Math: Should I Boost My Roll to Win This Duel?

Good morning gamers,

In our last post on dueling math, we talked about how to make a spreadsheet that computes your probabilities of winning a fight, given two sets of combatants. We talked about how to scale the number of dice being used - and even applying certain dueling penalties to your models. When we got to increasing your dueling rolls (usually with Might, could be with other rules), we decided to call it quits and start a new post. That's where we're picking up today.

Part 5: Handling Dueling Penalties and Boosts Better

It turns out that computing boosted dueling rolls requires re-thinking the way we've done the previous computations, since we can now have numbers going both up and down (and, of course, the reductions and boosts can be totally different). We also know that penalties are applied before Might is spent, so nothing can get below a 1 - and nothing can be higher than a 6. We know that reasonably we shouldn't see more than a -2 penalty applied to dueling rolls AND we shouldn't see more than a +5 applied to boosting rolls (to boost a 1 to a 6), so let's assume that (from a configuration perspective) this is what we need the tool to do. 

To begin with, let's take as our case study a duel between Angbor the Fearless (who gets a -1 penalty when he doesn't roll a 6) and a Morannon Orc Captain with shield. To compute Angbor's dueling probabilities, we will be setting up an extra set of numbers near the box that pre-computes the dueling penalties and boosts we're willing to make:

On the right side of the 6x6 grid, we have greyed-out numbers that apply the penalties and bonuses that we specify in our configuration section for the vertical character. You'll see something similar below the 6x6 grid for the horizontal character. These tell us what value we would ROLL in order to get the final result on the left-hand side (so Angbor's value for "1" on the left shows a "2" on the right side because the 2 will be degraded to a 1 with his broadsword). Similarly, he has no value for a 5 - any 5s will become 4s and any 6s will remain 6s.

The greyed out cells work as follows: =<original_value> - <dueling_penalty> - <boosting_bonus>. If you're thinking, "Wait, why am I subtracting a negative number and subtracting a positive number? I want my negative number to make my number go down and my positive number to make my number go up, right?" Well, yes . . . that's actually what we're doing, but it's not intuitive. Let's look at an example to make it a little clearer.

Suppose you suffer a -1 penalty from a two-handed weapon. Assuming you don't have Might to boost your dueling roll, you will end up with a 3 whenever you would have rolled a 4. In our system, to figure out how often you get a result of a 3 as your final value, you want the grey number to become a 4 in the line that the 3 is displayed in. Since the penalty is listed as a negative value, we need to subtract the negative from our original value in order for it to get increased properly.

Similarly, if you boost your dueling roll by +1 (usually with Might), all rolls of 2 will become a 3 after you boost it. So if we're trying to figure out how often we get an end result of 3, we would want the grey number to become 2 (which, again, involves subtraction).

Two of the greyed-out cells do special things: the one in the "5" slot has to check to see if both the dueling penalty box is set to preserving 6s AND no Might is being used to boost - when this is the case, it returns 0 (since you will never have a situation where you end up with a 5 high). In all other cases, it works just like all the other cells. Similarly, the cell in the "6" slot has to check to see if the dueling penalty box is set to preserving 6s - if so, it's set to 6. Otherwise, we use the exact same formula that we used in the previous boxes (original score - penalties - boosts).

Okay, so let's assume that we now have the right values being calculated so we know what our expected values are supposed to be. How do we translate these new values into our 6x6 grid? Well, we update our cells on the left and top sides of the grid - here's how . . .

Part 6: Calculating Expected Probabilities Given Adjusted Die Rolls

We previously had all kinds of values in the left/top cells to account for dueling penalties. We're going to simplify all of this now. We're going to have two special cells and four "normal" cells - let's start with the normal ones first.

In the image above, you'll notice that cell B5 (which computes our probability of getting a 2 high) is looking at J5 instead of C5 (our new computed value of "3" instead of the original value of "2"). This is because the two-handed weapon wielded by Angbor will make all 3s into 2s, as we've already discussed above. There is an "IFERROR" clause included, which allows us to get "0" if the number is not 0-6 (which would happen if we boosted our roll by 2 or more - since we'd never get a "2 high" that way). The cells for 2-5 all work this way - find the new value that you have and return that number. Note that for Angbor, the "5" value (Cell B8) is set to 0 because our "Preserve Natural 6s" penalty results in us not being able to get a "5 high."

