How Magical Really Is 4 Defense?

[vegietarian18]

For the entire time I've played Heroscape, I've heard a lot about the "magical" fourth defense die. It used to be a big talking point for why the Knights of Weston were meaningfully better than other melee bonding squads, why Grimnak/Nerak were so important to Heavies, etc. As a stats person, I've always been somewhat skeptical of the concept. In general rule, figures with low defense benefit more from getting extra defense dice than figures with high defense. But I'm also reluctant to discount heuristics like this, so I wanted to spend some time looking at the data and trying to figure out what makes the 4th defense die so magical.



For this article, I'll be using AliasQTip's probability table. Screenshot above. Soapbox for a moment here: it's so important to see Heroscape dice rolls not as independent attack and defense rolls, but rather two rolls together that form a "probability density function". In my head when I'm playing Heroscape, I never treat a one skull roll as "oh I'm not getting any damage". Let's look at a 3v3 attack, the most common attack in Heroscape, which is a 50.9% chance of hitting or blocking. People can whiff on 3 dice (or 4 dice or even more than that), and that percent chance of them whiffing against your 1 skull roll is meaningfully a part of the 50% chance of your 3v3 attack hitting.

1/3 skulls = 37.5% likely, 0/3 shields = 30% likely to happen, probability of both happening =.375 * .30 = .11. So before any dice are rolled there was an 11% likelihood of 1 skull met by 0 shields, so that's more than 20% of the time that your 3v3 attack goes through. Sure, 2 skulls met by one shield is more likely, and 2 skulls does damage on a whiff too, but the point is just not to discount your chickens before the ugly egg hatches.

Anyway, back to the original question at hand, let's look at how the percentages change as we roll across the probability table towards 4 defense from various angles.

For a squad figure blocking an attack of 3 dice, you have a 12.5% chance of surviving with 0 defense, a 25% chance of surviving with 1 defense, a 37.5% chance of surviving with 2 defense, a 49.1% chance of surviving with with 3 defense, a 59.3% chance of surviving with 4 defense, a 67.9% chance of surviving with 5 defense, and a 75% chance of surviving with 6 defense.

So let's look at how much percent chance of survival each additional die is giving you:

0->1 defense = 12.5% -> 25.0% = 12.5% gained
1->2 defense = 25.0% -> 37.5% = 12.5% gained
2->3 defense = 37.5% -> 49.1% = 11.6% gained
3->4 defense = 49.1% -> 59.3% = 10.2% gained
4->5 defense = 59.3% -> 67.9% = 7.6% gained
5->6 defense = 67.9% -> 75.0% = 7.1% gained

You can see the diminishing returns here; with every defense die after the second, you gain blocks a smaller percentage of the time, which makes 2 our inflection point. 2 is the inflection point of defense die gain to percent chance to block gaining, which surface level could be our magic number, but maybe there's more going on here. I think the heuristic of why 4 feels magical is clear looking at this table; 4 is the critical point when you become >50% likely to survive an attack of 3. More significantly, you start see the diminishing returns rapidly with the 5 defense die, going from gaining around 10% more likelihood to block per die to about 25% less than that, 7.6%. So I think the heuristic has some value.

Let's do the same math for 4 attack dice thrown at a squad figure, probably the second most common attack in Heroscape.

0->1 defense = 6.2% -> 14.6% = 8.4% gained
1->2 defense = 14.6% -> 24.3% = 9.7% gained
2->3 defense = 24.3% -> 34.5% = 10.2% gained
3->4 defense = 34.5% -> 45.4% = 10.0% gained
4->5 defense = 45.4% -> 53.7% = 8.3% gained
5->6 defense = 53.7% -> 62.0% = 7.3% gained

3 defense is our inflection point here, at which every defense die gives diminishing returns. But 4 is very close behind, only .2% less. 4 doesn't quite get you to 50% likelihood to survive an attack of 4, but attacks of 4 are very powerful. 4 gets you to 45%. 5 is the cutover to when you can survive attacks of 4 more than 50% of the time, but it takes a lot to get to get 5 dice, and the proportional gain relative to 4 is notably lower.

Let's take a look at 2 attack dice at a squad.

0->1 defense = 25.0% -> 41.7% = 16.7% gained
1->2 defense = 41.7% -> 55.6% = 13.9% gained
2->3 defense = 55.6% -> 66.6% = 11.0% gained
3->4 defense = 66.6% -> 75.3% = 9.7% gained
4->5 defense = 75.3% -> 82.1% = 6.8% gained

The inflection point here is 1, so every die is diminishing returns, since attacks of 2 are very likely to whiff. Interestingly, I think you can argue 4 is a powerful number against attacks of 2, but 2 is the number at which you cross over 50% chance to block. But attacks of 2 are usually pings that you have to block frequently, so getting to 3 and 4 dice against those attacks meaningful.

Going up to attacks of 5, 6, 7, the inflection points get much higher, but those giant attacks are so likely to kill squads and so often used against heroes that the math can't be applied the same way here. Against heroes what's meaningful is average damage, not likelihood to get one wound (until you get the hero down to 1 or 2 life remaining).

So against the meaningful range of Heroscape attacks, I think 4 defense does have some magical properties. Most notably against attacks of 3 and 4. What do you think?