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the math behind fighters

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original post from Deathskyz on the asia forum: http://forum.worldofwarships.asia/index.php?/topic/10326-usn-fighters-vs-ijn-fighters-a-detailed-analysis/

 

Welcome to World of MathShips!

"I will never use this silly Math things" - College Student

 

So this topic keeps popping up a lot. And we have some of the most heated discussions regarding this.

The topic of USN CVs vs IJN CVs. And how some will argue that its the USN CVs that are better, while others advocate that the IJN CVs dominate the sky.

 

So with the help of many people, we spent hours calculating via a Skype chat that effectively killed 80% of our brains.

 

So this is the result. Thanks to Amade, Haku, Footscray, IJN_Yura and many others who helped try to understand

 

This is how AA Mechanics and Fighter vs Fighter Mechanics work.

Firstly, it is NOT, repeat, NOT a Damage over Time.

So that Average DPS you see is NOT really Damage Per Second.

Secondly, ONE squad can only engage ONE squad at a time.

 

AA mechanics work like this. (Values here are all in top configuration, without skills or modules)

Every second, a calculation is performed with this formula:

Chance of 1 Squad to shoot down 1 Plane:

Total DPS / Survivability = (X)*100%/sec

 

So for example in Tier 4s, a Langley Fighter Squadron meets a Hosho Fighter Squadron.

We know that the Langley fighter squad has a average DPS of 31 (per fighter) so the Squad has a DPS of 186 and a survivability of 760 whereas the Hosho fighter squad has an average DPS of 28 (per fighter) so the squad has a DPS of 112 and a survivability of 780.

 

So when they vie in battle this calculation is performed:

 

For the Langley: 186(Langley Fighter's total DPS) / 780(Hosho Fighter's Survivability) = 23.85%/sec.

For the Hosho: 112(Hosho Fighter's total DPS) / 760(Langley Fighter's Survivability) = 14.74%/sec.

 

What does this mean? It means that in that engagement, the Langley's fighters has a 23.85% chance every second to shoot down one of the Hosho's fighters (That % increases the less planes there are in the squad) while the Hosho only has a 14.74% chance every second to shoot down one of the Langley's fighters.

So we can clearly see how in a 1v1 Squad Engagement, the USN CV will more or less win that engagement.

 

Ok so we got 1 Squad vs 1 Squad... But what happens if there is 1 Langley squad vs 2 Hosho Squads?

Well you would think that the two squads will combine their DPS, so simply multiply the chances by two right?

 

Well no... Apparently now, TWO rolls of 14.74% happens every second. This is because the stats of TWO squads are NOT cumulative. So neither their survivability nor their DPS are combined.

To calculate that - this formula is used every second:

Chance of N squad to shoot down 1 Plane:

1 - (1-X)^N

Where X is the % chance of plane shot down every second (1 squad vs 1 squad)

And N is the number of squads.

 

So once again we look at the Langley vs Hosho example, when both meets in battle (assuming at the exact same time)

This calculation is performed: 1 - (1-0.2385)^2 = 27.3%/sec.

 

So when 2 Hosho squads meets 1 Langley squad, the Langley still retains the 23.85%/sec chance to shoot down a Hosho Fighter.

But the 2 Hosho Squads will now have a 27.3%/sec chance to shoot down 1 Langley Fighter.

 

Bear in mind that this is calculations are ALL made with the assumption that both meet at the SAME time and both have FULL squadrons of 6(USN) or 4(IJN). The % values will differ when the planes are shot down during an engagement. This calculation is to show in engagement which squad will shoot down the other's fighter first.

 

With that, here are our calculations:

(No Crew Skills, No Ship Upgrades)

Tier
USN Fighters IJN Fighters
Individual DPS Total DPS Survivability Individual DPS Total DPS Survivability
4 31 186 760 28 112 780
5 35 210 940 35 140 970
6 42 252 1140 41 164 1150
7 53 318 1400 47 188 1360
8 63 378 1670 63 252 1620
9 73 438 1830 80 320 1820
10 80 480 2150 87 348 1970

 

  USN Squad vs IJN Squad IJN Squad vs USN Squad
Tiers 1v1 2v1 3v1 1v1 2v1 3v1 4v1
4 23.85% 42.01%   14.74% 27.30%    
5 21.65% 38.61%   14.89% 27.57%    
6 21.91% 39.02%   14.39% 26.70% 37.25%  
7 23.38% 41.30% 55.02% 13.43% 25.05% 35.12%  
8 23.33% 41.22% 54.94% 15.09% 27.90% 38.78%  
9 24.07% 42.34% 56.22% 17.49% 31.91% 43.82%  
10 24.37% 42.79% 56.73% 16.19% 29.75% 41.12% 50.65%

 

Alright lets understand this data...

Lets look at 12 Fighters vs 12 Fighters as an example, lets say a Lexington(211) fights a Shokaku(311)

​So there are 2 USN Squads vs 3 IJN Squads, each totalling 12 fighters.

