Wednesday, August 23, 2017

The Galaxy's Fate in My Hand: Excursus - "Deep breath and Focus"...

The Galaxy's Fate in My Hand

Excersus: "Deep breath and Focus," in which we predictably do a deep-dive into the Focus condition...

Welcome back to our semi-kinda-sorta-regular in-depth look at IA campaign weapons! Since we're about to launch fully into Tier III melee weapons (all of which have three dice in their attack pools), and we're all about quantifying the opportunity cost and credit-efficiency of various weapon choices, I thought it might be a good time for a brief excursus into the general value of adding a third dice to an attack pool.

The reason comes down to cost. Tier III weapons are always more expensive than Tier II weapons, and almost always much more expensive than Tier II weapons. And since most Tier II weapons have comparable (and in many instances, better) surge abilities than Tier II weapons, the difference in cost is almost always going to be largely based on the presence of that third attack die. So having some benchmark for measuring just how much "better" a three-dice weapon is over a two-dice weapon will provide us some useful data for making abstract comparisons, and also give us a good measuring stick for particular three-dice weapons (if adding a third die generally improves our damage by 1D, and a particular three-dice weapon improves it by 3D, then we may be willing to stomach a higher credit cost than if that particular weapon only improves our damage by 1D). 

At the same time, I'd like this abstract exercise to still have some practical use. If only there was a common way to add a third die to two-dice attack pools that we could measure for its impact on expected damage and surge results... 

[Photo credit: FFG &]

Why Focus?

I've settled on four basic reasons.

The first is ubiquity. Focus is everywhere. Campaign staples (healthy Gaarkhan [Rage], Diala [Battle Meditation], Gideon [For the Cause!], Fenn [Superior Positioning], and Verena [Master Operative]) and niche heroes (Mak [Jeswandi Training], Jyn [Get Cocky], Saska [Adrenaline Injector], Loku [Mon Cala Special Forces], and Onar [Hold Still]) all have abilities to either self-Focus or to Focus other heroes or friendly figures. So does the IP, and of course it's everywhere in competitive skirmish play.

The second deals with combat theory. While we could look at the impact of any of the four attack dice (and at some point, perhaps we will), the Green strikes the best balance between the base components that our attack dice contribute: damage and surges. It has less damage than the Red (but more than Yellow and Blue), less surge than Yellow (but more than Red and Blue), and when we get to ranged weapons, less Accuracy than Blue (but more than Yellow and Red). Swapping in any of those other dice will skew our results more favorably toward that die's strength (hence the beautiful flexibility of dice-swap mods like the Tier II Energized Hilt) . But if we're just trying to get a general sense of exactly how powerful adding a third die is to a two-die attack pool, the Green gives us the most balanced general impression. 

[Photo credit: thesmallman and AdrianT on boardgamegeek]

The third is utilitarian. If we add a Green die to our ten potential two-dice attack pools, we end up with a pretty good list of all the dice pools currently available in the game:

