Ship component balancing and math

why?  I am admittedly slow But

(keeping math simple) say a ship with 100 mass capacity and 10 components at 10 mass each

increase hull by 20%=120 capacity ; i can add 2 more components

vs

decrease component mass by 20% ; reduction of 2 eachx10 components= 20 less mas ;  once again i can add 2 more components

looks the same to me

With a 100 hull mass, increase it by 20% gives 120 hull mass → gives space for 12 components of mass 10.

The same 100 hull mass, decrease the component mass by 20% gives a mass of 8 → resulting in space for 12 (and a half) components of mass 8 .

No real difference so far. But now take 30% instead of 20%:

100 hull mass, increase it by 30% gives 130 hull mass → gives space for 13 components of mass 10.

100 hull mass, decrease the component mass by 30% gives 7 component mass → resulting in space for 14 (and a bit) components of mass 7. And we have got a difference of 1 extra component.

The larger the hull mass and the larger the percentage the more pronounced the difference becomes.

And it becomes really funny when the component mass is lowered by 100%, for then component mass becomes zero and you can mount an infinite number of them on your ship.  (This was a known bug, and should be fixed by the way).

Note:

It is a mathematical given that a negative percentage has a stronger influence than a positive percentage.

Proof:

Take an arbitrary number x increase it by a percentage p and then decrease it by the same percentage p and you end up with a result less (closer to zero) than your original number x. And it does not matter whether you first increase and then decrease or first decrease and then increase the result is the same.

example:

100 + 10% = 110 → 110 - 10% = 99 → 99 < 100
100 - 10% = 90 → 90 + 10% = 99 → 99 < 100

 

This is incorrect, or more specifically is only true for certain methods of stacking. A percentile penalty applied to the base value of an attribute is cancelled out entirely by a percentile bonus of equal magnitude which also applies to the base value of an attribute, and there is no reason why a percentile modifier cannot be applied to the base value (which results in additive stacking, which means that penalties and bonuses of equal magnitude cancel out exactly) rather than the current value (which results in multiplicative stacking, which means that penalties of equal magnitude as bonuses have greater influence than the bonuses). You can also use more unusual methods of stacking, but for many of those it is unreasonable to treat the modifier as a percentile modifier.

I would also point out that your proof is not exactly relevant to the topic of effective hull capacity; an average component capacity requirement modifier of -r over all the components on a given design does not modify effective hull capacity as (1 - r) but rather as 1 / (1 - r) = (1 + r') where r' > 0 for r < 1, which means that the effective hull capacity of a design having a base hull capacity of C with a direct hull capacity modifier of c and an average component capacity requirement modifier of -r has an effective hull capacity of C' = C * (1 + c) / (1 - r) = C * (1 + c) * (1 + r'); in most cases, c and r' will both be positive and so there is little need to concern yourself with how (1 + x) * (1 + y) behaves when sgn(x) = -sgn(y).

If you want to show that an average component capacity requirement reduction of -x is more powerful than a direct hull capacity bonus of x, you need to show that 1 / (1 - x) > 1 + x, which is true for x < 0 and 0 < x < 1. Proof:

Let f(x) = 1 / (1 - x) and g(x) = 1 + x. g(x) = f(x) gives 0 = x^2 so g(x) = f(x) at x = 0 and no other point; g(x) is continuous for all x, while f(x) has a discontinuity at x = 1 but is continuous on the intervals x < 1 and x > 1. Therefore, there are three intervals of interest - x < 0, 0 < x < 1, and x > 1.

For x > 1, f(x) < 0 and g(x) > 2, so for all x > 1, g(x) > f(x).

For x < 0, f(x) and g(x) are continuous functions which are not equal at any point. g(x) <= 0 for x <= -1 while f(x) > 0 for all x < 1, so f(x) > g(x) for x < 0.

For 0 < x < 1, f(x) and g(x) are continuous functions which are not equal at any point. f(x) approaches positive infinity as x approaches 1 from below while g(x) approaches 2 while x approaches 1 from below, so f(x) > g(x) on 0 < x < 1.

Granted, I skipped proving that f(x) is continuous on the intervals x < 1 and x > 1, and I skipped proving that g(x) is continuous for all x, but we shouldn't really need to do that for such simple functions as these.

