Debriefing Report: The Battle of Kagoshima: On the use of two-stage missiles

[Theokrat]

I like your chart, but i think you're waving this off a bit too much.
Final fire has a peak, but as long as there are good enough sensors and launchers, a competent AMM defense is almost entirely attritional in nature. That is to say, its very difficult to breach an AMM umbrella of equivalent technology until it runs out of missiles.

Well the beauty of the above formula is that it applies one per-time basis, just as much as on an attrition basis. I formulated it as X, Y, and Z being the number of missiles per minute that can launched/shot down/get through. If you assume that Y >Z, then you get into an attrition situation. Let’s put the argument differently then: Let z be the number of missiles that can be shot down by the enemy in total, z be the number of ASMs you can launch in total, then x=z-y is the number of missiles that get through in total. So it’s the same formula (unless we would start considering beam weapons), and importantly the number of missiles launched enters the formula just the same way.

One difference is of course that on a per-time basis, the number of missiles that can be launched goes with the inverse square of the missile size. E.g. one might be able to launch one size-2 missile every 10s (from one launcher) or two size-1 missiles every 5s (from two smaller launchers), giving either 1 or 4 (=2^2) missiles per 10s. In total one is limited by available magazine capacity, and thus the total number of missiles that can be launched goes with the inverse (not squared) of the missile size. E.g. one can either store one size two missile, or two size-1 missiles.

Thus while the formula stays the same, the variable takes a different value, and on an attrition basis smaller missiles do not look as good as on a per-time basis. I.e. by halving missile size one can only double z, while Z could have been quadrupled. But as you point out nicely, even on an attrition basis smaller ASM provide a significant benefit. Also, just to point this out I was being conservative when I made the graph and only used same-magazine size number of missiles. I.e. I compared nine size-1 missiles to one size-9 missile, while on a per-time basis one could also have compared eighty-one (!) size-1 missiles to one size-9 missile.
In other words the chart is based on the assumption of attritional missile warfare.

This attritional problem is why I don't like to use tiny missiles. They kind of break the missile balance, even without looking at how hard final defensive fire sucks against them.  The final nail in the coffin is that they are almost free to research while big missiles are hella expensive.

Well, from a game-design point of view I would hope that some economies-of-scale factors are included in missile designs. Right now you could do the equivalent of taking a present-day real-world ICBM and just scale it down by a factor of 1000, and get a miniature ICBM with the same individual characteristic (speed, range). That’s weird. For instance one could make the warhead more-than proportional to the Missile size points assigned to it, and thereby provide an inventive for larger missiles.

This is getting VERY nice!  Thanks for that chart!  I shall see to never using anything other than WH:4. 

Glad you like it. One point though: The graph depends on the rather arbitrary weighting of damage caused to different layers. This weighting is not based on any analysis, but simply a number I assigned based on subjective experience (spell: made up). As an anecdote: I once encountered an enemy that apparently liked to use two layers of armour. Against this enemy a size-9 warhead will cause internal damage with the first hit. This internal damage could be on battle-critical systems, and I actually had a first hit cause a magazine explosion that ripped the whole thing apart. So in this case layer-3 damage should be valued much higher than in my previous example. Actually this drove me to use a weighting of 25-5-1 for damage to layers 3, 2 and 1 respectively. And this weighting resulted in favour of warhead-9 missiles (which were awfully slow due to my short-term launcher size restrictions, but still performed extremely well, often causing engine damage with the first few hits, resulting in the enemy battle line falling apart).

Just to add more about AMM fire:
There are three factors that determine what gets through an AMM envelope. 

1) Hit Rate against your missiles. 
2) AMM range vs missile speed (aka. number of interception chances)
3) AMM ammunition count. 

Tiny WH:1 ASMs win on number 3.  They launch faster and in more numbers than AMMs can intercept them.  Fair enough, they break the game.  For getting through the most damage through a contemporary AMM umbrella, size 1s win hands down. 

Now let's talk about the other two.  This is why speed is so important. 