  • Red-Red: Wampa, Gammorean, T-1 Hand Cannon [C], T-3 DXR-6 [C]
  • Red-Red-Green: None
  • Red-Yellow: Royal Guard, Saboteur, 0-0-0, Gaarkhan [S], T-1 Gaffi Stick [C]
  • Red-Yellow-Green: Royal Guard Champion, Obi-Wan, Grand Inquisitor
  • Red-Green: Nexu, Tusken Raider, Wookie Warrior, Echo Base Trooper ("Front Line"), Bossk, Jabba, C1-10P, Diala [S], T-2 BD-1 Vibro-Ax [C], T-2 Double Vibrosword [C], T-3 Disruptor Pistol [C]
  • Red-Green-Green: Rancor
  • Red-Blue: Heavy Stormtrooper, Bantha Rider, Loku [S], T-2 434 Deathammer [C], T-2 Stun Baton [C], Reward Shu-Yen's Lightsaber [C]
  • Red-Green-Blue: Fenn [S], Onar [S], T-3 Electrostaff [C]
  • Yellow-Yellow: Jawa Scavenger
  • Yellow-Yellow-Green: Blaise, 3xp Shrouded Lightsaber ("Falling Leaf") [C]
  • Yellow-Green: Hired Guns, Alliance Smuggler, Dengar, Lando, Saska Teft [S], Davith [S], T-1 Armored Gauntlets [C], T-1 DH-17 [C], T-1 DDC Defender [C], T-2 Vibro Knucklers [C], 3xp Shrouded Lightsaber [C]
  • Yellow-Green-Green: Shyla [S], T-2 T-21 [C]
  • Yellow-Blue: Imperial Officer, Rebel Trooper, Sorin, Ugnaught, MHD-19 [S], Murne [S], T-1 DL-44 [C]
  • Yellow-Green-Blue: Luke Skywalker, Boba Fett, Terro, Dewback Rider, Hera, T-3 Pulse Cannon [C]
  • Green-Green: Weequay, Greedo, Jyn [S], T-1 VibroBlade [C], T-1 Vibroknife [C], T-2 EE-3 Carbine [C], T-2 DT-12 Heavy Blaster Pistol [C]
  • Green-Green-Green: None
  • Green-Blue: Stormtrooper, Trandoshan, Snowtrooper, Echo Base Trooper (Basic), Wing Guard, ISB-Infiltrator, Jet Trooper, Gideon [S], Verena [S], Vinto [S], T-1 E-11 [C], T-1 Vibrosword [C], T-2 A-280 [C]
  • Green-Green-Blue: Han Solo
  • Blue-Blue: Alliance Ranger, Mak [S], T-1 Tatooine Hunting Rifle [C]
  • Green-Blue-Blue: Kayn Somos, Elite Jet Trooper ("Fly-By"), T-3 DLT-19 [C]
  • Wildcard: Every IG-88 variant, every T-3 Ancient Lightsaber variant [C]

There are some dice pools missing of course, although most of them are for niche units and some high-end campaign items:

  • Red-Red-Yellow: Darth Vader
  • Red-Red-Blue: AT-ST: 
  • Red-Yellow-Yellow: T-3 Force Pike [C], 
  • Red-Yellow-Blue: E-Web, Chewbacca, SC2-Repulsor Tank, BT-1, Biv Bodhrik [S], 
  • Red-Blue-Blue: T-3 Valken-38 Carbine [C]
  • Single Yellow: R2-D2
  • Yellow-Yellow-Blue: Probe Droid, Leia, T-3 Sporting Blaster [C], 
  • Yellow-Blue-Blue: HK Assassin Droid
  • Wildcard: Some Wiess variants, T-3 Ryyk Blade variants [C], T-3 Modified Energy Cannon variants [C]

So by performing this exercise, we not only get a general sense of how a Focus impacts two-dice attack pools, but we also get some advanced intel on where almost every dice pool in the game falls on the power curve.

The fourth is click-bait value. No matter what form of IA you play (skirmish, campaign, both) don't you love Focus? Isn't it poweful? Have you ever wondered just how powerful? Are you still reading this?

My point exactly. :-D

Finding the "Average" Value of Focus

We'll start with the simplest metric for measuring the impact of a Focus die: how it impacts our average expected damage and surges. This calculation is pretty straight-forward, and not at all interesting: we look at the Green die, calculate its total damage (8 damage) and surges (3 surges), and divide both by the total number of sides (6 sides), for a grand total of 1.33 damage (8/6) and 0.5 surge (3/6).

In other words, in a vacuum, adding one Green die to our attack pool should increase our rolled average damage results by 1.33, and our rolled average surges by 0.5. The "rolled" distinction is important of course when evaluating that half-surge. Depending on the surge abilities we have on our weapon/deployment card, the value of this rolled surge may vary immensely, and thus may impact our total damage immensely. So for my own sanity, we won't be tackling those variables here. :-P

Of course, we know from personal experience that "in a vacuum" isn't particularly useful once we start rolling defense dice. We'd be much more interested in how a Focus die affects our average rolled damage and surges against those defense dice. Thanks to the new shiny Monte Carlo simulator my brother sent me this week, we can now adjust for this. :-D Here's what our basic two-dice attack pools roll (damage and surge) when we throw in defense dice:

