An analysis of 2.2+ Missile Warfare

[bean (OP)]

I've written a follow-up to the OP:

First, a mea culpa. I misread the patch notes when putting the last post together, and it turns out that allocation is on a per-missile basis, not a per-salvo basis. This had surprisingly little impact on the overall number of leakers to be expected,¹ but quickly led me down a rather interesting rabbit hole that conclusively resolved a long-running question.

Figuring out how to most efficiently allocate beam weapons to avoid overkill has long been an issue in Aurora, although often masked in my experience by gross overkill in available PD systems. Back in the VB6 days, each fire control would only target one salvo, so you needed lots of fire controls. For the pre-2.2 C# versions, this was per-weapon/turret, which was better, but still meant that sizing turrets took some thought. The current system gets rid of all that, but also gets rid of the ability to just make sure there are no leakers by having enough systems available, or at least makes that a lot less desirable as a solution.

After the last post went out, I started messing around with models of single-missile engagements, particularly seeing what happened as I investigated the tradeoff between rate of fire and chance to hit, and I got a very clear result. If you, say, replace full-size gauss guns with an equivalent HS of smaller gauss guns, you get worse performance against missiles without decoys. This is because even if the average number of expected hits stays the same, the increase in variability means you’re failing to get even one hit more of the time, while the fact that there’s only one missile means that you can’t capture any overkill.² This is particularly stark if the initial Ph is high, when the chance of a leaker might triple if you quadruple the number of shots.
Base Ph   0.67
Base Shots   3
Decoys   0
Base Leaks   3.59%
2x Leaks   8.65%
4x Leaks   11.08%
Base Ph   0.34
Base Shots   5
Decoys   0
Base Leaks   12.52%
2x Leaks   15.52%
4x Leaks   16.92%
Base Ph   0.25
Base Shots   10
Decoys   0
Base Leaks   5.63%
2x Leaks   6.92%
4x Leaks   7.57%

But this is all without decoys, and it turns out that decoys, because they offer the chance of capturing some of the overkill, can even the odds quite a bit. That said, even in the limit case, with enough decoys to absorb all of the overkill, the expected leaker percentage is exactly the same as it is for the full-Ph weapon. So even once we include decoys, you’re never going to be worse off with a full-size turreted gauss gun over the alternatives.³
Base Ph   0.67
Base Shots   3
Decoys   2
Base Leaks   33.00%
2x Leaks   37.05%
4x Leaks   38.79%
Base Ph   0.67
Base Shots   5
Decoys   2
Base Leaks   8.42%
2x Leaks   13.81%
4x Leaks   15.97%

The basic conclusion of all this is pretty simple. The way that beam PD allocation is handled now means that it makes sense to prioritize getting the highest reasonable Ph for your weapons, because the higher variance you get when trading Ph for more shots generally works against you and will never benefit you on net. In a lot of ways, the new system has actually made ship design easier. Previously, picking beam PD layout was a matter of figuring out what the best option was to try and minimize overkill without making things too complicated (I suspect that this was a major and overlooked advantage to railgun PD in original C#), whereas now, full-size gauss guns are the way to go if at all possible.

1 It turns out that when I re-ran the numbers, the higher percentage of single-missile saloves with leakers was almost entirely balanced by cases where larger salvoes had multiple leakers, leaving total leaker numbers the same. ⇑

2 If this doesn’t make sense, consider. I give you the option of flipping two quarters, or one quarter with two heads. If I offer to give you a dollar for every head, the two are equivalent in expected value. If I say “I’ll give you a dollar if you get any heads”, then the two-headed coin is clearly better because you get only one dollar for two heads, and have the possibility of zero dollars. ⇑

3 There are some obvious caveats to this, the biggest relevant to railguns being the assumption that the incoming missile is faster that the tracking speed for the gauss turret/FC, so the railgun is .25 the Ph of the gauss gun. If the missile is slower than that, the railgun is going to have an advantage. ⇑

[bean (OP)]

This is pretty obvious math.... one 100% shot is better than two 50% shots because two shot will occasionally produce overkill and waste shots. You only gain something if you can manage to increase the hit chance of the 50% above 50% so say two shots at 60% will be better than one at 110%. You can't really hit more with one shot as 100% is always a guaranteed shot as much as 110% from a game mechanic perspective.