That f(x) < g(x) for x > 1 doesn't really matter for two reasons. Firstly, the function given above for effective hull capacity only works under the assumption that the hull capacity required by the components on the ship is nonnegative, but x > 1 corresponds to a component capacity requirement reduction such that the net hull capacity requirement of the design is negative. Secondly, improving the average component capacity requirement reduction to the point where the average capacity required per component is negative isn't really any better than improving it to the point where the average capacity required per component is zero; in either case, you can fit infinitely many components on the hull, unless there's an underflow issue causing problems when you have too many of the components with negative hull capacities.

As Thecw notes, it doesn't really make much difference when the modifiers are low. However, let's consider end-game tech levels of hull capacity bonus and component capacity requirement reduction. Most component types can get a capacity requirement reduction of 60% or 70% by the time the tech tree is fully researched (or at least by the time their branch of the tech tree is fully researched). If the average capacity requirement reduction is 60% for all components on the design, this is effectively the same as +150% hull capacity; if the average capacity requirement reduction is 70%, this is effectively the same as +233% hull capacity. The maximum hull capacity bonus you can obtain from the tech tree is +70%; you can get another 20% from empire traits, about +50% from a Hyperion Shrinker (if you capture some from your opponents, they'll stack, so you could in theory get a much higher bonus from Hyperion Shrinkers), +5% per Helios Ore your empire controls, and +10% per time you took the Malevolent choice in the Design Revolution event (assuming it works; I'm pretty sure it did the last time I checked, but that was a while ago and there were some posts a little while back indicating that it does not). Assuming 1 Hyperion Shrinker, no Helios Ore, and no Design Revloution events, you can get a direct hull capacity bonus of around +120% (or around +140% if you have Dense 2). If you had to choose just one of these two ways of increasing effective hull capacity, which do you take?

Of course, these choices are not mutually exclusive - you can have significant component capacity requirement reductions and significant hull capacity bonuses simultaneously. They also stack multiplicatively, and so when you have both a +120% hull capacity bonus and a 60% reduction in average component capacity requirements you have a 450% increase in effective hull capacity.

Just throwing this out there, a well placed Shrinker can net 70-100% Bonus.

You allread got two good responses on this, let me illustrate a little further with a table. I will stay with the attack speed (AS) and attack cooldown (AC) example, but in principle it is the same for number of components and average component mass reduction respectively.

AC reduction -> equivalent AS increase

-10% -> +11% (I wouldn't write a post if the game was off by 1%)
-20% -> +25%
-30% -> +41% (probably throws of your balance calculations, but doesn't break the game)
-40% -> +67%
-50% -> +100%
-60% -> +150%
-70% -> +233%
-80% -> +400% (holy crap, this last 10% reduction gave over one hundred percent AS increase)
-90% -> +900%
-99% -> +9900%
-99.9% -> +99900%

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This also throws of the true value of buying a component.

If you buy -25% AC, you get + 33% AS
If you buy -25% AC, with rapid fire missiles researched you get +57% AS
If you buy -25% AC, with rapid fire researched and -50% ac allready installed, you get +1667% AS

So for the same cost you can get very different things, and if your +-% goes way beyond lets say 30%, you can't ignore that +% and -% are not the same thing.

 

Back to the topic of hull capacity

I changed the example, because mathing hull capacity is more invloved then attack speeds. Joeball allready introduced average component mass reduction (AMR), which is a variable that depends on what you put on your ship.

Example:
You want to equip a ship (100 capacity) with engines (20 mass each) and weapons (10 mass each).
With tech you reduced engine mass from 20 to 15 (-25%) and weapon mass from 10 to 5 (-50%).

Case A: you use 1 engine, 17 weapons

AMR = [15*(-25%) + 85*(-50%)]/100 = 46,25% (+ 86% effective hull capacity)

Case B: you use 3 engines, 11 weapons

AMR = [45*(-25%) + 55*(-50%)]/100 = 38,75% (+ 63% effective hull capacity)

And after all this we still ignored the production cost of the ship has gone up, so we will possibly have less ships/fleets...

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But on that note, I would consider discussions on maxing of effective hull capacity to be off topic. My real concerns were attack speed, possibly jamming and not working shield penetration.

