After "Fair Ship Rebels 2.0" (Biggs / Lowhhrick + Selflessness / Rex / Jess + DtF) started dominating tournaments in Store Championships immediately upon the weekend of its release, I decided to go back to the drawing board and do the pre-requisite math to figure out why. Eyebrow raising tournament results or strong playtesting results are always a good motivator to go back and look at the fundamentals. Unfortunately I didn't fully quantify this before release, but then I'm not getting paid to develop / playtest the game, so in a sense you get what you pay for, and besides that I couldn't have affected the outcome anyway. In general I prefer to predict game balance before launch to see that the theory has merit without confirmation bias, but we'll have to settle for a post-launch analysis in this case. In any event, I wrote up some show notes for the S&V episode #56 that I unfortunately wasn't able to be on. The notes didn't make it into the audio, but here's the pertinent snippet that have been making their rounds and I will lead in with:

Speaking of Biggs, the combination of Biggs' ability, Selflessness / Draw Their Fire, Lowhrick free evade, and Rex debuff has created a monster. Jess is a perfect fit to add as the DtF carrier with very solid jousting value. I wanted to actually be on the podcast to talk this point myself, but here’s the cliff notes anyway. This squad's ability to spread damage around fundamentally breaks the focus fire assumptions that apply to all other ships and squads, which gives it a huge advantage. [Warning: some sort-of-complex math to derive exactly how big that advantage is. We’re skipping that part here!] The individual pieces already have good jousting value, so spreading around incoming fire makes the squad worth around 115 - 120 points of total jousting value. The previous record holder was pre-nerf Parattanni at around 115 points, and we all know what that did to the game. To be fair, Parattanni has some other things going for it: PS9 arc dodger, stress control, and turret / auxiliary arc. But no squad should be worth well in excess of 100 points of jousting value unless it’s all shuttles with terrible dials. This new Biggs squad is so ruthlessly efficient that it's quantifiable power creep, and extraordinarily difficult for most lists to fight.

So now lets rewind and see how I got there. First we need to go back to the fundamentals. One of the things I did for MathWing 3.0 (not public) was to derive the closed-form equation for how much of a boost a squad gets if its damage goes down in quantized "steps" as ships are removed, vs the continuous time and steadily decreasing damage output assumptions that Lanchester's model uses. I haven't done a full literature survey yet to see if this has been previously derived elsewhere, but this is at least the first time that I am posting some of the results publicly. As an aside: I will eventually want to submit the full derivation along with other related findings (and other ongoing stuff I haven't finished yet) as an article to an applied journal of mathematics, although I haven't decided yet where would be best to publish.

Result #1:

Let us assume there are two opposing forces with perfect focus fire from both sides, and one side's damage output (which we shall call "Team X") continuously goes down as it receives damage, but the other side (which we shall call "Team Y") has damage output that goes down in staircase fashion as its combatants are removed in a quantized manner. Also let k_x and k_y refer to the lethality coefficient of each team's individual units. If the size of the two forces are such that both are mathematically perfectly balanced and they simultaneously kill each other, then it follows that:

X_N = ( (k_y / k_x) * (Y_N² + Y_N) )^0.5

where Team Y and Team X have Y_N and X_N units respectively. This leads to a closed-form solution (that I won't go into here) of the "curve fit" that I only obtained an approximation for in MathWing 1.0 and 2.0 via numerical methods. This result by itself however is relevant to the Biggs discussion.

Result #2:

If Team X's damage output is 100% defocused onto Team Y, then Team Y's damage output will remain at its initial maximum output until all Y_N units are simultaneously killed. Under this scenario, for Team X and Team Y to be equally matched, the size of Team X must be:

X_N = ( (k_y / k_x) * (Y_N² + Y_N) * 2Y_N / (Y_N + 1) )^0.5

I.e. if Team X has 100% defocused fire onto Team Y, then for the two sides to remain equally balanced Team X's size must increase by:

( 2Y_N / (Y_N + 1) )^0.5

The limit as Y_N approaches infinity is 2^0.5 = 1.41.