Take the Tanto;
47800km/s, maneuver 32.  It has 19% vs an 80 000 km/s missile and about 38% vs 40 000 km/s.
However, it also has a range of 7 million km. 

VS the 40kkm/s missile
The first interception occurs around 3.5 million km range.  2nd is at 1.75 million, 3rd at 875 thousand, 4th at 440 thousand, 5th at point blank. 
So it generates 38% chance to hit, across 5 chances to hit it.  At 3v1, the leak rate of this umbrella is ~24% per interception = 0.07% getting through all interceptions. 
This is the reason why my fleet suffered exactly one hit. 

VS the 80kkm/s Wakizashi
The first interception occurs around 2.33 million km.  2nd is at 777 thousand, 3rd at point blank. 
So it generates 19% chance to hit, across 3 chances.  At 3v1, the leak rate is 53% per interception meaning a whopping 15% go straight through!  Even at 5v1, the leak rate for all chances is still at 4%. 

You are quite correct in your analysis of the trade-of between the relative distribution of missile size points between the warhead and the engine. But, just to iterate this it does not touch on the optimal absolute size of the missiles. You can have large or small 80kkm/s missiles, and you can have large or small 40kkm/s.

Also you are assuming that the enemy can “use” all theoretical interception points. That might be true for the Wakizashi (with its indeed impressive ability to both dodge missiles and grant few interceptions). Assume the enemy devotes roughly the same space to AMM launchers as we do to ASM launchers. We are able to launch one Saya from the size-5 33%redction launcher at a time, while the enemy would be able to fire 3 AMMs from size-1 launchers (roughly the same space). However the Saya would only be fired every 500s, so the enemy would only ever face one salvo at a time. By the time the first salvo arrives at the target, the second salvo has not even reached the outer range of the AMMs.

Now consider the smaller missiles. These are not only launched in greater quantity, but also much more frequently. Indeed a new salvo can be launched every 5s. Consequently, every five seconds a new salvo will enter the engagement range of the AMMs. A salvo that contains just as many missiles as the enemy has AMM-launchers. So what does that mean for the number of interceptions? Funny, but the enemy will only ever be able to launch one AMM against each incoming ASM in the long run! So in fact, the expected number of interceptions is 1 in the long run! With only one missile!

So the Wakizashis are engaged by an average of 9 AMMs (as you nicely show), meaning that 15% get through at 19% individual hit chance. Yet the 40kkm/s missiles are only engaged by 1 AMM, meaning 62% get through in the long run! Funny, eh? (Ok, the long run means after an infinite number of salvos, which takes a while. But even with only two salvos, the average number of interceptions drops from 5 to 3 – the same number as with the Wakizashis). Of course this is not an effect of the lower speed (which is detrimental), but solely of the small size – the miniature Sho-Wakizashis would enjoy a similar effect.

… Going below size 4 means you will see diminishing returns on range … The flip side is that your smaller ASM's lack speed, which is killer…For example, if your facing an opponent using the small missile doctrine, range WILL be an issue…

No. Small missiles do not have a disadvantage in range. The range of a missile is determined by the fuel efficiency technology and the percentage of missile space devoted to missile size. A missile with x% of mass devoted to fuel will have the same range, regardless of whether it is size-1 or size-10.
In other words, a large missile can be reduced in size by decreasing all components by the same amount – and it will go just as far. Therefore range is never an argument for using smaller or larger missiles. Except, of course for warhead-1 missiles which can not be decreased in size.

It’s quite human to fall for this trap by saying “oh I could add 1 MSP of fuel to this missile to make it go further, but it will increase the missile size from 4 to 5”. So it looks like there is this trade-off between size and range. But there were really two things that were changed when 1 MSP was added to fuel- the size of the missile and the relative proportions of the various components. And it was the later, not the former, that made the range go up. You could again reduce all components by 20% and remain at the same range. So getting more range is a question of the relative proportions of components – but not of the total size. In other words: Range must be balanced against the size of the warhead, the attainable speed and manoeuvre ratings- but not against the size. Roudnign can be an issue, but it can go both ways.