Under ideal conditions, it's possible to get at least +110% hull capacity out of a single Hyperion Shrinker. The Krynn tech tree has a building which gives +4 levels to adjacent military improvements, and you can get another 9 levels between an Antimatter Power Plant, a Planetary Defense Dome, and an Elerium Defense Shield, after which I think you're stuck looking for +2 level improvements like Preparedness Centers or trade goods, and then you can round it all off by building the Hyperion Shrinker on a Caverns tile for the last 3 levels. It might be possible to do better than that, but I wouldn't know how without spending more time than I care to digging through the various tech trees to see exactly who gets what. I think the Thalan tech tree can manage +105% hull capacity from a Hyperion Shrinker, and I know that you can't get less than +100% hull capacity as the maximum bonus from a single Hyperion Shrinker under ideal circumstances since everyone gets Planetary Defense Domes (3 levels), Elerium Defense Shields (3 levels), Antimatter Power Plants (3 levels), and Preparedness Centers (2 levels), and it's possible to find a Cavern tile (3 levels) adjacent to a pair of military trade goods (2 levels each), though this is not necessarily likely.

Stacking the greatest possible number of levels on a Hyperion Shrinker that you can is however unnecessary to compete effectively with the computer, and it's quite easy to knock 4 or more levels off the maximum - missing Caverns drops the level by at least 1 (3 if you also miss out on Wastelands), not yet having Planetary Defense Domes drops the level by another 2 unless you have something that can replace the Planetary Defense System that will eventually upgrade into the Dome in the meantime and isn't good enough to already be in use elsewhere in the final ring of boosting structures, at least one more if you don't yet have the full ring terraformed to usability, etc. For estimating end-game maximum hull capacity bonuses, yes, I probably should have taken the Hyperion Shrinker to be worth +100% or so hull capacity, but +50% or so is probably more in line with what you'll get out of it most of the time in most games. There's also the possibility that for whatever you're doing maximizing the levels on a Hyperion Shrinker just isn't sufficiently valuable to justify spending a tile with a full ring around it, and it's also possible that the strategic resources necessary might be unavailable (no black holes = no antimatter, no nebulae = no elerium, and 6 levels that went into the maximum require you to have one of each of those, or you might for whatever reason place a higher value on the other things you can do with the elerium and antimatter).

There is one ship-only targeting component (a 0.25 which requires no thulium) and two fleet-wide targeting components (a 0.1 and a 0.25 which each require 1 thulium) as well as a specialization option available early in every tech tree which increases the accuracy of all weapons by 0.1. For 1 thulium invested in a fleet accuracy booster, a specialization tech available early in the game in every tech tree, and the single-ship accuracy booster, I can get an accuracy bonus of +0.6, which more or less counteracts the +0.65 evasion bonus you got for 1 thulium invested in a fleet booster, a tech that requires a specific tech tree to be taken (granted, it's a good tech tree), a two-point empire trait, and the single-ship evasion booster that doesn't use thulium. If you decide to counter with the other fleet evasion booster at the cost of another thulium, I can counter with the other fleet accuracy booster at the cost of another thulium, which leaves the situation unchanged. If you counter with the single-ship evasion booster that consumes thulium, I lose ground in expected hit chance but the thulium cost of your fleet shoots up significantly.

The tools necessary to counter even heavily evasion-focused empires with a reasonable degree of effectiveness are there, so I am not very inclined to agree that the attainable evasion values are actually a significant issue except in extreme cases (e.g. evasion-focused empire against an empire that dumped accuracy and uses mass drivers, or the side needing to counter the evasion-heavy empire has no real access to thulium while the evasion-heavy empire has enough for at least its primary fleets). The issue is getting the computer to use the tools available to counter your investment in evasion.

The Altarian tech tree also have 0.1 Jamming, the Thorians has +0.2. Just mentioning for completeness.Forgot about that, I usually take -10% mass which is a great pick, too, due to the way -% stacks. If we talk about traits, there is the +0.2/+0.1 accuracy boosting trait, too, ofc. However as a blind counter this is not a good choice ofc.

But I don't mean to argue about this here. Your analysis is essentially correct, there are proper counters to evasion stacking. It is not as bad as the AS stacking for missiles.

