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cheese_cake
02.17.2020 , 01:43 AM | #1
Preface:
The findings here are purely theoretical. They are not an accurate representation of practice. The purpose of this post is to provide insight and encourage practical testing. The calculations in this post may be in error. Everyone is invited correct errors, refine the calculations, and further explore the presented question.

NOTICE: THIS POST IS MATH HEAVY. THIS IS NOT A HOW TO JOUST GUIDE

Introduction:
This post explores two popular builds and compares their performance when they joust. The two builds that will be compared are the Ion Cannons and Heavy versus the Quads and Heavy Rycer/Starguards. It is generally understood that the winner of the joust between the two builds is determined by luck -all other things equal. This post seeks to quantify the probability (in other words luck) of each build winning the joust -i.e. the question posed by this post is: Which build does luck favour in a joust?

It is stressed that the findings in this post are purely theoretical. Nothing has been tested. In practice there are many more factors that go into determining who might win a joust. All this post merely seeks to do is to provide theoretical insight that would encourage practical experiments and discussion.

TL;DR:
Ions+Hvy has a higher theoretical probability to win the joust compared to Quads+Hvy at all ranges. At extreme close ranges, the difference is marginal; likely in favour of quads + hvy.

Assumptions:
Performance is determined by the probability the shots from each build required to eliminate the other hits. This is derived from the accuracy of the respective lasers. It is assumed both have full shields and hulls and every factor aside from the choice of lasers & resulting rotations are equal.

Both builds will assume everything is fully upgraded, Quick Charge shields with Large reactor, and the crew with the +evasion and +extra shield pool passives. It is assumed both ships use Damage Capacitor.

It is assumed that Ions are upgraded with engine and weapon power drain. The effects of Ion’s drain tracking penalty, the possibility that Heavies would crit, and shield piercing were not considered. Shield piercing is assumed to be accounted for in the hull damage calculations following full shield removal. Other assumptions will be introduced throughout the explanation.

Method & Findings:
The post explores the probability of each build to eliminate the other at the max range of ions, and the minimum range of Quads and Heavies.

Establishing the input values:

Both ships:
1525 hull points
12.5% evasion
Shield pool of 2338 in F4
Accuracy and damage are assumed to scale linearly with distance.
Damage to shields are in F4 while damage to hulls are in F1

Ion Cannons:
@range 5750m
Let the event that Ions hit their target be P(I)
P(I) = 101% - 12.5% = 89.5%
Rounds Per Minute (RPM) = 162
Damage Per Second (DPS) to shields = 1331.
Damage per shot to shields = DPS * 60s / RPM = 1331 * 60 / 162 ≈ 492.96
@range 525m
P(I) = 122 – (122 – 107) / (3450 – 500) * (525 – 500) – 12.5 ≈ 109.37%
DPS = 1879 – (1879 – 1746) / (3450 – 500) * (525 – 500) ≈ 1880.13
Damage per shot to shields = 1890.13 * 60 / 162 ≈ 696.34

Quad Lasers:
@range 5750m
Let the event that Quads will hit their target be P(Q)
P(Q) = 91 + (101 – 91) / (6038 – 3622) * (6038 – 5750) – 12.5 ≈ 79.69%
RPM = 162
DPS (shield) = 792 + (978 - 792) / (6038 – 3622) * (6038 – 5750) ≈ 814.17
DPS (hull) = 938 + (1159 – 938) / (6038 – 3622) * (6038 – 5750) ≈ 964.34
Per shot shield = 814.14 * 60 / 162 ≈ 301.55
Per shot hull = 964.34 * 60 /162 * 1.25 ≈ 446.45
@range 525m
P(Q) = 116 – 12.5 = 103.5%
DPS (shield) = 1053
DPS (hull) = 1247
Per shot shield = 1053 * 60 / 162 = 390
Per shot hull = 1247 * 60 /162 * 1.25 ≈ 577.31