This isn't actually true under the current mechanics.  The two 60% shots will leak 16% of the time, the single 110% won't leak at all. 

On the other hand you can gain some weight saving from multiple smaller gauss, so it is not entirely obvious which is better all the time.

That is at most a 6% saving going from single to quad (I just checked at 4x turret mechanism tracking speed), and the benefits from high Ph are striking, as discussed in the follow-up I just posted.

I would also like to comment on terminal guidance which is more efficient to use for ASM rather than AMM missiles, so terminal guidance do have a place in missile design.

Check the eighth paragraph, 3rd sentence from the end.

[Jorgen_CAB]

This is pretty obvious math.... one 100% shot is better than two 50% shots because two shot will occasionally produce overkill and waste shots. You only gain something if you can manage to increase the hit chance of the 50% above 50% so say two shots at 60% will be better than one at 110%. You can't really hit more with one shot as 100% is always a guaranteed shot as much as 110% from a game mechanic perspective.

This isn't actually true under the current mechanics.  The two 60% shots will leak 16% of the time, the single 110% won't leak at all. 

On the other hand you can gain some weight saving from multiple smaller gauss, so it is not entirely obvious which is better all the time.

That is at most a 6% saving going from single to quad (I just checked at 4x turret mechanism tracking speed), and the benefits from high Ph are striking, as discussed in the follow-up I just posted.

My point was that in a layered defence where you also include decoys, CIWS and shields it probably is not that much of a major issue as the size of the system actually might matter more as large expensive  modules also have more impact on the design choices and maintenance requirements as well as research investment etc...

The current system where you allocate shots first certainly does favour high accuracy weapons over low accuracy that is for sure. I'm just saying it is not the only thing to consider when designing ships.

[bean (OP)]

My point was that in a layered defence where you also include decoys, CIWS and shields it probably is not that much of a major issue as the size of the system actually might matter more as large expensive  modules also have more impact on the design choices and maintenance requirements as well as research investment etc...

The current system where you allocate shots first certainly does favour high accuracy weapons over low accuracy that is for sure. I'm just saying it is not the only thing to consider when designing ships.

I am not saying that there is literally no case where you should use anything other than size 6 gauss guns.  I am saying that if you use something else and aren't in an extreme edge case, all of the other components of your defenses are going to have to work harder.  How much harder depends on the base Ph you're dealing with, but it's going to be significant in a wide variety of cases.  And if a twin or quad turret is too big or expensive, use singles.  A lot of my ships do to preserve redundancy.  It's a good tradeoff.

[Jorgen_CAB]

My point was that in a layered defence where you also include decoys, CIWS and shields it probably is not that much of a major issue as the size of the system actually might matter more as large expensive  modules also have more impact on the design choices and maintenance requirements as well as research investment etc...

The current system where you allocate shots first certainly does favour high accuracy weapons over low accuracy that is for sure. I'm just saying it is not the only thing to consider when designing ships.

I am not saying that there is literally no case where you should use anything other than size 6 gauss guns.  I am saying that if you use something else and aren't in an extreme edge case, all of the other components of your defenses are going to have to work harder.  How much harder depends on the base Ph you're dealing with, but it's going to be significant in a wide variety of cases.  And if a twin or quad turret is too big or expensive, use singles.  A lot of my ships do to preserve redundancy.  It's a good tradeoff.

I agree in general terms... but you also need to account for the increased cost for larger more expensive components and more expensive research costs, everything will have a cost for different reasons in different ways. A large quad 100% gauss turret will have a significant impact on maintenance failures for example and overall maintenance cost over time. It is the same issue with large versus smaller engines for example.