Problems I would point out here would be:

1) How bad is it uncountered?

Against missiles and let's say you took -10% capacity tech, in order to balance for the low dps/mass ratio of mssiles you get 0.9 acc - 0.65 evasion = 0.25 Hit chance = +200% effective hp
Even against Beams with the accuracy tech, jamming stacking still gets you: 1.1 acc - 0.65 evasion = 0.45 hit chance = +122% effective hp
If I used shields instead I would get, lets say two times 16 (about the same mass cost as the jammer on each ship), which would be 64 effective HP (against beam only!), that's 128% effective HP on a small ship (50 base hp), 64% on a medium ship (which you will most likely use), or 25.6% on a large one.

...come to think about this, I think this is the first argument, why I would pick ship hp over hull capcity as specialication, since it stacks better with jamming (+get an earlier shrinker, then I use to).

2) Making choices in general

If you had to choose between the jamming 20% and the hp 20% trait which would you choose?
Against an 100% acc enemy jamming 20% = +25% effective hp
With lower acc it is more. At higher acc you would be better off without it (but you still might lock your enemy out of some weapon choices).
Is there a scenario where you would pick -15%/-30% acc?

3) The AI... well, you can make it smarter so it counters properly, or you can balance the game better, so that the exploit doesn't nullify the AI. Both can be viable ways. If I got to choose, I would make the AI smarter, because I love how you can design ships the wreak entire fleets solo. I included this because this here primarily to make stardock aware, that the same problem as the AS-stacking is potentially within other mechanism, too. It is basicly everywhere in gal civ 3 (maintenance vs. income production, production cost vs. manufacturing) and I would like to know, that missile AS is only a slip and they are actually aware of where their mechanic can potentially cause problems.

Concerning the efforts to maximize the Hyperion Shrinker:  I use the Thalan tech tree and can get +2 from the Mother Hive and +3 from the Gaia Vortex.  It is interesting to see the variations available.  Good work by the devs!

updated op with two remarks:

- actual attack speed boost numbers for kinetics and beams
- remark that the Snathi tech tree can not counter jamming due to lack of the respectivee tech

Unless you turn off tech trading, you can usually pick up both the -10% component mass and the +0.1 accuracy techs. If you decide you need it, you can usually get it.

But yes, I agree that the accuracy bonus is not my first choice unless I happen to know I need it, or unless I've checked out what the neighbors have and can get the component mass reduction from them.

Yep. That scenario is called "handicapping myself."

As far as designing a faction not intended to handicap myself, though? No, I cannot see a good reason to take an accuracy penalty for additional points, not when penalties to soldiering, influence, income, manufacturing, research, and tourism provide plenty of points at relatively little cost. Invasion is almost an auto-win even with a soldiering penalty; the influence, income, manufacturing, and research penalties are all more or less -1 early structure of the appropriate type and so are easily counteracted (and if you spend the points gained on a food bonus, you more than get the income, manufacturing, and research back), and tourism applies to a number which can be and often is very small by comparison to standard income.

Hull capacity is a lot more versatile than HP is, and a fleet that combines defense-heavy escorts with offense-heavy capitals takes very little damage even without any evasion whatsoever. Increasing effective HP by stacking standard defenses and evasion is more than good enough without any need for additional HP to absorb what little damage leaks through, even with escort designs that use generalized defenses rather than defenses specialized for a particular opponent.

Also, if you happen to be playing with the Arcean tech tree, there are a couple of components that cost 5 + 5% hull capacity, a one-per-ship +20 and a repeatable +15, which don't require strategic resources. For that specific tech tree, or if you want to spend resources on Durantium Hull Reinforcement, you can probably convert hull capacity bonuses to HP bonuses of about the same magnitude as the direct HP bonus from specialization techs.

If you do nothing more to counter moderate-heavy evasion stacking than taking a single minor accuracy bonus, it should be bad; if it's not, the existence of the counters becomes pointless and evasion itself becomes useless. If your opponent invests strategic resources into their fleet and you do not, their fleet should be at an advantage over yours. Your scenario involves both moderately heavy stacking of evasion and investment of strategic resources into a fleet by only one side with the other doing nothing more to counter the evasion stacking than getting a single minor accuracy bonus.