This has multiple implications for X-wing, from both a player's tactical perspective and a game designer's balance perspective. As it related to this discussion, this means that if you have N ships in your squad, if you can force your opponent to completely defocus their fire, you gain an effective points advantage of:

The point multiplier for the theoretical 1-ship case is obviously 1.00 (i.e. no advantage) because the damage output of the one ship already starts at maximum and never decreases until it is completely dead, and focus fire vs defocus fire is meaningless when there is only one ship to shoot at.

A 4-ship list can see a maximum theoretical improvement of 26% if it completely defocuses incoming fire. In practice no squad will reach this upper limit, but the Biggs list can get pretty close. I need to do some more fundamental research and gather empirical data to quantify exactly how much of an improvement the Biggs list sees, but for now I have been estimating the improvement at 15% - 20%.

Now lets look at the specifics of the Biggs squad. The base of the list is 96 points, and the last 4 points in the list can be filled out in a variety of ways. For this example we will use Paul Heaver's list, which fills it out with Wookiee Commandos, Autothrusters, and Primed Thrusters. The backbone of the list is a highly durable Biggs behind Lowhhrick's auto-evade and R4-D6, and splitting incoming fire via Draw Their Fire (once per attack) and Selflessness (once per game, then discard).

• Biggs + R4-D6 + Integrated Astromech (26)
• Lowhhrick + Selflessness + Wookiee Commandos (30)
• Captain Rex (14)
• Jess + R2-D6 + Draw Their Fire + Autothrusters + Primed Thrusters (30)

Now lets look at the jousting value of each of these pieces individually. Note that I will be quoting two jousting value numbers: one is an "absolute" number that assumes simultaneous fire among all contestants (not reasonable), and the other is a PS-rated jousting value, where higher PS gains an advantage for removing ships before it can shoot. The PS-rated jousting values are normalized to PS2, so a PS2 Z-95's absolute jousting value and PS-rated jousting value are, by definition, the same. Deriving the PS-rated jousting value is an in-depth discussion for MathWing 3.0 and/or academic journal article that I won't get into here, but suffice to say it is a function of where the ship sits in the PS pecking order, and how much of a glass cannon it is.

Biggs

We're going to approximate him as having a 3/2/2/4 statline, completely ignoring R4-D6, and we'll give him a free evade every round via Lowhhrick's ability. With the parameters I have dialed in for MathWing 3.0, this works out to 28.7 points of JV in the absolute sense, and about 31 - 32 points at PS5 if we assume a flat distribution of pilot skill across the meta.

Lowhhrick

Lowhhrick's action economy is straightforward to model because he always reinforces, and gets a soft mod on offense via Wookiee Commandos. If his reinforce triggers while he's getting attacked, then he has an absolute JV of 24.0 points, and about 26 - 27 at PS5.

Captain Rex

Rex is a little trickier to value because he only has ATT2, but he's also lowering someone else's attack, likely from ATT3 to ATT2. So, in net, his total contribution is more as if he has an ATT3 attack: ATT2 from his own gun, and then the rest equivalently from debuffing an enemy. It can be reasonably argued that his incremental debuff is actually worth more than as having an offensive boost as modelled here due to Bigg's damage mitigation. For simplicity's sake however we will treat him as having a 3/3/3/0 statline, with the additional bonus that since he's not getting attacked for most the game, he gets to use his focus on offense, so his damage output will be modelled as having a 3/0/x/x statline. (MathWing 3.0 models a full action economy, based on stat lines and abilities.) This gives Rex an absolute JV of around 16.7, and 18-19 at PS4.

Jess Pava

We'll treat Jess here as having a 3/2/3/3 statline, plus autothrusters, and assume that she always gets full rerolls on her attack via her ability, so she's essentially got a free target lock each round. Like Rex, we will assume that she uses her focus on offense, so her damage output is modelled as having a 3+TL/0/x/x statline. Her absolute JV is 27.2 and about 29 at PS3.

Adding it all up

So lets look at it all together. All the pieces summed together provide 96.6 points of raw jousting value, before considering pilot skill. Jousting values are normalized to pilot skill 2 (by my own convention), so assuming a flat distribution of PS in the meta (not entirely accurate, as PS1 and PS2 are largely non-existent) this is worth around 105-106 points of jousting value when PS is considered.