Not everything can be calculated with efficiency in combat, it all depends on many different factors.

Therefore you will need to at least acknowledge that there are trade-offs that will show in other ways outside combat that you need to account for as well. Everything will impact the decision of what components you develop and why. In general a quad 100% gauss is always worse than a single due to how the maintenance mechanic works, you will never really save neither cost nor space with a larger turret.

[Steve Walmsley]

Therefore you will need to at least acknowledge that there are trade-offs that will show in other ways outside combat that you need to account for as well. Everything will impact the decision of what components you develop and why. In general a quad 100% gauss is always worse than a single due to how the maintenance mechanic works, you will never really save neither cost nor space with a larger turret.

It's worth bearing in mind that four single gauss weapons will fail 4x more often than a quad turret (because each weapon checks for failure, not each shot), so while the quad is more expensive when it fails, it might not be the most expensive over time.

[Jorgen_CAB]

Therefore you will need to at least acknowledge that there are trade-offs that will show in other ways outside combat that you need to account for as well. Everything will impact the decision of what components you develop and why. In general a quad 100% gauss is always worse than a single due to how the maintenance mechanic works, you will never really save neither cost nor space with a larger turret.

It's worth bearing in mind that four single gauss weapons will fail 4x more often than a quad turret (because each weapon checks for failure, not each shot), so while the quad is more expensive when it fails, it might not be the most expensive over time.

Is it not so that the quad is also four times more expensive when it does fail so the cost would be the same. I suppose it has been changed as before each weapon in the turret was rated separately so a large turret had four times the chance to fail, but I think you changed this a while ago.

[nuclearslurpee]

Is it not so that the quad is also four times more expensive when it does fail so the cost would be the same.

I think this was a C# change, but multi-weapon turrets have modest reductions in tonnage to be a bit more cost and size-efficient than an equal number of single-weapon turrets. See here. No explicit cost decrease is mentioned but there is a reduction in crew requirement, so the cost is indirectly reduced if nothing else. Possibly the reduced size also affects the armor calculation, I don't know.

[Jorgen_CAB]

Is it not so that the quad is also four times more expensive when it does fail so the cost would be the same.

I think this was a C# change, but multi-weapon turrets have modest reductions in tonnage to be a bit more cost and size-efficient than an equal number of single-weapon turrets. See here. No explicit cost decrease is mentioned but there is a reduction in crew requirement, so the cost is indirectly reduced if nothing else. Possibly the reduced size also affects the armor calculation, I don't know.

When comparing two ships with smaller gauss but more turrets you will save maintenance supplies and need less maintenance facilities for the same maintenance life value. You also need less research and it will be easier to fit turrets into designs as the components are smaller.

A turret with a single weapon will still occupy more space due to decreased crew requirement and that it is smaller but will still require less maintenance supplies over time.

Not sure of smaller turrets will cost more due to failure in combat as even if they fail more often they also are cheaper, so to me it seems as if the cost will be toughly the same.

[Pedroig]

1)  Let's assume that for the same number of GUNS there will be the same failure rate/cost.  So the single/multiple repair cost portion goes out the window over the long term. 
2)  Less research is a non sequitor, if one is going pure smaller, then there is no "less research" and if one is going for "more advanced" then once again, the research is going to be done regardless.  Comparing different tech levels to one another is rather frivolous, especially in hindsight, either one wants/needs a tech or one doesn't, so there is no "research cost savings" ever.
3)  Having more weapons means more chances of failure to occur to the ship, but the impact of that failure will be less overall.  (Extreme example is if one loses one of four single turrets during an engagement, the leak chance goes up, but is still less than 100%, whereas if a ship loses its only quad turret then the leak chance technically becomes 0% because all missiles will be getting through).
4) Per Bean's numbers, lowering chance to hit, REGARDLESS OF SOURCE of reduction, INCREASES leakers for any given number of shots. 
5)  Given a layered defense philosophy; this would mean wanting to have the highest hit chances on the  innermost layer.  So something like having a 50%/75%/100% Ph onion, with 12/8/6 guns all taking up the same effective space (with minor variance for crew) would be a redunancy max kill focused doctrine based upon either relative or absolute displacement usage.  (CIWS nor AMM included in this example, both can be used to "tweak" the numbers, but have other considerations as well)  Ideally each layer will cover at least one full five second tick of incoming movement.