I'd also point out that rate of fire boosters are also soft counters to high evasion, in addition to simply making the ships that have them better. For that matter, as it appears as though all weapons of a single damage type on a single ship hit or miss as a group rather than hitting or missing independently of whether the other weapons of the same damage type hit or miss, or at any rate that an evasion rate of 20% is a 20% chance of fully evading each salvo rather than a 20% chance of evading each individual shot, having multiple weapon types on the ship is also a soft counter to high evasion as it increases the number of attacks per time period, though as the attacks are individually less powerful it's not necessarily a straight improvement.

Which I actually allways do x) ... I like that aspect were you actually make a choice with consequences, where +range can lock you out of -component mass. But yes, another good point about why I might perceive some things differently.

Besides being quite a bit better then what their tooltip suggests, no, they are not a counter mechanic to evasion, they just make the ship better and are not different from just adding more weapons (assuming there is no overkill damage against the evasion ships).

Rate of fire boosters and mixing weapons are great once you have reached a certain amount of damage so their % bonus/cost is better then another weapon and if increasing your weapons would just be wasted damage e.g. your opponent only uses lots of 50 health smalls (carriers/ drone carriers).

But it is not a counter, it is just optimizing your dps and abalncing it against your ships durability, which you one should do anyways.

I can't really think of a better way to quantify and compare how jamming and other things behave uncountered right now (most counters are defense and not offence based).

 

I guess the minimum message of this thread would be:

#1 fix attack speed boosters for missiles

#2 make the AI counter jamming

#3 fix shield leech

The expanded message would include things that help the AI, due to being a more controllable model

#1 fix attack speed boosters of all types by actually using an attack speed variable and make associated threat values scale with the number of weapons affected

#2 make the AI counter jamming and make its fortitude scale reasonably with hit points, defences etc.

#2b consider moving to an evasion model that is linear in effective hp instead of the evasion effect

Let us say that in a period of time T you can fire a number of shots N which each have a probability p of hitting the target. You can compute an expected number of shots E(p, N) which will hit the target over the period of time T. If we modify the weapon in such a way that the probability of any one shot hitting the target is p' > p without affecting the rate of fire, we increase the expected number of hits over some time period T. If we modify the weapon so that the number of shots fired over T time is N' > N without affecting the probability that any individual shot will hit, we also increase the expected number of hits over some time period T. Rate of fire increases that do not impact accuracy are similar in effect to accuracy increases which do not impact rate of fire. If increasing accuracy is a counter to evasion, so is increasing rate of fire.

Consider a weapon which can fire once per time period T has a 50% chance of hitting the target per shot, and a weapon which fires twice as fast with the same chance of hitting on any given shot. Over any given time interval T, the weapon which fires more slowly has a 50% chance of hitting the target and a 50% chance of missing the target. Over the same time interval T, the weapon which fires faster has a 25% chance of doing nothing, a 50% chance of hitting once, and a 25% chance of hitting twice. The faster-firing weapon therefore has a 75% chance of damaging the target over a time interval T whereas the slower-firing weapon has only a 50% chance of dealing damage to the target over the same period.

Next, realize that the way the game handles weapon accuracy, adding more weapon components does not appear to increase the likelihood that any given salvo will hit. If you doubt this, consider that the probability that a salvo of 5 missiles which each have a 10% chance of missing the target will not produce any hits on the target is 0.001%. It is therefore highly unlikely that you would ever see a ship with 5 or more missile components ever miss its target if the probability that any given missile will hit is 90%. Missile salvos miss the target sufficiently often that it does not appear to be the case that the probability that any given missile will hit the target is 90%; rather, it would appear more likely that the probability that any given salvo will hit the target is 90%. With that being the case, increasing the number of weapon components on a ship very definitely does not have the same effect as increasing the overall rate of fire; rather, adding more weapon components looks like you're replacing the ship's armament with an increasingly-powerful single weapon.