Now we need to multiply that by the benefit from "Defocus Fire" introduced earlier. If we assume the benefit is 15% to 20%, then we we have an absolute Jousting Value of 111 - 116, and around 121 - 126 when a flat PS distribution is assumed.

So, how much jousting value is the squad really worth? It's hard to tell without doing a more precise analysis. Some of the approximations are slightly uncertain:

• I'm assuming the squad sees a 15% to 20% benefit of Defocus Fire, vs the maximum theoretical of 26%. But given empirical results from many games in which all four of the ships end up heavily damaged but not dead, 15% to 20% seems like a reasonable first order approximation.

Some of my approximations are overestimating the jousting value:

• Rex and Jess both get full benefit of "The Biggs Bubble" for offense in my model here, but eventually they will get shot and need to use their focus on defense.
• There's a little bit of double-counting going on in valuing ships' PS-rated jousting value and the Defocus Fire benefit. Specifically: if a squad saw the maximum 26% increase then it will still see some of the PS benefit because it can still remove ships in discrete time before they shoot, but not as much as the approximations here, because the biggs squad ships can't actually get PS killed until they're all dead. More fundamental analytical research required to tie this up and put a bow on it.
• The PS race doesn't favor the biggs squad quite as much as approximated here, because PS1 and PS2 are essentially non existent.

However, some of my approximations are undervaluing the squad:

• I'm putting zero value on R4-D6, which is in reality a tremendous value. If we assume it grants an additional 2 shields, then Biggs becomes 33.5 points of raw JV and 36 at PS5, pushing the entire squad up to about 110 points of jousting value at its PS, and that's before considering the Defocus Fire benefit of an additional 15% - 20%.
• I'm not putting any extra value on Lowhhrick's 180 degree arc.
• I'm putting zero value on Jess's primed Thrusters.

So, considering all of the above, I am very comfortable saying that the (Biggs / Lowhhrick + Selflessness / Rex / Jess + DtF) squad is worth around 115 to 120 points of jousting value, which should be a new record for the game.

And now, if you will excuse me, I will return to working on my House Rules to balance the entire game for a "Community X-wing Mod" that sozin and I are spearheading. Not coincidentally, I have changed Biggs' ability:

Once per round, the first ship that declares an attack against another friendly ship at Range 1 of you and could have targeted you instead, receives a stress token.

Thanks for showing us this! It's really in-depth. I like your idea for a Biggs change- mine was going to be "other friendly ships at range 1 of you count as being obstructed when defending".

It's bugging me that you said 'peak' instead of "peek" in the title though.

Out of curiosity, what's lowhhricks jousting value without the free evade when being attacked (since it's also being assumed that he's using his token to give Biggs an evade every round)

Not sure this can be done with a succinct answer, but why zero value on R4-D6? It really is, as you say, tremendous value to assign 0 value to it (if I follow, and I am trying to as the formulae are really interesting).

Mostly just curious, if you don't want to divulge, I understand. Some nice math here.

On 7/29/2017 at 1:29 PM, MajorJuggler said:

This new Biggs squad is so ruthlessly efficient that it's quantifiable power creep, and extraordinarily difficult for most lists to fight.

You said it. While I am enjoying the challenge of trying to compete with it, man, "extraordinarily difficult" describes it to a 'T'. Yeesh. I'd almost rather face Paratanni.

10 minutes ago, Kaptin Krunch said:

Thanks for showing us this! It's really in-depth. I like your idea for a Biggs change- mine was going to be "other friendly ships at range 1 of you count as being obstructed when defending".

It's bugging me that you said 'peak' instead of "peek" in the title though.

Oops. Fixed. I'm an engineer, we can only do math, not spell!

37 minutes ago, VanderLegion said:

Out of curiosity, what's lowhhricks jousting value without the free evade when being attacked (since it's also being assumed that he's using his token to give Biggs an evade every round)

Same as quoted above, because his action economy for attack doesn't change (he always reinforces), and then you use his actual durability (reinforce applies to an otherwise 0 action x/1/6/3 statline). If Biggs gets attacked first in the round and then Lowhhrick gets attacked, then his reinforce is gone, so his numbers would be lower. [Edit: he drops to abs JV = 19 and PS JV = 21.2] You can file that under 'overestimate' with a couple of other items.