Some interesting things here.
  • First off, the Red-Red actually ends up with negative surges on average. Against a Black die, it will roll a single surge past the defense approximately 28% of the time, and two surges roughly 3% of the time, but the Black die still manages to roll more Evades than Surges roughly 10% of the time. The White die reduces the odds of keeping 1 surge to around 16% (and 2 surges down to just over 1%), and rolls more Evades than surges a whopping 33% of the time. Hence the net-loss overall when we average them together. This isn't necessarily a bad thing, since a "negative" surge result means that the defender has "wasted" at least 1 Evade (and wasted defense results are always good for attackers), but it does mean that spending credits on a mod that adds a surge ability to a Red-Red weapon probably isn't the best use of our credits (there are exceptions--the Tier II Plasma Cell is an obvious one, though it has nothing to do with the fact that it adds a surge ability :-P ).
  • Our top surge rate is Yellow-Yellow, which isn't surprising. What is surprising is that it tops out at just 1.26 surges on average, which on its own isn't great, and when paired with its abysmally low average damage (0.57) is downright awful.
  • Red-Red nets our best average damage (again, not surprising), and every Red-die variant has higher average damage output than every other two-die variant (a Red-Yellow rolls an average of 1.92 damage, substantially better than its nearest challenger, the Green-Green at just 1.62 average damage). Red dice are that consistent when it comes to dealing damage.
So what happens when we add a Focus die to these pools? I've charted the Average Damage and Surge with Focus below, and put the value of the increase in a callout above, just so we can get a sense of what the Focus die is contributing across the board. The results were simultaneously surprising and unsurprising:

First, the unsurprising: if we were to scan across our table, we'd find that adding a Green Focus die doesn't change where any of these dice pools fall in the average damage or average surge hierarchy: Red-Red-Green is still the best damage and the worst surge, Yellow-Yellow-Green is still the worst damage and best surge, all the Red pools still have better damage than all the non-red pools, and so on and so forth.  In fact, our damage improvement across the board is pretty uniform: somewhere between +1.28 (Red-Yellow, Green-Green) and +1.38 (Red-Blue). And our surge improvement falls somewhere between 0.48 (Yellow-Green) and 0.52 (Green-Green, Blue-Blue). And if we were to average this all out, we'd come up with an average increase of +1.32 damage and 0.51 Surge.

Or almost exactly what our simple "Average" calculation said we'd find (+1.33 damage, +0.5 surge)...

So much for all my advanced calculations. :-P

So there's actually a very simple explanation for this (that just took me four hours of calculations to spot): a single defense die only impacts our attack pool once, no matter how many dice we add to it. If a Black defense die rolls the dreaded 3 Blocks, it takes damage out of our base attack pool. Same with an Evade on a White Die. And as we saw from our lead-in table, that defense die can do a number on our attack stats depending on the dice pool we've brought. But that also means that when we add a Focus die on top of an existing dice pool, we get to layer its extra rolled damage and surges on top of what we'd normally expect, without having to account for defense dice at all (even the White Dodge), because our initial attack pool has already accounted for that defense die.

Which means we can apply this principle not just to a Focus die, but any additional attack die, and get some pretty quick data on how adding any attack die will impact what we roll on average:

So adding a single Red die (Gaarkhan's Rage + Ferocity, or Onar's Don't Make Me Hurt You, for example) improves the average rolled damage of an equipped weapon by more than 2 damage... and improves our rolled surges by about 1/6th of a surge. I've also added accuracy to this chart, though we (usually) don't care about that for melee weapons. But it could be very important for ranged weapons, depending on the weapon our hero is using.

"Consistency" isn't always "Uniformity"

Having said all that, there are some limitations to looking at averages. By its very nature, an "average" outcome falls between some extremes, and unless we know something about those extremes, knowing a weapon's average damage output may not help us assess its predictability.