Range is not the only variable with this behaviour. Speed and maneuver Rating are also functions that are only dependent on the relative proportions, but not on the absolute size of the missile. Something that does not share this is the active sensor bit. Indeed there are good arguments for making self-guided missiles rather big. And this is a good reason for the size of your designs, as well as the rounding issues. But not for range purposes.

[Havear]

There's two things I'd like to point out. 1) Agility points are independent of missile size. 2) Warhead strength is independent of size.

Yes, it *is* indeed possible to get a smaller missile with the same speed and range, since those are proportional to size. However, you then have to lower either your warhead or agility ratings, which are not proportional, and thus are flatly lower. It's indeed possible to get missiles with better performance in *very* small packages, but actually doing decent damage when they get there is another story.

[TheDeadlyShoe]

Agility ratings are divided by missile size, and thus are proportional.  Warhead strength is completely independent of size; 4 strength 1 warheads are equal to 1 strength 4 warhead, except insofar as damage grouping/cratering is concerned. 

However, damage grouping is irrelevant versus shields, and even strength 4 warheads dont do much clustering to write home about against decent (5+) armor layers.  It takes strength 9 or higher warheads to do meaningful clustering in my experience.

The only things that small missiles cant do are ecm and sensors.  Even armor scales down to some extent, but you're almost always better off with more engine instead of armor.

[Theokrat]

There's two things I'd like to point out. 1) Agility points are independent of missile size. 2) Warhead strength is independent of size.

Yes, it *is* indeed possible to get a smaller missile with the same speed and range, since those are proportional to size. However, you then have to lower either your warhead or agility ratings, which are not proportional, and thus are flatly lower. It's indeed possible to get missiles with better performance in *very* small packages, but actually doing decent damage when they get there is another story.

At the risk of repeating a bit what TheDeadlyShoe said:

1) The Maneuver rating is, like range and speed, a function of only the proportion of size devoted to agility- but not of the total size. x% of missile space towards agility results in the same maneuver rating, regardless of whether its a size-1 or a size-10 missile. Thus decreasing the size of a missile does not change the agility the tiniest bit.

2) The Warhead is the only component other than sensors and ecm that is propotional to the total size devoted to it (and not to the proportion of the missile).

But of course one should not compare missiles on a per-unit basis. Sure one size-1 missile is worse than one size-2 missile. But that comparison is meaningless, as it does not consider the constraints involved. Players are limited by the industrial costs of producing missiles, by the magazine space on their ships, and by the weight of ships that can be devoted to missile launchers. All of these factors scale proportionally to the size of a missile, thus one size-2 missiles must be compared to two size-1 missiles (or even four when considering missiles per second in non-attrition situations). But therefore "splitting" a missile into a number of smaller ones does not decrease the total sum of warheads, it merely leads to a different damage profile caused by these. And for strongly-armoured ships this is almost certainly a second-order effect at best.

[sloanjh]

A few comments (many of which many people are already aware of):

1)  I think a lot of people upthread have been using "size" when describing warheads rather than "strength".  When I say "strength 4 warhead" below, I mean a warhead that does 4 points of damage.

2)  If you stick with the basic systems (engine, warhead, agility, fuel) and don't put "dead weight" systems (e.g. armor, sensors) onto your missile designs, then missile capabilities scale exactly.  In other words, a size-4 missile, with a strength-4 warhead will have exactly the same range, speed, and hit probability as a size-1 missile with a strength-1 warhead if you pick the percentages of basic systems to be the same (and the size and cost will be 4x greater).  This means that, in terms of expected damage per (non-intercepted) missile, the size-4 missile and 4x size-1 missiles cost the same and occupy the same amount of magazine space.  I'll assume that missiles of different sizes have been tuned in this way (to have same performance characteristics) in the discussion below.

3)  Launch rate is proportional to missile size, and launcher size is inversely proportionaly.  This means that (assuming no interceptions) the expected damage per salvo for size-4 missiles is the same as the damage per salvo for size-1 missile.  It also means that the rate of damage is inversely proportional to missile size, since salvos are launched more quickly for smaller missiles.  Since it costs the same to intercept a size-4 missile as to intercept a size-1 missile, sandblasting with size-1 missiles (which might have warhead strengths greater than 1!) is vastly superior if damage templates are ignored.