This calculation is therefore performed:

USN: 1 - (1-0.2333)^2 = 41.22%/sec

IJN: 1 - (1-01749)^3 = 38.78%/sec

 

So we can prove that in the initial engagement, the USN Squads has a higher chance to shoot down the first plane, as a result, their next calculation after the Shokaku squads loses 1 fighter squad will have a higher %/sec to shoot down the next plane.

 

In summary, during squad engagements

In a 1USN vs 1IJN engagement, the USN will usually get the first enemy plane down. Their %/sec being 61%(Langley) - 50%(Midway) higher than their IJN counterparts

In a 1USN vs 2 IJN engagement, the IJN are likely to get the first enemy plane down. Their %/sec being 22%(Ryujo) - 33%(Taiho) higher than their USN counterparts

In a 2USN vs 2IJN engagement, the USN will win. Their %/sec being 65%(Ranger) - 43%(Midway) higher than their IJN counterparts

In a 2USN vs 3IJN engagement, the USN theoretically has the upper hand. Their %/sec ranges from 5%(Indy) - 18%(Ranger) higher than their IJN counterparts. Even when both sides have equal numbers, the USN has the upper hand.

 

I hope that this thread gives you an understanding of the chances at play when Fighter squadrons engages other Fighter Squadrons.

 

This thread assumes that

  • Squads will focus the same enemy squad
  • Squads are at full strength (6 for USN, 4 for IJN)
  • All squads meet at the same time

 

This thread does not account for

  • USN Planes being faster
  • Crew skills and/or Ship Upgrades
  • Ships AA Auras
  • IJN re-arm times
  • Catapult Fighters

 

tl;dr: In equal and sometimes less numbers, the USN fighters will win.

 

Wery interesting info, and it seems that the 20% upgrade HP module for fighters is really worth it.

 

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There is a slight math problem with the multiple squads , otherwise it might be a very solid piece of info. Now i cant say if its mistake of the poster or both WG and poster in implementing calculations. From what I've seen so far, wg calculations are actually correct (multiple accounts of more then one plane dying simulatnously), but without really large sample i cant be really sure (we are talking in probabilities here, we need large sample). However in both cases there are some interesting implications.

 

1) Lets assume WG implemented multiple squads correctly from the mathematical point of view then:

 

 

The chance for multiple squad to get down a plane are really 1-(1-x)^N, but there is also a chance to get multiple planes every second. Its not a huge chance, but it can change the results on certain tiers. In fact for the simplicity of calculations you might just assume the chance is really x*N - as it will give you realistic fighters/second value. 

 

For example at tier 6, for Japanese 3v1 it gives us 37.25% chance to shoot a plane (which is correct), however in reality the chance is:

 

37.25% is a chance to shoot AT LEAST one plane.

which comes down to:

 

0.3% chance of shooting down exactly THREE planes at the same time

5.3% chance of shooting down exactly TWO planes at the same time

31,6% chance of shooting down exactly ONE plane

 

they simply add those numbers together for the 37.2% (rounding error), and continue as if it was always one plane being shot down. In reality there is a significant chance of shooting multiple planes, so we can say that on AVERAGE there is 0.43 US plane being shot down every second, as opposed to the 0.3725 they posted. 

 

in their example on t8, yes 2 US groups have higher chance of shooting down 1st plane than 3 IJN groups, but there is a lot higher chance that when they do IJN will shoot down double or even triple plane from US flight group.

Complete chances are:

 

US:

41.22% chance of shooting down any number of planes including:

5.44% chance of shooting exactly TWO planes

35,77% chance of shooting exactly ONE plane down

for a total of 0.47 planes shot down/sec

 

IJN

 

38.78% total chance including:

0.34% chance of shooting down THREE planes

5.8% chance of shooting down TWO planes

32.6% chance of shooting down THREE planes

 

meaning 0.45 planes shot down per second on average. 

 

The difference is there, but not as big - the US got around 14% increase in effectiveness while IJN got 17% increase. On this tier it gets both groups within 4% difference from 6.3% 

 

 

 

2) What if WG made the same mistake as in the post. In this case there is a very tangible (up to 25%) difference in fighter effectiveness if you send them on single target - and a negative one. It means you should SPLIT the dps of your fighters over as many groups as possible - it will give you considerably more kills/sec. It actually gives even higher advantage to USN, because they can under most circumstances have "enough targets".

 

On a side note, this math they posted suggest you should probably be splitting groups in fighter engagements ANYWAY. 

In the variant 1), it makes no difference to the outcome, but you can hope he will be left with multiple groups with 1 plane should you lose, and have to land/rearm them, as opposed to having to rearm only one group and have others start even during the dogfight anyway (and sometimes even rejoin the fight).

In the variant 2), it actually can win you engagements you would otherwise lose by upping your dps (each consecutive fighter group on the same target suffers multiplicative 15-25% penalty to dps - depending on tier).

 

There is no benefit to focusing fire on fighters, IF there was hp on fighters focusing would help you avoid leaving multiple enemies at low hp without actually killing them - but that is not the case apparently - there is no hp at all. Its an open question if there is a reason to focus fire on bombers - im not sure if rear gunners have fixed TOTAL dps (in which case focusing fire would limit your loses as only one group of enemy bombers would be firing at the same time), or fixed PLANE dps, attacking every fighter on it simultanously (in this case focusing fire is just bad as it will allow them to keep up their speed and some will slip away). 