Heavy Lasers:
@range 5750m
Let the event that Heavies will hit their target be P(H)
P(H) = 101 + (106 - 101) / (7849 – 3622) * (7849 – 5750) – 12.5 ≈ 90.98%
RPM = 120
DPS (shield) = 845 + (902 - 845) / (7849 – 3622) * (7849 – 5750) ≈ 873.30
DPS (hull) = 913 + (974 – 913) / (7849 – 3622) * (7849 – 5750) ≈ 943.29
Per shot shield = 873.30 * 60 / 120 ≈ 436.65
Per shot hull = 943.29 * 60 /120 * 1.25 ≈ 589.56
@range 525m
P(H) = 111 – 12.5 = 98.5%
DPS (shield) = 1006
DPS (hull) = 1087
Per shot shield = 1006 * 60 / 120 = 503
Per shot hull = 1087 * 60 /120 * 1.25 = 679.375

Calculating the probability of one build to eliminate the other

@range 5750m

Required number of ion shots: 2338 / 492.96 ≈ 4.74
5 ion shots are required to strip shields with negligible overflow damage to hull.
Let the event that all 5 ion shots hit & remove shields be P(Si).
Each shot is independent from each other.
Ergo,
P(Si) = P(I∩I∩I∩I∩I) = P(I)P(I)P(I)P(I)P(I) = .895^5 ≈ 57.43%

Pile driving stage to strip shields:
4th Quad with an overflow of 178.15 hull damage.

This calculation for this is derived from http://www.swtor.com/community/showthread.php?t=944782
Here's the dcap piledriver, with a rotation that seems to be possible much of the time:
0: Heavy1
0+small: swap, Quad1 (at this point you are waiting with your finger over 1)
0.37 Quad2, swap, begin spamming 1
0.5 Heavy2
0.67ish, the spammed swap happens
0.74, Quad3
.97ish, the spammed swap happens
1.00, Heavy3. This is the end of the normal piledriver burst window. But if we continue following:
1.11 - quad comes off cooldown, but you are stuck on heavies.
1.27, the spammed swap happens, firing Quad4
1.50 - heavy comes off cooldown, but you are stuck on quads
1.57, the spammed swap happens, firing Heavy4

Let the event that all shots up to the 4th Quads stage hits be P(Sq).
Ergo,
P(Sq) = P(H∩Q∩Q∩H∩Q∩H∩Q)
= P(H)P(H)P(H)P(Q)P(Q)P(Q)P(Q)
= .9098^3 * .7969^4
≈ 30.37%

Heavy shots required by the ion build to eliminate hulls:
1525 / 589.56 ≈ 2.59
3 heavy shots are required.
Let the event that all shots from the ion build to eliminate hulls hit be P(Hull i).
P(Hull i) = P(H∩H∩H) = P(H)P(H)P(H) = .9098^3 ≈ 75.31%

Pile driving stage to eliminate hulls:
By the second Quad shot or by the second Heavy shot. I.e. H,Q,Q or H, one of the quads, H
It is assumed that the pile driving rotation reset by the next Heavy shot. The overflow of 178.15 hull damage is included here. While that number is derived from damage per shot to shields, the increase of damage to hulls does not affect the number of shots required by this build to eliminate hulls.
The reason for the option is that both events would happen by the same time.
Let the event that both options occur be P(Hull q).
P(Hull q) = P((H∩Q∩Q)∪((H∩H)∪Q))
P(H∩Q∩Q) = P(H)P(Q)P(Q)
P((H∩H)∪Q) = P(H)P(H) + P(Q) – P(H)P(H)P(Q)
P(Hull q) = P(H)P(Q)P(Q) + P(H)P(H) + P(Q) – P(H)P(H)P(Q) – P(H) P(Q)
= .9098*.7969^2 + .9098^2 + .7969 - .9098^2 * .7969 - .9098^3 * .7969^3
≈ 81.78%

Now that the probability of eliminating hulls has been calculated within its own sample space i.e.
P(Hull∪!Hull) = 1

We need to find the probability of eliminating hulls in terms of the entire sample space -i.e full elimination of the target. The entire sample space includes the probability to strip shields and its complement.

The probability of eliminating hulls is a sequential event contained within the probability to successfully strip shields.
i.e. P(S|(Hull∪!Hull) ) = 1
Ergo after stripping shields, either hulls are eliminated, or they are not.

Let the event that hulls were eliminated within the entire sample space be P(E)
Therefore,
P(E) = P(Hull∩S) = P(Hull|S) * P(S)

Or in other words P(E) is a proportion of P(S).