36 minutes ago, Scopes said:

Not sure this can be done with a succinct answer, but why zero value on R4-D6? It really is, as you say, tremendous value to assign 0 value to it (if I follow, and I am trying to as the formulae are really interesting).

Mostly just curious, if you don't want to divulge, I understand. Some nice math here.

It's a first-order approximation, and sets the "floor" for what we can expect the squad to do. In this case the lowball estimate probably more or less offsets some of the other highball estimates. There's some variance in the certainty of the final answer, but I'm comfortable stating its at least 115 - 120.

10 minutes ago, MajorJuggler said:

Oops. Fixed. I'm an engineer, we can only do math, not spell!

Let's all be thankful you didn't try to skirt anything.

3 minutes ago, Verlaine said:

Let's all be thankful you didn't try to skirt anything.

You mean Squirt, am I right?

@MajorJuggler why has FFG not made you one of their lead playtesters yet? Or better still, one of their lead designers? Seriously...

Just now, Herowannabe said:

@MajorJuggler why has FFG not made you one of their lead playtesters yet? Or better still, one of their lead designers? Seriously...

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

What if you took the zero value things, like Primed Thrusters, and instead changed Commando crew to Chewbacca crew? Would the overall value be even higher? Negating an additional damage card *and* recovering another shield seems pretty hilarious in this already annoying damage mitigation build.

EDIT: I suppose to get the other 2 points you'd have to change AT on Jess to IA as well, so might be a wash

3 minutes ago, piznit said:

What if you took the zero value things, like Primed Thrusters, and instead changed Commando crew to Chewbacca crew? Would the overall value be even higher? Negating an additional damage card *and* recovering another shield seems pretty hilarious in this already annoying damage mitigation build.

EDIT: I suppose to get the other 2 points you'd have to change AT on Jess to IA as well, so might be a wash

Lowhhrick really wants to keep his damage output as high as possible. Other variations include Rey crew instead of Commandos. So long as you have the Rey focus token stack, you have an even better attack economy. The problem is that after a few rounds it runs out and you're better with Commandos.

Glad you're finally coming around to looking at squad values, and loss exchange ratios, rather than specific ship values -- though I do think that trying to norm to single comparative values (JV) leads to very rough approximations where better explicit results are obtained by comparing two specific opposing forces. Then you can judge relative LER changes based on tactics (focused fire or not, for example) or squad element adjustments.

We use force coefficients (a more general term for your jousting values) quite a bit in high level conflict modeling, but the problem with them is that so much has to be aggregated in order to normalize forces to values that can be compared, that you lose granularity of detail. So you end up with either a nice model whose numbers look good but can't reliably account for the effects of small changes in unit composition, or models which address small changes well but don't scale or generalize. You end up in a position where you need several scales of conflict model for insight -- and even then it's at best informative, not conclusive -- and in the end well-calibrated PMJ is almost as reliable as the models in terms of qualitative conclusions; the quantitative conclusions have meaning in terms of direction and order of magnitude but differences below a level of magnitude are essentially meaningless.

Will be encouraged to hear what else you learn.

26 minutes ago, Hawkstrike said:

Glad you're finally coming around to looking at squad values, and loss exchange ratios, rather than specific ship values -- though I do think that trying to norm to single comparative values (JV) leads to very rough approximations where better explicit results are obtained by comparing two specific opposing forces. Then you can judge relative LER changes based on tactics (focused fire or not, for example) or squad element adjustments.

We use force coefficients (a more general term for your jousting values) quite a bit in high level conflict modeling, but the problem with them is that so much has to be aggregated in order to normalize forces to values that can be compared, that you lose granularity of detail. So you end up with either a nice model whose numbers look good but can't reliably account for the effects of small changes in unit composition, or models which address small changes well but don't scale or generalize. You end up in a position where you need several scales of conflict model for insight -- and even then it's at best informative, not conclusive -- and in the end well-calibrated PMJ is almost as reliable as the models in terms of qualitative conclusions; the quantitative conclusions have meaning in terms of direction and order of magnitude but differences below a level of magnitude are essentially meaningless.