4)  When thinking about damage templates, there are two limiting regimes: the warhead can penetrate the targets armor and do internal damage with a single hit, or it can't.  Consider a ship with thick armor (e.g. 12, or 20, or even 6) and a strength-9 warhead, which has penetration 3 and so needs to hit a patch of armor only 2 units thick in order to do internal damage.  Because of the way statistics of large numbers works, you should assume that you're going to have to grind away essentially all of the armor before you're able to do internal damage.  Because of the increased RoF, size-1 sandblasting is most effective here; even in comparison to size-2 missiles it will grind away twice as fast (and that's before the effects of interception are taken into account).  Now consider a really big freighter with armor-1.  In this case, the size-4 missile will have twice the penetration (and an even bigger improvement in the ratio of internal-to-armor damage) of the size-1 missile, and you don't have to wait for a later hit to randomly hit a hole you've made in the armor before you can do internal damage.  So for thinly armored ships (e.g. armor less than  4 or 5) you want big honking warheads.

In my designs, I tend to have two missile sizes: size-4 ASM missiles, which give a good balance between RoF and salvo size vs. penetration in the early game and size-1 missiles.  For the size-1 missiles, I usually have a warhead strength-1 designed optimized for AAM work, and an ASM "slug" design with the biggest warhead size I can manage (typically 2 or 3 in the early game).

John

EDIT - Ninja!!  (I got pulled away while typing ).  Theokrat's post is spot on.  One caveat: when you start talking about missile armor the advantage goes to the big missiles.

[TheDeadlyShoe]

4)  When thinking about damage templates, there are two limiting regimes: the warhead can penetrate the targets armor and do internal damage with a single hit, or it can't.  Consider a ship with thick armor (e.g. 12, or 20, or even 6) and a strength-9 warhead, which has penetration 3 and so needs to hit a patch of armor only 2 units thick in order to do internal damage.  Because of the way statistics of large numbers works, you should assume that you're going to have to grind away essentially all of the armor before you're able to do internal damage.

I think you are underballing damage clustering.   Check this out:

http://en.wikipedia.org/wiki/Birthday_problem

let's assume you have a 6,000 ton warship being attacked by missiles, which is 29 armor columns.  The second missile has a 1/29 chance of landing on the same location. The third missile has a 2/29 chance of landing on either previous location. The fourth has 3/29, and so on. It rapidly becomes more probable to have landed two hits on the same location than not. That's for *direct hits* on previous impact craters.   What this basically means is that battle-damaged ships armor will be mountains and valleys - even on the thickest armor, you're likely to have undamaged columns while other sections have been completely penetrated. 

or to put it another way: if you hit something with 7 missiles, there are 21 distinct chances for hits to cluster. on a 29-column (or 6000 ton) hull that works out to about a 52% chance that 2 missiles will hit directly on top of eachother.  So lets suppose that the hull is armor rating 5, being attacked by strength 9 warheads - you have a 52% chance of internal damage after taking only 63 points of damage out of 148 armor strength.  Bear in mind that this significantly underballs that chance, because it does not take into account damage outside the 'central column' of the pyramid.

If you hit that same ship with 10 missiles, you have about an 80% chance that two of your hits will be on top of eachother. 15 missiles is a 98% chance. 

Just for kicks... looking at a 40,000ton freighter being hit by strength 1 warheads.  That's 104 armor columns, but with 20 missile hits you have a 84% chance of having scored internal damage. 

EDIT - Ninja!!  (I got pulled away while typing  ).  Theokrat's post is spot on.  One caveat: when you start talking about missile armor the advantage goes to the big missiles.

Yeah - now if only armor scaled up with technology.