Note: in the previous post, I had the dueling penalties listed in the Data Validation pane directly, but because of all the moving pieces now, I made a little table for them in cells L18:M21.

When we want to compute the value for a 1-high, we need to include the possibility that we might get a 1 by rolling either a 1, 2, or 3 as our high result, depending on how many dueling penalties we have (a Moria Goblin Prowler within range of an Angmar Shade would be forced to have a -2 penalty to its dueling roll, which would cause any rolls of 1, 2, or 3 to result in a "1 high"). As such, our formula for the "1" value will need to add the values for 1/2/3 based on how high that grey number is:

Similarly, we need a special cell for the "6" value - which is actually the simplest equation we have so far: because we know we can handle all situations where we get a 1-5 high, we know that whatever is left over will result in us getting a 6. As such, our formula for the "6" value is =1-SUM(<other_five_cells>). This means we can avoid doing all kinds of complicated formulas (and technically, the greyed-out cell for "6" is irrelevant and not being used - I've chosen to keep it in for clarity). Whatever you've done for the vertical computation now needs to be done for the horizontal computation. 

Okay, so let's say that we have all of this made up - how can we use it to compute our probabilities of success in combat? Glad you asked - let's dive into the Elrond/Cave Drake fight, mentioned by Sharbie.

Note: because of the sheer number of dice that are about to be rolled, I moved the newly-created roll modifier table in L18:M21 below Row 25 so I could extend the probability table to the right - if you're following along, I recommend you do the same, cutting and pasting the last column of the probability table to some place further to the right and then copying/pasting the computation cells and adding/changing the header row values.

Part 7: Elrond vs. a Cave Drake

Sharbie's situation is an interesting one on a number of levels - both Elrond and the Cave Drake appear to be underrated slayer models and both are capable of dealing LOADS of damage. So how well would Elrond do battling a Cave Drake? Well, we just built a tool that can handle that math, so let's take a look at our contestants:

Contestant #1: Elrond, Master of Rivendell with heavy armor and horse (190pts)

• Fight 6 with an Elven-made hand-and-a-half sword
• 3 Attacks + 1 Attack when charging while mounted + 1 reroll from Lord of the West
• 3 Might points with Heroic Strike/Defense

Contestant #2: Cave Drake (150pts)

• Fight 6 with mundane claws and teeth
• 4 Attacks + Monstrous Charge
• 1 Might point with Heroic Strength

Situation #1: Mounted Elrond charges the Cave Drake

This is a pretty likely scenario on a number of levels - first and foremost, Elrond has 10" movement on his horse, while the Cave Drake only has 8" movement (and a much larger base). As such, Elrond can play the avoidance game until he's ready to fight the Cave Drake where/when he wants to, simply because he's faster. 

Secondly, Elrond has Foresight points that can allows his player to get priority more often than the Moria player - and with a single Might point, the Drake would need another hero nearby to call a Heroic Move in order for the Drake to even have a chance of charging Elrond (which, again, the Drake would have to be in charge range for that to work).

So, let's assume for this scenario that Elrond gets the charge - here's how the match-up stands:

• Elrond will be rolling (effectively) 5 dice to win the duel at F6 with an Elven-made weapon, while
• The Cave Drake will be rolling 4 dice to win the duel at F6 without an Elven-made weapon.

Here's how that plays out:

As you can see, without Striking, Elrond is supposed to win about 3-out-of-5 fights against the Cave Drake - not bad. The issue Sharbie raised in his article is that Elrond might win a lot of the duels, but he's not going to wound very well - here's the break-down:

• Elrond's S4 will wound the Cave Drake's D7 on 6s - with 4 Attacks and a reroll (effectively 5 dice), Elrond is expected to deal ~0.833 wounds each round. Elrond is expected to kill the Cave Drake (6 Wounds + 1 Fate) after he successfully wins ~8 rounds.
• The Cave Drake's S7 will wound Elrond's D7 on 4s - with 4 Attacks and no charge, he's more likely to Rend on 3s, which will deal ~2.667 wounds to Elrond each round. The Cave Drake is expected to kill Elrond (3 Wounds + 3 rerollable Fate) after he wins ~2 rounds.

