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#1




How Magical Really Is 4 Defense?
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? Comic Battle Reports / Gencon Battle Reports  Probability Calculator App  Reverse the Whip Army Archetypes "It's all about the game."  Sgt. Ernie Calhoun Last edited by vegietarian18; May 15th, 2021 at 05:44 PM. 
#2




Re: How Magical Really Is 4 Defense?
Great post!
I think the other thing to take into consideration is just how "easy" it is to get to 4 defense. After 4 defense, you start paying a lot more for those extra defense dice, and those extra defense dice aren't as big of a deal as just getting to 4 in the first place (compare 4man melee bonding like Heavies, Knights, etc. to Hounds, Minions, etc.). 
#3




Need More Research
I feel like there's something potentially meaningful here with a derivative or something but I haven't looked deeply enough to see if that's anything useful beyond what you've already done.
~Dysole, also stats geek My Twitch Channel where I play Scape and other things My YouTube Channel where the games get uploaded later This sentence is seven words long. This sentence is not seven words long. 
#4




Re: How Magical Really Is 4 Defense?
What I have found (that ties into the survivability concept) is early loss. Being able to survive the first attack with most of your positioned pieces for a counter is what makes 4 defense dice so magical (in my mind). I was playing someone who had Quahon against my three defense Specters. His comment that if you could just roll 2 out of 4 on the skulls (11 out of evey 16 attacks), he would win. This thought was supported by the game. He rolled 2/4 the first three times and killed 2.5 Specters on average (plus one more with spiders). This completely stopped any forward movement I tried to establish. If you could have just reached some level of height (getting that 4th die), I would have been in the game. Those 4 defense dice gives me much more confidence that I can advance and hold position to establish my game plan. I know that this is not a statistical study, but it sure seems to play out a lot in the games (Veggie, you bombing my 3 defense mezzos with your elf wizard, for example).
the four defense dice allow you to "hold the line" against first attacks and advance the rest of your army. What GenCon taught me was to assume only one skull on three dice! 
#5




Re: How Magical Really Is 4 Defense?
Cost is clearly a factor. I think melee squads that have 4 def or ranged that easily get e def + 1 on height are not much more expensive then 3/2 squads. The 4th is a good example.

#6




Re: How Magical Really Is 4 Defense?
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#7




Re: How Magical Really Is 4 Defense?
And quickly realizes the math shows his extra defense dice are redundant and would have been better served by a deeper life pool.


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