[Jorgen_CAB]

2)  Less research is a non sequitor, if one is going pure smaller, then there is no "less research" and if one is going for "more advanced" then once again, the research is going to be done regardless.  Comparing different tech levels to one another is rather frivolous, especially in hindsight, either one wants/needs a tech or one doesn't, so there is no "research cost savings" ever.

If you use smaller gauss you need less turrets and the gun itself are cheaper to research so you need to research less number of turrets and the turret is less costly to research over the equivalent version. So you will save in research cost.

For example you research the 100% gauss and then one single, twin and quad turret in order to fit into designs properly with different sizes. Or.. you just research a 33% gauss and then a single quad turret. The single quad 33% turret are likely enough to fit into any design in some number to save on both maintenance and research costs.

One example ship I created with 4 100% quad turrets and one with 12 33% quad turrets used 20% (1200 vs 1500 supplies) less supplies and was 200t smaller, less crew and slightly cheaper to build. That is why it is not a straight answer to what is best.

[bean (OP)]

2)  Less research is a non sequitor, if one is going pure smaller, then there is no "less research" and if one is going for "more advanced" then once again, the research is going to be done regardless.  Comparing different tech levels to one another is rather frivolous, especially in hindsight, either one wants/needs a tech or one doesn't, so there is no "research cost savings" ever.

If you use smaller gauss you need less turrets and the gun itself are cheaper to research so you need to research less number of turrets and the turret is less costly to research over the equivalent version. So you will save in research cost.

I will grant you that this is technically true, but it's also not that much for the benefit you get.  I just ran the numbers, and in a fairly high-tech game (gauss ROF 5/range 5) I would spend less than 2000 RP to deliver a quad 100% gauss from scratch.  Extra turrets will be under a thousand each.  Oh, and you've spent 134,000 RP in basic research to get to those gauss guns.  So the trade here is spending an extra couple thousand RP (maybe 3% of what you've spent on gauss so far in the game) in exchange for 20%+ fewer leakers (or, you know, 60%+ fewer leakers with good FC).  Now, the best option I can see for that kind of 20% reduction in (effective) leakers is another level of missile jammer, which is, oh, 80,000 RP.  Yeah, I think I'll spend the extra research.

One example ship I created with 4 100% quad turrets and one with 12 33% quad turrets used 20% (1200 vs 1500 supplies) less supplies and was 200t smaller, less crew and slightly cheaper to build. That is why it is not a straight answer to what is best.

Let's say that your FC has a 50% inherent to-hit, and you shoot 4 v 1 at 50% Ph with the basic ship.  I get 6.25% leakers.  Now, if I shoot 12v1 with the 33% ship, I get 11.22% leakers.  You have to shoot something like 15v1 to get the same leaker rate (technically, the slightly-higher 6.49%).  To put it another way, if you're bound by the total number of guns and want a constant leaker rate, you need 25% more ships.  So the full-size gauss ship needs the same number of supplies, and is cheaper everywhere else.  I'll take that one.

(Edited to fix basic math error in my last paragraph.)

[Jorgen_CAB]

2)  Less research is a non sequitor, if one is going pure smaller, then there is no "less research" and if one is going for "more advanced" then once again, the research is going to be done regardless.  Comparing different tech levels to one another is rather frivolous, especially in hindsight, either one wants/needs a tech or one doesn't, so there is no "research cost savings" ever.