If the Cave Drake is expected to win 2-out-of-5 fights, he'll probably kill Elrond before Elrond can kill him - and that wouldn't be good for the Rivendell player. So what happens if Elrond is willing to boost his dueling roll with 1 Might point? Will this increase his likelihood of winning the duel enough to tip the scales? Let's find out:

Just by being willing to spend 1 Might to boost his roll, Elrond increases his likelihood of winning the duel to 0.762, so he should win 3-out-of-4 fights. Our wounding probabilities haven't increased at all, but Elrond is expected to win 6/8 fights and will be more likely to kill the Cave Drake before the Cave Drake kills him. Still, Elrond isn't likely to win 8 duels before losing 2, so Elrond will still probably die before he slays the Cave Drake. 

So here's the question: what happens if we have Elrond go two-handed against the Cave Drake (boosting our wounding potential) AND being willing to boost our die roll with Might?

We'll actually be back where we started with a 62/38 split favoring Elrond, but we'll increase our wounding abilities:

• Elrond's S4 with a two-handed weapon will wound the Cave Drake's D7 on 5s - with 4 Attacks and a reroll (effectively 5 dice), Elrond is expected to deal ~1.667 wounds each round. Elrond is expected to kill the Cave Drake (6 Wounds + 1 Fate) after he successfully wins ~4 rounds.
• The Cave Drake's S7 will wound Elrond's D7 on 4s - with 4 Attacks and no charge, he's more likely to Rend on 3s, which will deal ~2.667 wounds to Elrond each round. The Cave Drake is expected to kill Elrond (3 Wounds + 3 rerollable Fate) after he wins ~2 rounds.

This brings Elrond the closest to killing the Cave Drake - and he could do it if he casts Renew on himself once or twice. Still, it's a tough match-up. Having paid Elrond all of the love and attention, let's see what happens if we have the Cave Drake do the charging . . .

Situation #2: The Cave Drake charges Elrond

Our second scenario involves the Cave Drake charging Elrond. We already talked about how Elrond is more likely to be charging, but if the Cave Drake pivots on his base and the Rivendell player underestimates how big that base is, you could do it with a Cave Drake (or if Elrond has already committed to a combat and you're charging into him). However it happens, let's look at the match-up:

• Elrond will be rolling (effectively) 4 dice to win the duel at F6 with an Elven-made weapon, while
• The Cave Drake will be rolling 5 dice to win the duel at F6 without an Elven-made weapon.

While the Cave Drake has more dice than Elrond, that Elven-made weapon comes in clutch for Elrond because both models have a high chance of getting a 6 - here's the break-down:

So interesting thing: without Elrond boosting his roll, even without the charge, he's got a roughly 50/50 chance of winning the duel. IF he wins, he's only slightly less likely to wound than he was on the charge (since he's only down 1 die). The Cave Drake, however, gets quite a bit better when he's on the charge:

• Elrond's S4 will wound the Cave Drake's D7 on 6s - with 3 Attacks and a reroll (effectively 4 dice), Elrond is expected to deal ~0.667 wounds each round. Elrond is expected to kill the Cave Drake (6 Wounds + 1 Fate) after he successfully wins ~10 rounds.
• The Cave Drake's S7 will wound Elrond's D7 on 4s - with 5 Attacks thanks to Monstrous Charge and the knockdown, he will deal ~5 wounds to Elrond each round. The Cave Drake is expected to kill Elrond (3 Wounds + 3 rerollable Fate) after he wins ~1 round.