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Yep, there was an addendum:

 

Q&A from both:

  • Does AA Mechanics on Ships work the same way?

Yes. Ship AA works the same way. Ships have 3 types of AA Aura's

​Long Range AA: 88mm, 127mm etc. etc.

Medium Range AA: 40mm Bofors, 55mm Gerat etc. etc.

Close Range AA: 12.7mm Brownings, 25mm AA, 20mm Oerlikons

These 3 AA Auras are Cumulative, so the 3 different DPS will combine to form the total DPS to be used in the formula

TOTAL DPS / SURVIVABILITY = (X)*100%/sec

Multiple ships do not accumulate, rather their own separate rolls are made.

 

  • Can two planes (or more) be shot down at the same time?

Yes. Although the chances are very low and we did not include this in the original thread as the it would complicate things. But if you are interested, the chances for example of 2 Hosho Squads shooting down 2 Langley fighters at the same time are 0.147∩0.147 = 2%

 So aparently each squadron and ship have their own "fight" with each squadron, nothing is cumulative. But with more squadrons/ships focusing one squadron, you have better chances to kill planes faster.

But if you are really unlucky, you can have a 3 fighter squad loose to 1 fighter squad (the one squad have the same fight with each of the 3 squads, dps don't cumulate)

 

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So it seems they kinda dropped the ball at the last part. They "didn't include" it because it would "complicate things" , but the difference between taking it into account and not, is greater than difference between USN and IJN! You cannot just drop 8-29% (depending on tier/setup) difference in dps because it would "complicate things" - you turn your whole math rather useless. They go to conclusions about different matchups, and in many cases they get wrong results. They can say 2% is low, but if the "base" is 28%, 2 percent POINTS actually make almost an 8% difference - which can get a LOT higher with 3 or 4 squads (by order of x5-x10). If 4 t10 squads attack one target, they give the %/sec at 50.65%, but in fact it is 64.76% - thats undervaluing Hakuryu by 29% ! In fact they could simplify it by simply making dps cumulative, because its a LOT closer to reality then their convoluted math. 

 

Also multiple ship auras indeed don't accumulate in a very literal sense - but a 2 ships with same guns total as 1 ship will shoot the planes exactly as fast (just in a different "rhythm" - 1 ship will have slightly shorter gaps between air kills, but 2 ships will compensate it with ability to score 2 simultaneous kills). So for all purposes you can say the accumulate - the expected time to kill lets say all 6 bombers coming their way is EXACTLY the same as with 1 ship. 

 

All in all the whole fighter vs fighter math is not very useful, not past what someone with basic math skills could assume already. But there is one extremely valuable piece of information in this post - that planes have no HP. Which results in following actual tips:

 

1) In fighter vs fighter engagement if you both control more then 1 group, its actually beneficial to split your targets. You dont lose anything in terms of performance, and very often it will be easy to get enemy down to multiple groups of 1 plane that he will have to fly all the way back to CV and land, instead of instantly replenishing. If you fly lets say 2 jap groups vs 2 US as with Hiryu vs Ranger - you will lose. But if you focus on one group you can take it down. However in many scenarios (the notable exception is when your strike group is close enough for them to intercept right away and you want to get rid of a group) its vastly preferable to leave him with 2 low ammo 3 plane group, rather then 1 full 6 fighter group and 1 dead (that he will relaunch in 10 seconds as opposed to minutes required to land/resupply/start/fly etc time). 

 

2) If you fly your planes (regardless fighter or bomber) nearby to an AA aura and you manage to pass them without suffering any loses (lets say you fly for a BB in backline and you go past a cleveland 4km to the side on the way), the AA did NOTHING. Your planes when they arrive at actual target are as good as new. So unless the AA source is strong enough that you expect them to shoot planes down, there is NO REASON to avoid them and go the long way around. 

3) Tip for playing AA - if multiple groups attack YOU, manual targetting is usually not worth it - it way more reduces your damage on OTHER groups, that it increases on your target one. However if the planes pass you for other targets, manual targetting is definitely worth it - if you dont, you will "damage" lots of planes but not shoot down any, if you do you can get a kill or 2 - given how you can say the planes "heal to full" every second, the latter is better. 

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3) Tip for playing AA - if multiple groups attack YOU, manual targetting is usually not worth it - it way more reduces your damage on OTHER groups, that it increases on your target one. However if the planes pass you for other targets, manual targetting is definitely worth it - if you dont, you will "damage" lots of planes but not shoot down any, if you do you can get a kill or 2 - given how you can say the planes "heal to full" every second, the latter is better. 

 

I think I read somewhere, that AA doesnt affect all planes in your aa-area but only the most dangerous one. So it will rather focus the loaded torpedo bomber than the empty one, rather focus the empty dive bomber than the fighter. If you manual target, then you increase your chance of killing planes, which is totally worth it.

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