Will be encouraged to hear what else you learn.

One of the nice things about X-wing is that it is a relatively straightforward combat model compared to real life engagements, which you are far more familiar with than I am! The upshot is that the idealized equations hold up pretty well for X-wing, and reality can't stray very far away from the Square Law's result of X = Y*(k_y / k_x)^0.5, or in this case X_N = ( (k_y / k_x) * (Y_N² + Y_N) )^0.5.

The trouble arises when modeling the action economy, because it is highly nonlinear. I wasn't modeling a real action economy at all in v2.0, so it didn't differentiate action economies between various stat lines, let alone pilot abilities (hello Soontir Fel!). v3.0 is better in that it models it, but still has to use a linearized approximation with a blender-style approach. You're right that more accurate results can certainly be obtained with another model that truly looks at squad A vs squad B in all its nonlinear glory. I don't know if I'll ever get to that point though. (knock on wood). Last I knew, Vorpal Sword switched his PhD dissertation over to analyzing X-wing, presumably using a more direct squad approach as you're describing. Don't know anything about the details.

There's more work that I could do on mixed-unit squads with discrete units vs each other, at which point I need to check the literature again so I don't reinvent the wheel. Obviously it always makes sense to kill the glass cannons first and leave the tanks for last, assuming a simple linear system (non-linear value of regen can flip this on its head).

But, again, even a first-order approximation of the action economy in 3.0 is going to get you pretty close to reality, the Square Law makes it impossible to be radically wrong unless you seriously messed up your assumptions somewhere. With almost zero playtesting and just pure theory it's still sufficient to demonstrate that the list itself is extraordinarily powerful.

The tendency of Square Law and similar approximations in deterministic solutions to wash out details is why with our major modeling tools we've gone almost entirely stochastic. Plus that allows some input to variations that purely attrition-based models can't account for (effects of maneuver, for example). Feeder elements in tools like CASTFOREM and COMBAT XXI have deterministic elements, but sometimes there's just no substitute for brute force and thousands of model runs when you have to aggregate.

(As an aside -- just the engagement priority problem in an interesting one. Using the real world scenarios I work with -- most of our models are "tanks first, infantry later" models. But when you change how the infantry are equipped, or how AFVs protect themselves, some of the assumptions invert and you end up with very interesting, and counter-intuitive results. That assumes you model the defensive systems adequately, which is a challenge we're dealing with now. Attempting to translate some of thse problems to X-wing -- how does a change in apparent capability affect the decision of a player to choose which target to prioritize? The math can lead to a more optimal solution, but can't always account for the game state under which the decision is made -- so how do you use the math to better inform the intuition of the player so he or she can make a better decision under the more uncertain real-world conditions?

Edit: Put another way -- for example, how do you account for the choice of whether or not to shoot Dengar in arc, or when to shoot and risk a return shot from Quickdraw? Done well, the model should be able to suggest the conditions under which shoot is preferrable to don't shoot, but I think the player needs to understand the assumptions beyond just a heuristic that says "if X, do Y".)

Even X-wing is a bit too complex, I think, for deterministic precision -- as I recall just adding indirect fire into the conventional vs. insurgent differential models was a huge breakthrough back in the '60s, and at a tactical level X-wing is far more complex than the very simple assumptions those models were reduced to.

I think one thing you need to do is caution your audience -- these models, no matter how advanced, still hinge so much on underlying limitations and assumptions that while they provide insight, insight isn't an exact solution.

I'm not going to bother with the math as it might as well be the gibbering from monolith from 2001 A Space Odessy to me

Instead, I just wanna know why in the blue **** it's called "fair ship rebels"

3 minutes ago, ficklegreendice said:

Instead, I just wanna know why in the blue **** it's called "fair ship rebels"

Irony would be my guess

1 minute ago, LordBlades said:

Irony would be my guess

That's so lame, though

Also confusing due to all the cries over miranda

Unless miranda, low and Biggs also qualify?