One round, huh? That's . . . pretty scary. Basically, you're giving Elrond a gamble on the 50/50 dueling roll. Would Striking help much in this situation (so you outright win tied fights)? Not really:

Winning 2-out-of-3 fights is good, but we're not going to beat him 10 times before he kills us, now are we? We could boost our dueling roll to make it higher, but when it comes down to it, Elrond would be better off spending his Might on Heroic Defense instead of calling a Strike and "tank it":

• Elrond's S4 will wound the Cave Drake's D7 on 6s - with 3 Attacks and a reroll (effectively 4 dice), Elrond is expected to deal ~0.667 wounds each round. Elrond is expected to kill the Cave Drake (6 Wounds + 1 Fate) after he successfully wins ~10 rounds.
• The Cave Drake's S7 will wound Elrond's D7 on natural 6s with Heroic Defense - with 5 Attacks thanks to Monstrous Charge and the knockdown, the Cave Drake will deal ~1.667 wounds to Elrond each round. The Cave Drake is expected to kill Elrond (3 Wounds + 3 rerollable Fate) after he wins ~3 rounds.

Still not great, but if we call Heroic Defense whenever the Cave Drake charges us, we can make it less likely that we die when he charges us than if we charge him. Okay, one last scenario . . .

Situation #3: Mounted Elrond + Help charges the Cave Drake

My comment to Sharbie (and in my previous post) was that Elrond probably has help - how often do you let one of your most important pieces fight alone? Let's assume that Moria is surrounding our forces and Elrond gets the following help while using his the rest of his army to shield the Goblins away from the fight:

• Elrond is on the charge on his horse, wielding his sword two-handed, but willing to boost with Might;
• A Rivendell Knight is also on the charge, contributing to a trap;
• A High Elf Warrior is engaged in the fight (fighting normally, not going two-handed), backed up by a High Elf Warrior with shield, spear, and banner.

This comes out to the Rivendell side getting effectively 5 dice from Elrond, 2 dice from the Knight, and 3 dice from the High Elf Warriors (one of which gets a -1 penalty). With 10 dice to win the fight, let's see what the probabilities are (note: this is why I had to extend the probability table a bit to make this work in my tool):

Well look at that - we're at a 75/25 split. Why aren't Elrond and his warriors crushing the Cave Drake in this fight? Because ties on 6s (which will happen a lot) are a roll-off still. What HAS changed is how easy it is for Elrond and his guys to translate winning fights into dealing wounds:

• The High Elf Warriors' S3 will wound the Cave Drake's D7 on 6s - with 4 Attacks, they are expected to deal ~0.667 wounds each round.
• The Rivendell Knight's S3 with a lance will wound the Cave Drake's D7 on 5s - with 4 Attacks, he is expected to deal ~1.333 wounds each round.
• Elrond's S4 with a two-handed sword will wound the Cave Drake's D7 on 5s - with 3 Attacks and a reroll (effectively 4 dice), Elrond is expected to deal ~3.333 wounds each round. 
• Between all of your models, you're expected to kill the Cave Drake (6 Wounds + 1 Fate) after Elrond and his men successfully win ~1 round.

Interestingly enough, Elrond is packing enough damage here that he could kill the Drake in two turns, so if you can just cage the Drake in, you could trust to him taking the Drake alone. So, lesson learned: if Elrond is ever up against a Cave Drake, make sure you pile in a few helpers (or at least trap the beast). :) 

One final note: our tool doesn't handle multiple models well if there is a mix of dueling penalties (our third scenario above didn't account for the banner penalty - we actually assumed EVERYONE was going two-handed and could up their roll with Might). If you wanted to accommodate this correctly, you would want an input for number of dice that have no penalty, number of dice with a -1 penalty, etc. - each with the Might for boosting rolls option (and even then, you'd probably want four columns - one for heroes that fight without penalties, one for heroes that fight with penalties, one for warriors that fight without penalties, one for warriors that fight with penalties). No matter how you do it, you'd need to compute the total number of cases where each group gets a 6-high, 5-high, 4-high, etc. and then find their overall expected likelihood of getting that roll (it gets complicated). Our current method is simplistic, but functional (and I think it can be explained).

The next time we revisit this topic, we'll talk about the wounding game - I've done the hand-waving, simple calculations here, but there are actually several ways you can compute your probability of wounding, depending on the question you want to answer (such as "how many wounds am I expected to deal?" or "How likely am I to deal X Wounds?"). We'll tackle all of these in the posts on wounding - including some interesting analyses about Elrond himself against various foes of varying sizes. Until then, happy hobbying!