Dice Choice and Reliability

This morning I was thinking about monster design, and how different choices affect play.

Specifically, I was thinking about how to balance attacks. Is it better to have two attacks each doing 2d6 damage, or a single attack doing 4d6? Or, making the situation a bit bigger for examination, one attack for 8d6, two for 4d6, or four for 2d6.

Examination

Let’s look at all three configurations, at three different attack success rates (50%, 25%, 75%).

Attacks Successful 50% of the Time

Simplest math is when there is a 50% chance of each attack hitting.

Successes	1 Attack	Mean Dmg (8d6=28)		2 Attacks	Mean Dmg (4d6=14)		4 Attacks	Mean Dmg (2d6=7)
0	50%	0		25%	0		6.25%	0
1	50%	14		50%	7		25%	1.75
2				25%	7		37.5%	5.25
3							25%	5.25
4							6.25%	1.75
Total		14			14			14

Turns out mean damage is the same either way. Is there a difference if the chance of hitting changes?

Attacks Successful 25% of the Time

Let’s see what happens when there is a 25% chance of each attack hitting.

Successes	1 Attack	Mean Dmg (8d6=28)		2 Attacks	Mean Dmg (4d6=14)		4 Attacks	Mean Dmg (2d6=7)
0	75%	0		56.25%	0		31.64%	0
1	25%	7		37.50%	5.25		42.19%	2.95
2				6.25%	1.75		21.09%	2.95
3							4.69%	0.98
4							0.39%	0.11
Total		7			7			7

Same mean across the board.

Attacks Successful 75% of the Time

Finally, when there is a 75% chance of each attack hitting.

Successes	1 Attack	Mean Dmg (8d6=28)		2 Attacks	Mean Dmg (4d6=14)		4 Attacks	Mean Dmg (2d6=7)
0	25%	0		6.25%	0		0.39%	0
1	75%	21		37.50%	5.25		4.69%	0.33
2				56.25%	15.75		21.09%	2.95
3							42.19%	8.86
4							31.64%	8.86
Total		21			21			21

Again, same mean across the board.

Interpretation

So, in terms of raw damage, all three options work out the same. Over time, all three options (at each success rate) will approach the mean damage.

And yet, there are reasons to pick one over the other.

The single attack will hit hard, when it hits at all. You might favor this one when you need to exceed a target amount (such as damage reduction in D&D 3.x), or if you want something that is devastating but unreliable.

Four attacks will individually hit for much less damage, and odds are against maximizing the damage on any attack.. but missing entirely is less likely also. This is a good choice if you want to reliably do some damage, especially if there are additional effects when you hit (poison/bleed are common in video games). You might prefer this one if you want something more reliable, if less dramatic.

Changing Success and Damage

Not shown above, you can also adjust the damage and get consistent output. For example, the first table (50% success) doing 4d6, 2d6, or 1d6 points of damage per attack will have the same mean as the second table.

Successes	1 Attack	Mean Dmg (4d6=14)		2 Attacks	Mean Dmg (2d6=7)		4 Attacks	Mean Dmg (1d6=3.5)
0	50%	0		25%	0		6.25%	0
1	50%	7		50%	3.5		25%	0.875
2				25%	3.5		37.5%	2.625
3							25%	2.625
4							6.25%	0.875
Total		7			7			7

Action Economy

All the above assumes they consume similar amounts of action economy. In D&D 3.x, taking multiple attacks in a round is almost always a full-round action, while single attacks are usually a standard action — leaving the option of a move equivalent action as well. In that case, I’d lean toward a single attack. It might not be as reliable in a particular round, but being able to do other stuff as well can be very valuable. If all options have the same economy/cost, the decision is not as simple.

Closing Comments

In the absence of other considerations, attacks * success * mean damage is a reasonable starting point when examining DPS (damage per second… or damage per round in D&D-based games). You can manipulate any of the three and keep DPS the same.

However, while DPS might be consistent there can be other considerations that lead to preferring one over the other. Action economy, need to overcome damage thresholds, or stackable results on any success regardless of damage, all can influence which option is most attractive.