**EDIT**

I got to thinking. What about strength 12 warheads? The damage shape means that you essentially have 3 chances(one slight off left, one direct, one slightly off right) of scoring a stacking hit on a previous impact rather than 1.  So a strength 12 warhead needs only 4 hits (48 damage) to hit ~50% internal damage chance, and 6 hits (72 damage) to hit ~80%.  Hrm.  Same applies to strength 6 vs armor 3, or strength 20 vs armor 7.

[Havear]

All good points. I'll admit, I've not used reloadable launchers for some time. Even on box launchers, I've been considering moving from size-6 back to size-4, as I'm not really noticing much more then slightly increased range (pretty useless for fighters) and some ECM, which doesn't seem to match the 50% density tradeoff.

[jseah (OP)]

Bigger ASMs could get better performance than smaller ones by having less than 1 Warhead per MSP. 
You are also presented with the tradeoff of getting slightly more than 1 warhead per MSP. 

Eg. the Wakizashi could be a size 4 carrying WH:3 (assuming sandblasting is going to happen)
It could also be a size 3 carrying WH:4

These proportions are inaccessible to a size 1 ASM. 

[sloanjh]

Just for kicks... looking at a 40,000ton freighter being hit by strength 1 warheads.  That's 104 armor columns, but with 20 missile hits you have a 84% chance of having scored internal damage.  

And if those 20 points of damage were instead from 5 strength-4 hits you'd have a 100% chance of doing at least 5 points of internal damage, with even more internal damage done if the str-4 hits overlap.  The intuition here is that the 4th point of damage (on the 29 column warship) has a 100%, rather than 3/29 chance of clustering.

The detailed math is really yuckky, but I'm pretty sure that clustering actually helps big warheads more than it helps small warheads.  The reason for this is that, for a large number of hits, the percentage fluctuation of hits in any one column (i.e. the relative size of the mountains and valleys) will go like 1/sqrt(NHits).  So the damage profile from a large number of big warhead hits should be more jagged than from a set of small warhead hits that do the same amount of total damage.  For example, imagine a ship with 100 columns of armor that is hit with with 10,000 str-1 hits or with 2,500 str-4 hits.  In the str-1 case, each column is likely to get 100 +/- 10 hits; in the str-4 case each will get 25 +/-5 str-4 hits.  The average damage per column is easy for the str-1 case - it's just 100.  For the str-4 case, it's 2*25 for the column being considered + 25 each spillover from each adjacent column, for a total of 100 (this is as it must be, since both cases did the same total damage).  The fluctuations in the str-4 case are very interesting though.  You need to add uncertainties in quadrature, so it sqrt(2^2+1^2+1^2)*5, or sqrt(6)*5, which is sqrt(1.5)*10.  In other words, the fluctuation in the penetration depth for str-4 warheads will be sqrt(1.5) larger than for str-1 warheads.  The reason that str-4 (stronger clustering) doesn't win in this regime is that the str-1 missiles can be fired 4 times as fast, and this affects the central (average) value, not just the fluctuations.

My gut feel is that if your warhead can penetrate with 2 or maybe 3 hits, then you're probably not in the sandblasting regime, i.e. large warheads are better because they penetrate (and cluster) better, but anything more than that and you're in sandblasting mode.

John

[Brian Neumann]

My gut feel is that if your warhead can penetrate with 2 or maybe 3 hits, then you're probably not in the sandblasting regime, i.e. large warheads are better because they penetrate (and cluster) better, but anything more than that and you're in sandblasting mode.

John

Another important factor is how big the crater is vs how large the target.  When you start getting craters that are around 1/3 of the total columns on the ship the chances of getting an early internal hit go up much faster than what the sandblasting will get you.  A quick example with a smaller ship (17 columns and 6 points of armor each) and size 16 warheads.  (Size 16 gets a maximum of 4 levels of penetration and damages at least 7 columns.)  By the fourth hit you will already have some damage overlaping.  On that fourth hit you have a small chance to get internals.  If you didn't then by the fifth you are pretty much garrunteed hitting somewhere that has already lost enough armor to have that hit get through.  In comparison the same total damage (64 or 80 points for 4 or 5 missiles respectivly.) from size 1 warheads will probably have penetrated the first 3-4 rows with about half of the next row being down as well.  You still need about 20 more hits before you are likely to get any internal.  Of course by the point that a sandplasting attack is getting internals the target is pretty much dead.  Flip this example around and have a much bigger target with 40 columns and it takes a lot more hits to start getting your clustering.  In effect you are having to partially sandblast the ship anyway before you get to the point that the larger warheads are actually helpfull.