If you use smaller gauss you need less turrets and the gun itself are cheaper to research so you need to research less number of turrets and the turret is less costly to research over the equivalent version. So you will save in research cost.

I will grant you that this is technically true, but it's also not that much for the benefit you get.  I just ran the numbers, and in a fairly high-tech game (gauss ROF 5/range 5) I would spend less than 2000 RP to deliver a quad 100% gauss from scratch.  Extra turrets will be under a thousand each.  Oh, and you've spent 134,000 RP in basic research to get to those gauss guns.  So the trade here is spending an extra couple thousand RP (maybe 3% of what you've spent on gauss so far in the game) in exchange for 20%+ fewer leakers (or, you know, 60%+ fewer leakers with good FC).  Now, the best option I can see for that kind of 20% reduction in (effective) leakers is another level of missile jammer, which is, oh, 80,000 RP.  Yeah, I think I'll spend the extra research.

One example ship I created with 4 100% quad turrets and one with 12 33% quad turrets used 20% (1200 vs 1500 supplies) less supplies and was 200t smaller, less crew and slightly cheaper to build. That is why it is not a straight answer to what is best.

Let's say that your FC has a 50% inherent to-hit, and you shoot 4 v 1 at 50% Ph with the basic sheet.  I get 6.25% leakers.  Now, if I shoot 12v1 with the 33% ship, I get 11.22% leakers.  You have to shoot something like 16v1 to get the same leaker rate (technically, the slightly-higher 6.49%).  To put it another way, if you're bound by the total number of guns and want a constant leaker rate, you need 33% more ships that are (being as generous as possible) 20% cheaper.  Hmm.  1.3333*.8=1.06667.  Again, better hit rate comes out the winner.

Every little tech points will count, especially earlier on... I spend allot more time in the game at lower tech levels anyway. When you play with rather slow tech progression and restricted scientist labs you can be rather short on Missile/Kinetic scientists. You also likely are using your least skilled scientists to research components and the more skilled ones to research more expensive technologies. That means that the components become relative to other tech more expensive.

I rarely see leaking missiles as much of an issue when shields and decoys are used. If you don't use the shields they are also wasted space.  If there are 6 or 10 missiles leaked is not very important when the decoys and shields stop it anyway. If your PD is overwhelmed then it does not really matter much anyway as it is the overkill that produce the leaking difference.

What you save on time from research and on used supplies over time can be quite substantially important.

No point to deny that 100% gauss will produce less leaking missiles, but it does come at a different cost. You also need to judge how much you want your resources spent on shields and decoys to matter as well as supplies and time in research

[bean (OP)]

Every little tech points will count, especially earlier on... I spend allot more time in the game at lower tech levels anyway. When you play with rather slow tech progression and restricted scientist labs you can be rather short on Missile/Kinetic scientists. You also likely are using your least skilled scientists to research components and the more skilled ones to research more expensive technologies. That means that the components become relative to other tech more expensive.

An RP is an RP.  That's how this works.  Sure, I'll grant you that the percentage is higher at lower tech, but again, the gains are substantial, and I think make sense by any metric, particularly if you're willing to invest in smaller-than-quad turrets.

I rarely see leaking missiles as much of an issue when shields and decoys are used. If you don't use the shields they are also wasted space.  If there are 6 or 10 missiles leaked is not very important when the decoys and shields stop it anyway. If your PD is overwhelmed then it does not really matter much anyway as it is the overkill that produce the leaking difference.

