Heroscape Probability 120

[vegietarian18]

This is my thread for my various Heroscape probability posts. We'll see how many of these I actually write; will probably depend on if there's any audience.

Heroscape Probability 100 is simple. There are three skulls on each die and only two shields. You are more likely to roll skulls than shields. You can compute the outcomes for how often you will roll on X skulls out of Y dice using a

binomial distribution

binomial distribution

.

Heroscape Probability 110 is AliasQTip's Probability Table, or my Android app Sterilizing Pear. These give the probability that an attack of X dice vs. a defense of Y dice does at least one wound, and also gives the average damage from each attack. Using series of binomial distributions, you can compute all possible outcomes for a single attack.

Heroscape Probability 120 goes one level up from 110. Once you know the odds of individual attacks, you can do some fun things. How many Redcoats does Q9 kill when he's Queglixing with 3-3-3 vs. 2-2-2-2-1, and what's standard deviation there? What are the odds Cyprien or Jotun can defeat Krug in 1v1 combat? These are the kind of questions we're trying to answer in 120.

Please keep in mind that at all levels, Heroscape games are way too complex to reduce down to probability calculations. There's always real gameplay and choice by either side that lies uncomputed in the math. And even just computing everything is too hard. However, I do think that knowing this math can increase your skill at the game. It helps solidify existing intuition about correct plays, and helps create new intuition.

Probability Post 1: Me-Burq-Sa vs. Attacks of Three
Probability Post 2: Major Q9 vs. Blade Gruts

[vegietarian18]

For today's lesson, we're going to be using an example from the VCheese 3 tournament game between dok and infectedsloth. Do Blade Gruts have enough in the tank to kill Q9? This is a fun example game because Q9 ended up dying on the exact last possible attack he could have faced, a 3v8 from Raelin that ended up being a 3 skull 2 wound killing blow. If Q9 survives that attack, he easily cripples Raelin enough for the remaining Rats to finish her off. So we have a very good measure of the "tank" of Blade Gruts.

As always, note that this game was heavily influenced by factors not quantified in the math. The largest is that infectedsloth's Grimnak successfully Chomped dok's Raelin while on 4 wounds, to majorly open up the door for the Blades to crack open Q9. However, following that, Q9 won a big initiative to be able to return to his perch, which majorly changed the amount of dice he faced from that point onwards. Quantifying the relative likelihood and relative impact of either of those two major swings, along with other gameplay swings, is very difficult. What we are going to try to quantify here is "should Q9 have died to the attacks that he faced?"

The attacks that Q9 faced are listed below:
12 attacks of 3v10 (Grimnak boosted Blades attacking into Q9 with height and Raelin's aura)
1 Leaving Engagement attack (Q9 leaving engagement with Kuthnak to shoot Grimnak)
1 Attack of 4v7 and 1 Attack of 3v7 (A Kuthnak boosted Blade and a normal Blade attacking Q9 with height after Grimnak chomped Raelin)
15 attacks of 2v8 (Kuthnak and the last Blades swinging desperately into Q9 after he returned to his perch)
1 Attack of 3v8 (Raelin coming in for a final swing, the blow that killed the Major)

The traditional method of approximating hero survivability is average damage, which is somewhat flawed. You can read my first post, linked in the OP, for a more in depth explanation, but essentially, average damages gives credit for overkill. If Q9 is already at 4 wounds, the fact that your excess attacks continue to wound him doesn't matter, but in average damage approximation method, that overkill would give credit as if you were speeding up your average Q9 kill time. Average damage says Q9 should have taken 3.82 wounds here, which makes it sounds like he pretty well should have died.

The more accurate way to measure a figures survivability is through a probability state diagram. Major Q9 can either be at 0, 1, 2, 3, or 4 wounds. If he's at 4 wounds he's dead, and no attacks after that matter. What we need to ask is after N attacks, what is the likelihood that Q9 is in each of these four states? And, as we transition to the N+1th attack, what is the relative likelihood of him changing from each state to the others?



The chart above is the completed state diagram, but we'll look it in close detail to explain the math. For example, by 3v10 attack #3, there is a 77% chance that Q9 is at zero wounds. As we transition to 3v10 attack number four, Q9 has a 91.7% chance to remain in the 0 wound state, a 6.3% chance to transition to the one wound state by taking one wound from that attack. 6.3% of 77% is 4.8%, but you'll notice that between #3 and #4, Q9's odds of being in the 1 wound state only went up by 3.6%. That's because in the alternate worlds where Q9 entered that attack in the 1 wound state, he similarly had a chance to leave that state and progress further up in the tree: the missing 1.2% is distributed up the tree into the 2, 3, and 4 wound states.

The final cell, in the bottom right corner, is the probability that Q9 takes >=4 wounds total from all of these attacks, and the answer my math gave is 50.1% percent, which I think is kind of a beautiful answer. Either it happens or it doesn't. Obviously again, this does not mean Blades beat Q9 50% of the time. There's so many complexities around whether Q9 would have taken more or less LEAs if he took more/less wounds, how he would have positioned, that even simple quantification of the unknown elements can't answer it.

I think the lesson here is that the Heroscape 110 probability way of looking at this situation with "Raelin only had a 3% chance to put the killing blow on Q9" is a little reductive. After all of the attacks Q9 had faced, there was only a 54% chance that Q9 was living at that moment. Q9 is one of the harder figures to think about killing, because he's just so tanky that it can feel like the attacks are doing nothing at all, but every attack has a chance to do something, and at the end of the day those chances add up if you keep taking them.

[S1R_ART0R1US]

Great post! Love to see these probability examinations of in-game scenarios. I forget which game it was, but somebody went for Wannok with a 2 life Q9 to close out a round and the lone Morgan's riflemen took him out with a height attack. I like seeing the breakdown (your's?) afterwards comparing the odds of going for Wannok versus attacking on even ground.