Brian

[Theokrat]

Another important factor is how big the crater is vs how large the target.  When you start getting craters that are around 1/3 of the total columns on the ship the chances of getting an early internal hit go up much faster than what the sandblasting will get you.  A quick example with a smaller ship (17 columns and 6 points of armor each) and size 16 warheads.  (Size 16 gets a maximum of 4 levels of penetration and damages at least 7 columns.)  By the fourth hit you will already have some damage overlaping.  On that fourth hit you have a small chance to get internals.  If you didn't then by the fifth you are pretty much garrunteed hitting somewhere that has already lost enough armor to have that hit get through.  In comparison the same total damage (64 or 80 points for 4 or 5 missiles respectivly.) from size 1 warheads will probably have penetrated the first 3-4 rows with about half of the next row being down as well.  You still need about 20 more hits before you are likely to get any internal.  Of course by the point that a sandplasting attack is getting internals the target is pretty much dead.  Flip this example around and have a much bigger target with 40 columns and it takes a lot more hits to start getting your clustering.  In effect you are having to partially sandblast the ship anyway before you get to the point that the larger warheads are actually helpfull.

Brian

You are very much underestimating the effects of sandblasting. It is extremely unlikely that the small missiles will neatly only touch the outermost undamaged armour layers like you suggest.

I have run 100 simulations on the situation you described: Let an armour layout of 17 (width) by 6 (layers) be hit by 80 strength-1 explosions. The results are very different from your assertion that "You still need about 20 more hits before you are likely to get any internal".

Internal damage was dealt in _all_ 100 simulations. Indeed the most frequent case saw 6 points of internal damage (20 cases), and it went as far up as 13 internal damage (one case). The median was 6.5 internal damage.

[blue emu]

I've found that people generally have poor intuition regarding statistics and probabilities... even statisticians themselves often fall prey to this.

[Bunga]

I've found that people generally have poor intuition regarding statistics and probabilities. . .  even statisticians themselves often fall prey to this. 

You may be right, statistically speaking of course.

[TheDeadlyShoe]

Data point:  10,000 ton (41 armor column) survey cruiser with 4 layers of armor (and thus 160+points of armor) took 11 hits from strength 6 warheads. This is just 66 points of damage - less than half its total armor strength, and needing 3 hits on the same location to breach - yet it took 4 points of internal damage and has a 5-column wide total breach in its armor. 

[sloanjh]

You are very much underestimating the effects of sandblasting. It is extremely unlikely that the small missiles will neatly only touch the outermost undamaged armour layers like you suggest.

I have run 100 simulations on the situation you described: Let an armour layout of 17 (width) by 6 (layers) be hit by 80 strength-1 explosions. The results are very different from your assertion that "You still need about 20 more hits before you are likely to get any internal".

Internal damage was dealt in _all_ 100 simulations. Indeed the most frequent case saw 6 points of internal damage (20 cases), and it went as far up as 13 internal damage (one case). The median was 6.5 internal damage.

Did you try the same experiment with 20 str-4 and 5 str-16 explosions? How about 17x60 with 800 str-1, 200 str-4, or 50 str-16 (i.e. everything 10x)?  My expectation is that you should (on average) always see better effects (in terms of internal damage) from large warheads than from small, because of the larger relative fluctuations in column depth.

The reason that sandblasting wins is because the damage for str-1 warheads is done 4x as fast as for str-4 warheads (assuming same performance).  This just made an interesting idea occur to me - if you're limitted by ammunition stocks, i.e. you're going to run out of ammo before you run out of targets, then large warheads also have an advantage in this case.  Of course this will be offset if the target has any anti-missile defences.

John