, I set up the table of all possible damage rolls, and number of ways I can get that damage (I’ll not bother listing the actual events):
1 damage in 1 way
2 damage in 3 ways
3 damage in 5 ways
4 damage in 7 ways
5 damage in 9 ways
6 damage in 11 ways
Note the number of ways adds up to 36—it’s a cutesy trick. Anyway, I take expectation:
(1/36) * ( 1 * 1 + 2 * 3 + 3 * 5 + 4 * 7 + 5 * 9 + 6 * 11) = 4.47
Gee whiz, the new mechanic is almost identical to just adding 1 point of damage (since the expected roll of a D6 is 3.5).
Roll D8 twice, take the highest
The great thing about seeing the pattern is it really speed up calculating:
1 damage in 1 way
2 damage in 3 ways
3 damage in 5 ways
4 damage in 7 ways
5 damage in 9 ways
6 damage in 11 ways
7 damage in 13 ways
8 damage in 15 ways
With 64 possible outcomes of rolling a D8 two times, it’s good to have a trick. Anyway, the expected damage is 5.8125—well over 1 extra point of damage now.
It’s very clear that as the die gets bigger, the new mechanic becomes more powerful.
When we get to “D10, take the highest”, the expected damage is 7.15—1.65 extra damage. Nice, but hardly game breaking.
Finally, with “D12, take the highest” the expected damage is 8.49—almost two whole extra damage! Woohoo!
While nobody rolls a D20 for damage (I had monsters rolling it in my 15th level 4e game, though), the new mechanic would give an expected damage of 13.825, well over 2 points. One might have guessed it would be closer to 4 points, since Advantage is basically a +4 to hit, but the usefulness of a bonus varies with how hard it is to hit, unlike damage which is always just, well, damage. Note that Advantage for saving throws is thus more like a +2.
While not the most amazingly powerful mechanic from a game standpoint, it is at least one less modifier to add to a single die roll, and getting the most out of it hardly requires any game mastery—everyone knows when its time to bash monsters, you always want to roll the biggest die you can grab, after all.
Dungeons and Dragons 5E has a few “roll two dice, take the highest” mechanics. The most popular, for good reason, is the Advantage mechanic for rolling a D20 to hit, which is fairly close to a +4.
There are also a few ways to get a “roll two dice, take the highest” effect when rolling damage. Before looking at how this mechanic works for damage, let’s first take a quick look at how one figures the average damage for a single die.
When rolling a single die for damage, you take an expectation. For a D4, for example, you take 1/4 times 1 + (1/4) times 2 + (1/4) time 3 + (1/4) times 4…which adds up to an expected damage of 2.5. I could also just factor out the (1/4), and get (1/4) * (1 + 2 + 3 + 4), a slightly easier way to calculate.
Rather than do that kind of calculation, most players, when dealing damage, just add up the lowest and highest number on the die, and divide by two. That works for a single die
So, the average damage for a D6 is 3.5, (1 + 6)/2. This trick works for a single die roll, but not when rolling multiple dice or dealing with a a strange new mechanic, although taking an expectation still works.
Let’s take a look at expected damage with this mechanic for various dice:
Roll 2 D4, take the highest
Let’s consider the four possible values, and the number of ways you can get those values—each way is equally likely amongst the 16 possible outcomes from rolling a D4 twice.
1 damage is achieved in 1 way, rolling (1,1)
2 damage is achieved in 3 ways, rolling (1,2), (2,1), or (2,2)
3 damage is achieved in 5 ways, rolling (1,3), (2,3), (3,2), (3,1) or (3,3)
4 damage is achieved in 7 ways, rolling (1,4), (2,4), (3,4), (4,3), (4,2) (4,1) or (4,4)
Note how the number of ways of rolling a damage number increases by two each time the damage increases by one—this pattern works for larger dice as well.
Note also how when I add up all the ways, I get 16—I’ve accounted for all the possible outcomes of rolling a D4, two times.
So now I take expectation.
(1/16) * ( 1 * 1 + 3 *2 + 5*3 + 4 *7) = 3.125.
Thus the new mechanic isn’t much of an improvement for a D 4—you basically expect to do an extra 0.6125 point of damage per attack. Whoop de do.
Roll D6 twice, take the highest
Again, I set up the table of all possible damage rolls, and number of ways I can get that damage (I’ll not bother listing the actual events):
1 damage in 1 way
2 damage in 3 ways
3 damage in 5 ways
4 damage in 7 ways
5 damage in 9 ways
6 damage in 11 ways
Note the number of ways adds up to 36—it’s a cutesy trick. Anyway, I take expectation:
(1/36) * ( 1 * 1 + 2 * 3 + 3 * 5 + 4 * 7 + 5 * 9 + 6 * 11) = 4.47
Gee whiz, the new mechanic is almost identical to just adding 1 point of damage (since the expected roll of a D6 is 3.5).
Roll D8 twice, take the highest
The great thing about seeing the pattern is it really speed up calculating:
1 damage in 1 way
2 damage in 3 ways
3 damage in 5 ways
4 damage in 7 ways
5 damage in 9 ways
6 damage in 11 ways
7 damage in 13 ways
8 damage in 15 ways
With 64 possible outcomes of rolling a D8 two times, it’s good to have a trick. Anyway, the expected damage is 5.8125—well over 1 extra point of damage now.
It’s very clear that as the die gets bigger, the new mechanic becomes more powerful.
When we get to “D10, take the highest”, the expected damage is 7.15—1.65 extra damage. Nice, but hardly game breaking.
Finally, with “D12, take the highest” the expected damage is 8.49—almost two whole extra damage! Woohoo!
While nobody rolls a D20 for damage (I had monsters rolling it in my 15th level 4e game, though), the new mechanic would give an expected damage of 13.825, well over 2 points. One might have guessed it would be closer to 4 points, since Advantage is basically a +4 to hit, but the usefulness of a bonus varies with how hard it is to hit, unlike damage which is always just, well, damage. Note that Advantage for saving throws is thus more like a +2.
While not the most amazingly powerful mechanic from a game standpoint, it is at least one less modifier to add to a single die roll, and getting the most out of it hardly requires any game mastery—everyone knows when its time to bash monsters, you always want to roll the biggest die you can grab, after all.
Under the Hood: Roll 2 Dice, Take The Highest
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