Depending on the die we add to our existing attack pool, our weapon's damage floor or ceiling may or may not be impacted. To illustrate this point, we'll take the simple Gaffi Stick:

[Photo credit: FFG &]

If you've followed our series on melee campaign weapons, you know that we're pretty familiar with this weapon by now. In its basic form, it's got a great Red-Pierce 1 combo, but no surge abilities that add or impact damage. This gives it a fantastic damage ceiling for a 200 credit weapon if it rolls its ideal 3D on the Red and a Damage/double-surge on the Yellow (4D, Pierce 1, Weaken, Recover 1), and an almost-as-abysmal damage floor if it rolles its worst possible single-damage Red and single-surge Yellow (1D, Pierce 1, Weaken or Recover 1). So what happens when we add a third die of our choice? Well...

So clearly, the damage ceilings go way up. By 2 damage in most cases (3 if we add another Red die). All is as it should be. But what about our damage floor? Well, only the Red die actually increases our worst-possible damage output (from 1D to 2D). For a Yellow, Green, or Blue die, there's no appreciable impact on our damage floor whatsoever (beyond the fact that we could now both Weaken and Recover... big whoop...). In fact, for this particular weapon, we'd have a more measurable impact on our damage floor if we added either a +1 damage bonus (from something like the Tier III Shock Emitter) or an ability to surge for extra damage (like the +2 damage from the Tier II High-Impact Guard) than if we took any die but the Red one. And, of course, adding the Tier III Vibrogenerator improves our damage floor by +2D.

The general theory of our campaign weapon series has been that we get more consistent damage results if we can compress the space between our damage floor and damage ceiling. If my best possible roll is 7D, and my worst possible roll is 2D, and I'm attacking a figure with 4 health left and a Black defense die, how confident am I that I will actually defeat that figure? Probably less confident than if my best possible roll was 7D and my worst possible roll was 4D, right? And if that's what we're after, then spending credits/xp/etc. to add "free" damage bonuses to our weapons, or surge abilities that make our weapon more surge-efficient, seem to be a more predictable path to higher damage than just adding a third die.

Having said that, it's important not to fall into the same trap that we tried to avoid when we were looking just at the average impact of adding a third die. We were a bit leery of our "average rolled" statistics because it's looking at just the mean of a data set (and doesn't account for its fringes). Damage floors and damage ceilings do the opposite: it looks at the outer fringes of what a weapon can do (best possible roll, worst possible roll), without necessarily accounting for the probability of those "best" and "worst" rolls. Yes, there's a chance that our extra Blue die will only roll a single surge, but it isn't high. And that Green die will rarely roll that single-surge. (It is, unfortunately, far more common on the Yellow die :-P ). So if we want to know what sort of likely impact adding a third die will have on our damage output, we should probably look at "at least" damage totals.

*Cue the huge colored graph*

I'll reiterate that this is still purely measuring rolled damage at this point (the surge chart is below, and we're not counting for damage from cashed in surge abilities), and how much of that rolled damage gets past adjusted defense dice (the average of damage rolled past 1 black die and damage rolled past 1 white die). Above each bar is a box that shows by how many percentage points the original two-die roll was improved by adding the Green Focus die. Boxes of the same color represent the same series (Black is 1D+, dark gray is 2D+, medium gray is 3D+, and so on).

So what, if anything, do we learn? Or at least notice (if "learn" is too strong)? :-P Here are some preliminary thoughts that suggested themselves to my admittedly not-mathy brain.

First off, every Focus pool outperforms its non-Focus pool at every damage band. This should be a no-brainer, but now we've confirmed it with figures, so hooray us! :-D

Second, not every Focus pool outerpforms every non-Focus pool. The top two-dice attack pool (Red-Red) has an 89.9% chance of dealing at least 1D past defense dice. By contrast, only three of our ten Focus pools cross that threshhold (Red-Red-Green, Red-Green-Green, and Red-Blue-Green). All the other three-dice pools have lower damage odds than what we have from the Red-Red. And that's not just true of our odds at 1D+ (where there's probably a diminishing returns issue for at least some of these dice pools). It holds true at every damage band for the Red-Red die, from 1D+ through 6D+ (except for the Red-Yellow-Green, which does pull ahead in the 5D+ to 6D+ bands, and has an outside chance at 7D which a simple Red-Red pool can't reach). This surprised me, at least.