Expected hits is not the variable you are looking for. It would be if you could kill a ship in one hit. In most situations, it does not. The only thing higher attackspeed at equal damage per second does, is reducing the uncertainty of the outcome. But the expected damage is the same. consider your second example, where weapon 1 fires once for 2 damage per T with a 50% hit chance and weapon two fires twice for 1 damage and a 50% hit chance. This results in:

Weapon 1: 50% chance to do 2 damage, 50% chance to do 0 damage. Expected damage: 1

Weapon 2: 25% chance to do 0 damage, 50% chance to do 1 damage, 25% chance to do 2 damage. Expected damage: 1

The only difference is that the spread around the expected damage becomes less for higher attack rate. It is the same for

The variable you are really interested in is expected time to kill a target T_K  or effective damage per second DPS_E. This is given by the targets effective hit points HP (jamming is considered in the dps and not in the hp, here) divided by the used weaponrys efective damage per second DPS_E. Your DPS_E is determined by the damage per shot D, the attack speed AS (D*AS = DPS) and the probability to hit P which in turn is determined as the difference of the accuracy ACC and the evasion EV. To summarize:

DPS_E = D*AS*(ACC-EV) = DPS*(ACC-EV)

and

T_K = HP/DPS_E

Therefore increasing the attack speed by 100% is equivalent to increasing the damage per shot by 100% regardless of evasion values. Since DPS is independent of EV it should allways be maximized anyways. Therefore it is not a reaction to evasion.

If (ACC-EV) is very low, it will be the determining factor in your dps, regardless of how high your DPS is.

It is the same bogus math used in Sins of a Solar Empire, which means it is almost certainly intentional.  Some things never change.

I find it hard to imagine, that they intended to put something that broken into the game (missile as booster stacking).

Is there something in sins, where it stacks up that badly?

Did I say that the theoretical DPS of the two weapons was the same? No, I did not. Does the game require the theoretical DPS of the two weapons to be the same? No, it does not.

Also, consider how attack speed, accuracy, and weapon damage affect expected DPS against a target which the weapon is not guaranteed to hit. Increasing attack speed increases the number of attacks, thereby increasing expected number of hits per time period. Increasing accuracy increases probability that a given attack will hit, thereby increasing expected number of hits per time period. Increasing weapon damage increases expected damage per hit but does not impact expected number of hits per time period. Increasing accuracy and increasing rate of fire improve expected average DPS in the same manner, a manner which is dissimilar to the way in which increasing attack damage improves expected average DPS. In other words, against a target which is not guaranteed to be hit by a weapon's attacks, increasing accuracy and increasing rate of fire do the same thing.

If we are talking about balancing the game then we either need to convert in the thalan tect tree the gai vortex and hive bonus to total manufacturing or give the non resource power plants bonuses to total base production. Also on the matrixes with the yor the adjancency bonuses need to work both ways or they become useless unless you build it on a island or conected to the capital.

I feel a little like you didn't read my previous post thoroughly, sorry if I was not clear enough, I will try to address things I can think of where you might ahve misunderstood me.

I am fully aware, that different weapons of the same type will be counted as one attack by the game.

Is there confusion as to my use of the term dps? I mean the literal meaning of dps: damage per second = damage per hit * hits per second (or what ever time interval suites you, it does not matter for our problem).

If you are only talking about the AS boosters that are in the game, then we are talking about different things. I talk about the fundamental mechanical impact of general AS and damage per hit increases and whether it is a mechanic that counters evasion. By that I mean it is a mechanic that makes more sense in a situation where there is lots of evasion in place. The answer to this is clearly no, because expected dps reacts in the same way to an +X% increase to either AS or damage per hit (see reply 13, but I also point it out in this answer again).

Overkill damage. I mentioned before, that I ignore that. There are some effects which are interesting in regard to overkill damage, but I ignore them for the sake of this discussion, because I assume, that ships will take at the very least 5 actual hits to die. Overkill effects are not that important in that scenario.

Other then that I really can't find where we might have misunderstood each other.

On to the point where we are talking about the actual weapon boosters in the game:

For every quantity of weapons there is a point were increasing attack speed is more efficient then adding one more weapon. This is independent of the evasion stat. Therefore it is not something you should change in your ship layout as a reaction to evasion. It is something you should do regardless on whether you face evasion or not. Therefore, still no counter.