I don't think this is true in the general case, or if it is, it speaks to weakness in the AI.  I agree that some number of leakers are a fact of life, and recommend shields to deal with them in the OP.  But to a first approximation, all of your inner-ring systems are going to be able to handle a specific number of leakers and anything past that is going to cause problems fast.  (Yes, I know that decoys are a little more complicated, but that's a complication I'm ignoring.)  Optimizing your beam PD means that you can handle a larger number of incoming missiles without getting overwhelmed.  For instance, comparing your two ships, the 100% one will have 48 shots if I assume that the ROF is 3, while the 33% one will have 144.  Let's say that you have a squadron of 4 of them, and base Ph is 0.5.  If I fire 96 missiles, my math says that the average number of leakers will be 24 for the 100% ships and 32.15 for the 33% ships.  For 64 incoming missiles, I get 8 and 12.4, respectively.  Both seem like places where you might plausibly find a cliff in response, and the 100% ships could be fine while the 33% ships are taking damage.  Sure, it's possible to get around this by being good enough to beat most AI and slightly overpaying, but you are overpaying.

[Jorgen_CAB]

Every little tech points will count, especially earlier on... I spend allot more time in the game at lower tech levels anyway. When you play with rather slow tech progression and restricted scientist labs you can be rather short on Missile/Kinetic scientists. You also likely are using your least skilled scientists to research components and the more skilled ones to research more expensive technologies. That means that the components become relative to other tech more expensive.

An RP is an RP.  That's how this works.  Sure, I'll grant you that the percentage is higher at lower tech, but again, the gains are substantial, and I think make sense by any metric, particularly if you're willing to invest in smaller-than-quad turrets.

I rarely see leaking missiles as much of an issue when shields and decoys are used. If you don't use the shields they are also wasted space.  If there are 6 or 10 missiles leaked is not very important when the decoys and shields stop it anyway. If your PD is overwhelmed then it does not really matter much anyway as it is the overkill that produce the leaking difference.

I don't think this is true in the general case, or if it is, it speaks to weakness in the AI.  I agree that some number of leakers are a fact of life, and recommend shields to deal with them in the OP.  But to a first approximation, all of your inner-ring systems are going to be able to handle a specific number of leakers and anything past that is going to cause problems fast.  (Yes, I know that decoys are a little more complicated, but that's a complication I'm ignoring.)  Optimizing your beam PD means that you can handle a larger number of incoming missiles without getting overwhelmed.  For instance, comparing your two ships, the 100% one will have 48 shots if I assume that the ROF is 3, while the 33% one will have 144.  Let's say that you have a squadron of 4 of them, and base Ph is 0.5.  If I fire 96 missiles, my math says that the average number of leakers will be 24 for the 100% ships and 32.15 for the 33% ships.  For 64 incoming missiles, I get 8 and 12.4, respectively.  Both seem like places where you might plausibly find a cliff in response, and the 100% ships could be fine while the 33% ships are taking damage.  Sure, it's possible to get around this by being good enough to beat most AI and slightly overpaying, but you are overpaying.

I agree that you will get less leaking missiles but you are still ignoring that time, research, design flexibility, maintenance supplies and weight saving is all something you also need to consider over time. Do your ship spend more time in combat or paying maintenance for example. If you have the shields they are useful for defence in beam combat as well as against missiles... so you can rely on them to soak incoming missiles as well as part of the total investment. If you don't use the shields then they are as much dead weight as anything else on the design.

In order to manage this you have a layered defence of AMM, PD, CIWS, Decoys and Shields and you want to balance their use as much as possible when needed.

If you tend to use relatively small ships in large numbers then leaking missiles is more of an issue than if you tend to use bigger and fewer ships for example. Against the AI I rarely actually rely much on PD as protection to begin with (only a minor part) to be honest but that is a completely different discussion. I also tend to favour designs that save cost rather than efficiency as overall long time costs is a worthwhile consideration. This is probably why I accept a larger degree of leaking missiles on my design for reduction in deployment costs. Avoiding being shot at in the first place obviously being the number one priority to start with. But if I'm shot at by the AI I tend to have overwhelming force or being completely outmatched, so then saving cost are much more important. I rarely find myself in situations where I'm evenly matched, such situations should be avoided at all costs.