Now, again, let's not overblow this. We already know that this data isn't accounting for rolled surges (and we expect other dice pools to benefit when we account for that), and it's not adding damage for surge abilities. So this data doesn't mean that every weapon with a non-red three-dice attack pool will be worse than a weapon pool with a Red-Red or Red-Green or Red-Blue attack pool. But it does suggest that we can't just rely on adding a third Green die to any old weapon and expect it to be automatically better than any two-dice weapon. The make-up of the weapons--their particular dice pool, surge abilities, and our mod and xp ability choices--are still extremely important decisions that we'll have to make.

Third, while the "average" damage bonus from a Focus die is ultimately pretty static across the board (right around +1.33 damage), this doesn't mean that our damage at every damage level goes up by +1.33--or even uniformly.
  • The Yellow-Yellow, Yellow-Green, Yellow-Blue, and Blue-Blue dice pools all receive a pretty big bump right at the start: between 17 and 28 percentage points in their odds at dealing 1D+. The Green-Green and Green-Blue are more tepid, seeing a bump of between 12 to 14 percentage points, and the Red dice pools (even those with Yellow and Blue) see barely any increase (a max of +8.2% to Red-Yellow).
  • Sticking with those Yellow- and Blue- pools, they use up most of their steam getting over the 3D+ hump. The Yellow-Yellow is almost completely shot after crossing that threshold, contributing only another 15 percentage points total before it fizzles out. The others have one last push bast 4D+, in the 17-22 percentage point range, before they trail off.
  • The Green-Green and Green-Blue still have a double-digit push left in them when they hit 5D+, before they final peter out.
  • The Red- pools, meanwhile, are still going strong at 5D+ (although the Red-Yellow is about to hit the fan), with gains of between 17 and 26 percentage points. Red-Green, and (surprisingly) Red-Blue have 9-11 percentage points left at 6D+, before they finally give up the ghost. Red-Red ends 6D+ with an impressive 25 percentage point boost, before it inevitably succumbs (at a whopping 8D+).
Graphically, we could represent these various phenomena with something like this:

Represented like this, Yellow clearly hits its peak at 2D+, and it's all downhill (actually, steeply downhill) from there. Yellow-Green and Yellow-Blue level off from 2-3D+, then begin their descent. Blue-Blue and Green-Green are still ascending from 2-3D+, before they begin their decline. Red-Yellow levels off from 3-4D+, and then drops, while Red-Green and Red-Blue ascend steadily until they hit 4D+. Red-Red doesn't stop ascending until it hits 5D+.

Cool-looking chart aside, what does all this mean in gaming terms? My initial reaction is that we may have stumbled on something akin to diminishing returns. Adding a Focus die always (or nearly always) improves our probable damage output, but at some point along the way, it reaches its maximum impact (improving our expected output by roughly 30-35 percentage points). From then on, while it still contributes to higher damage probabilities, its impact grows noticeably weaker, until it fizzles out entirely. This point varies widely, as we've seen. A Green Focus die dramatically helps a YellowYellow attack pool roll 1 or more damage past defense dice (an improvement from 47% to 75%), and pushes its punchers chance of dealing 2 or more damage (~22%) to a more respectable probability (~56%). But if we're adding that Green die because we want our Yellow-Yellow dice pool to roll 4 or more damage, we're wasting it (as it improves our probability from 0.9% to a much-improved-but-still-laughably-bad 12.8%). On the contrary, adding that same Green die to a Red-Red dice pool barely affects our ability to deal 1D+ (from 89.9% to 90.9%), but offers us excellent odds of 4 rolled damage (41.2% to 70.3%), and converts our odds of rolling 5 or more damage from roughly 1-in-6 (16.1%) to almost 1-in-2 (48.3%).