Finally we get:
The probability that the ions build will eliminate its target:
P(Ei) = P(Hull i)P(Si) = .7531 * .5743 ≈ 43.25%

The probability that the quad+hvy build will eliminate its target:
P(Eq) = P(Hull q)P(Sq) = .8178 * .3037 ≈ 24.84%

Luck appears to favour the ions build at maximum ion range. This is due to the low accuracy of the quads and heavy build where more shots matter to strip its target’s shields -Ions wins more at removing shields than quads + hvy wins at eliminating hull.

Here are the timelines of both events:
Ions:
0s – First ion shot
1.48s – Fifth ion shot + swap to heavies
1.48s – First heavy shot
2.48s – Third heavy shot. Target eliminated.

Quad + hvy
0s – First heavy shot
1.27s – Fourth quad shot and shields stripped
1.57s – Fourth heavy / first heavy in reset rotation
2.0s – Sixth quad or swapped fifth heavy. Target eliminated
Q
H 2.5s – extra shots able to be squeeze in within the same time ions would eliminate quads.

The extra shots quads and heavy are able to dish out within the same timeframe would definitely help in improving its probability of stripping shields. As exploring this area requires finding a plethora of combinations and unions in appropriate proportions for shield & hull damage, I am unsure how to calculate that at the moment. More help would be appreciated for this part. However, because so many shots need to be chained for quads + hvy to strip shields, I am confident that P(Eq) <= P(Ei) regardless.

The best scenario where quads + hvy might beat ions is by ensuring more shots hit. This can be achieved by closing the distance between the targets. In the next part we will explore the other bound where quads and heavies are most accurate. If within the range between quads as least accurate to quads as most accurate quads + hvy beats ions, it is worthwhile exploring a breakpoint.

@range 525m
Fore note: this part gets wacky due to over 100% accuracy. Setting events where the probability exceeds 100% to 1 will be explored after.

Shots required for the ion build to strip shields:
3 ions, then 1 heavy shot (swapping on the third ion). Overflow of 254.02 hull damage.
P(Si) = P(I)P(I)P(I)P(H) = 1.0937^3 * .985 ≈ 128.86%
Rather than making four ion shots, it’s more efficient to have the fourth ion shot be a heavy shot that overflows to hull damage.

Pile driving stage for quads + hvy to strip shields:
Third heavy shot with an overflow of 341 hull damage.
P(Sq) = P(H)P(Q)P(Q)P(H)P(Q)P(H) = .985^3 * 1.035^3 ≈ 110.87%

Shots required by the ion build to eliminate hulls:
2 heavy shots
P(Hull i) = P(H)P(H) = .985^2 ≈ 97.02%

Pile driving stage for quads + hvy to eliminate hulls:
Fourth heavy shot.
P(Hull q) = P(Q)P(H) = 1.035 * .985 ≈ 101.95%

P(Ei) = P(Si)P(Hull i) = 1.2886 * .9702 ≈ 125.02%
P(Eq) = P(Sq)P(Hull q) = 1.1087 * 1.0195 ≈ 113.03%

Theoretically the ions build still beats the quads and heavy build. Now let’s calculate it in a way that makes more practical sense:

P(Ei) = P(I)P(I)P(I)P(H) * P(H)P(H) where P(I) = 1
P(Ei) = P(H)P(H)P(H) * 1

P(Eq) = P(H)P(Q)P(Q)P(H)P(Q)P(H) * P(Q)P(H) where P(Q) = 1
P(Eq) = P(H)P(H)P(H)P(H) * 1

Since P(H) < 1, P(Ei) > P(Eq).

But for explicitness sake:
P(Ei) ≈ 95.57%
P(Eq) ≈ 94.13%

As a practical argument, it is worth noting that Ions has a significantly lower tracking penalty than quads.

Timelines:

Ions:
0s – First ion shot
1.11s – Third ion shot + swap
1.11s – First heavy shot. Shields stripped.
2.11s – Third heavy shot. Target eliminated.

Quads + hvy:
0s – First heavy shot
1s – Third heavy shot. Shields stripped.
1.57s – Fourth heavy shot. Target eliminated.
2.07s – Extra HQQH combo.

A HQQH combo at such high accuracy is definitely worth exploring. Alas I’ll leave it here for now. I may continue the exploration after. If others are interested in exploring this calculation, be my guest! Let’s work on this problem together.