Doing good work. Thank you MJ.

Any speculations on area of effect stuff? Assault missiles vs this? Bombs/bomlet

12 minutes ago, ficklegreendice said:

I'm not going to bother with the math as it might as well be the gibbering from monolith from 2001 A Space Odessy to me

Instead, I just wanna know why in the blue **** it's called "fair ship rebels"

http://meta-wing.com/ship_combos/2508

The 1.0 version of the list is using an ARC, a TIE, a T-65 and a T-70. Sounds like a lot of fair rebel ships. And the 2.0 version is just an update of that archetype bringing it to unfair levels based on the new stuff. Though even the 1.0 version was a lot of fun to fly, I loved thane in that list.

I faced a Miranda Nym bomber list with my variant today.

Biggs with R2-D2 and IA

Rex

Wookie Defender with Jyn and Jan

Lowhhrick with Selflessness and Chopper.

He got Rex early with a bomb, 2 hits + Sabine love this game. I eventually chased down Nym and killed it.

I realized that I may have been better off just running all game and going to final salvo. Maybe not, I'd probably die to too much plink damage from TLT and I can't run away and get proc Jyn.

IDK, just accept the 50~ MoV loss in that matchup and hope you don't face it too often? With your 180 degree arcs, you do have some good anti-Miranda tech, idk.

So I just did a quick test run of the heaver version against my squad of choice at the moment. I played loose and quick, and obviously playing against yourself kinda skews results.

None the less it was a very close game, though I did lose. I killed rex and lowhrick, and had had jess at 2 hull, and biggs at 2* hull (*because of IA).

I might have even won if my 5 dice cruise missile attack at the beginning didn't get 2 hits/crits total after mods....

51 minutes ago, Hawkstrike said:

The tendency of Square Law and similar approximations in deterministic solutions to wash out details is why with our major modeling tools we've gone almost entirely stochastic. Plus that allows some input to variations that purely attrition-based models can't account for (effects of maneuver, for example). Feeder elements in tools like CASTFOREM and COMBAT XXI have deterministic elements, but sometimes there's just no substitute for brute force and thousands of model runs when you have to aggregate.

....

I think one thing you need to do is caution your audience -- these models, no matter how advanced, still hinge so much on underlying limitations and assumptions that while they provide insight, insight isn't an exact solution.

Incidentally my first foray into modelling any sort of combat was to write a script to calculate the exact stochastic solution of a pair of Axis and Allies armies fighting each other, which I quickly found out yields a classic bimodal distribution. So a whole other way of looking at the problem is to determine when one side has a 50% chance of winning vs the other side, looking at the entire distribution of all possibilities. The results should not be too far off from the deterministic method, but it absolutely would be better to look at both.

If you're trying to fit ship cost to a power curve as a game designer, then the curve itself should not be stochastic, but the ship values certainly are. (Soontir rolled 5 blanks through a rock! Ha! sucker!). The game designers obviously can't use a stochastic variable for cost -- they have to choose one value, which hopefully turns out to be in the Goldilocks zone in real-world play.

Certainly every model has its limitations. If time and resources were unlimited, I could direct a team of programmers to go implement a bunch of ideas, heck, we could work together on it for kicks. But its just me in my spare time, so I have to come up with something that's good enough to be useful, but not so good that it's my entire life's work. Arguably it already takes up far too much of my spare time as it is already!

To use a golf analogy, it's like we're discussing how to best land the ball directly on the green from the tee on a Par 3, and what we need to do to go from sticking it +/-30 feet from the hole, to +/- 10 feet from the hole. Then meanwhile FFG takes a swing and not only misses the green and the fairway, but they shank it wide left and end up 50 yards into the woods.

36 minutes ago, ficklegreendice said:

I'm not going to bother with the math as it might as well be the gibbering from monolith from 2001 A Space Odessy to me

Instead, I just wanna know why in the blue **** it's called "fair ship rebels"

I think "Fair" sounds like "Four", and the list is, well, actually not very fair, so it's a bit of an intentionally ironic name.

If there's four ships, perhaps that could be considered a...fair amount