But wait, there's more! If we take the total percentage points we gain from adding a Focus die and add them together, we get an interesting measure of how much that Green die is improving our base attack pool's overall damage probabilities:

This is an admittedly-crude metric, and it's quite possible (given margins of error, the limited sample size of 2000 dice rolls per pool, and of course all those surge abilities we aren't accounting for) that the final order would shift around somewhat if we ran the simulation multiple times. What I believe is unlikely to change is the incredible disparity between the Yellow-Blue and Yellow-Yellow attack pools and all the others (especially the Red-Blue, Red-Green, and Red-Red). It really appears as if a Yellow-Yellow dice pool or Yellow-Blue dice pool aren't equipped to take advantage of the extra damage from a Green die, whereas the Red-Blue, Red-Green, and Red-Red press it to their full advantage.

Not to be outdone, let's do the same thing now for surges. Here's how adding a Green Focus die impacts our "at least" probabilities of rolling X number of surges:

So unlike the percentile increases we saw on the damage chart, these aren't exactly blowing us away. The single-largest percentile gain is +28% (the Red-Red's odds of rolling 1 or more surge), and there are only three other gains of twenty percentile points or more across the entire graph (Red-Green 1+ surge, Blue-Blue 1+ surge, and interestingly, Green-Blue 2+ surge). And there are ton of gains that only rise to single-digits.

Part of this is probably tied to the volume of what we're adding. A Green-Green die adds 1.33 damage on average, but only half a surge. So it makes sense that we'd seen more consistency in getting higher damage results than we would in higher surge results: you can't spend half a surge, after all. And when you pair that fact with the fact that the White die in particular can remove surges with tremendous frequency, the potential impact is reduced still further. After all, our best two-dice pool (Yellow-Yellow), for all its surge potential, still has less than an 80% chance of rolling a single surge past defense dice (78.1%). Adding a Green die gives us what ought to be a very surge-friendly pool (Yellow-Yellow-Green), and indeed, it does have a very solid 87.2% chance of rolling 1 or more surges through. But it's odds at 2 or more surges are still a paltry 60%--much better than anything else on the chart, but still much lower than we'd hoped for. Since we already know this pool's rolled damage is abysmal, we need surges to push some damage through. And odds are we'll have just one, and maybe two, to work with.

And just like the damage chart, we find here that a poor surge pool remains a poor surge pool even after we add a Green Focus die. That Red-Red pool improves its odds of rolling a single surge past defense dice from roughly 1-in-5 (20.3%) to just under 1-in-2 (48.3%). Which is an improvement, but certainly not one we can count of for reliable damage or keywords. In fact, adding a Green die pushes just two surge probabilities above an 80% confidence threshold: Yellow-Yellow's odds of 1+ surge (78.1% to 87.2%) and Yellow-Green's odds of 1+ surge (69.1% to 81.4%). If we're feeling extra generous and lower our probability threshold down to 70% or better, we can pick up a few more: Red-Yellow's 1+ surge (55.8% to 74.2%), Yellow-Blue's 1+ surge (62.8% to 76.6%), Green-Green's 1+ surge (58.3% to 75.3%), and (again, surprisingly) Green-Blue's 1+ surge (49.7% to a close-enough 69.3%).

Laying out the composite percentile increases confirms that only a handful of attack pools (Yellow-Yellow, Yellow-Green, Yellow-Blue, Green-Green) are still on the ascension when we hit 2+ surges, and all of them are on the way down after that (though Yellow-Yellow's descent is much more gradual).

And if we're looking to get maximum (surge) impact out of our Green Focus die, the board is much more level:

In theory, we could also run these simulations on our other three attack dice (Red, Yellow, Blue, or another 30 combinations total). And it would be interesting to see whether the pools that benefit most from them change, how far those increases to damage spread, and so on. But for the most part, those are just niche calculations (Onar for Red, Davith for Yellow, Diala perhaps for Blue), so we'll leave them for another time. ;-)

We now return you to your regularly scheduled programming...

[Photo credit: FFG &]

Inevitable post-posting edits: Apart from eliminating those awful strikethroughs by moving the post to another website... and fixing disappearing photos... not much. :-P 

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