How else would you compare the effect of increasing attack speed and increasing damage? Are you comparing two situations like this:

1) you have 4 weapons and add a 50% attack speed booster (+50% attack speed => +50% expected dps)
2) you have 4 weapons and add one more weapon (+25% damage per hit => +25% expected dps)

Of course AS booster will perform better then. And it even is, if there is no evasion! If this is what you do after seeing the enemy fielding evasion, you just played inefficiently before he did.

I pretty much exactly did that. It has all three influences on expected (I did mix up effective and expected in the quote, but from the math formula it should have been clear, that expected was meant) dps DPS_E: damage per hit D, attack speed AS, and accuracy ACC. It also clearly illustrates, that increasing attack speed is phenomenologicly not different in terms of expexted dps thenincreasing damage by the same amount. +X% AS or D both give +X% to DPS_E. ACC is the one that is different, because increasing ACC by +X% does not necessarily translate to +X% for DPS_E. if ACC-EV was 5% and you do +10% ACC you get +200% DPS_E. If ACC-EV was 90% before and you do +10% you get +11.1% DPS_E. If you do +25% in the last example you still only get +11% DPS_E.

On to hits per second.

This stat is meaningless without giving the associated damage values. 1 hit per 10 seconds that kills a ship is better then 10 hits per 10 second if those don't kill the ship. Hits per second doesn't have direct information about how a battle will go. You need the damage per hit information in order to know on whether you can kill something or not. Damage per hit and hits per second together give you: damage per second = dps.

 

The equation given is linear in damage per shot, rate of fire, accuracy, and evasion over the intervals where expected hit chance is between the minimum and maximum effective hit chance. The only difference is that when accuracy is taken to be the variable, the formula has a nonzero constant offset.

Also, neither [accuracy] - [evasion] = 0.9 => [accuracy] - [evasion] = 1 nor [accuracy] - [evasion] = 0.05 => [accuracy] - [evasion] = 0.15 is necessarily a 10% increase in accuracy; these are instead 10 percentage point increases in accuracy. It might be a 10% increase in accuracy, if initial accuracy was 1, but this is not guaranteed. When you compare a 10 percentage point increase in accuracy to a 10% increase in damage per shot or rate of fire, you are not comparing scenarios where you have modified the variable in question by the same relative magnitude. This is, quite simply, a flawed comparison.

You can also recognize that for [accuracy] - [evasion] >= 0.1 (the minimum hit chance defined in GalCiv3GlobalDefs) and [accuracy] + [change in accuracy] - [evasion] <= 1, [change in accuracy] = [change in hit chance]. As long as you're within the given interval, it is entirely valid to treat a change in absolute magnitude of a ship's accuracy as an equal change in the absolute magnitude of the ship's expected chance to hit a target, and a change in chance to hit of a given relative magnitude has exactly the same effect on expected DPS that a change in damage per shot or rate of fire of the same relative magnitude would have.

Most single-ship booster components cost roughly the same capacity as a single weapon. You would therefore want to compare a ship of N weapons without a booster to a ship of (N - 1) weapons with a booster, with N being determined by the ship and weapon in question (e.g. assuming no hull capacity bonuses or component capacity requirement reductions, a medium-hulled ship can fit five Disrupters, or four and a Rapid Recharger), or preferably you'd want to work out exactly what adding the booster component would cost you before doing the comparison rather than simply assuming that it'll cost you a weapon component. Fleet-wide rate of fire boosters are a bit more problematic to evaluate, as such a component would most likely be included on a support-role ship, but as support-role ships are rarely directly involved in the fighting they can usually have space available for fleet boosters at little direct cost, and unless this particular booster is the only reason that the support ship is included in the fleet it may not be directly possible to evaluate the component by comparison of N line ships to N - 1 line ships and a support ship. Moreover, even if the fleetwide booster is included on a line ship rather than a support ship and so comes at a more direct cost to damage per shot, you have to involve an assumption about the size of the fleet in which the ship with the booster component is included.

Good find, thanks.Ok, guilty as charged. I talk to Stardock about the meaning of percentages and then I am inconsistent in in my own definitions. Let's math again:

The duration of a ship shooting (i.e. from the point in time of it starting to shoot until it dies or the battle ends) is denoted T (ignore range).
We will judge the value of a ship by the the total damage DT caused by its components over the time T (ignore fleet boosters, ignore logistics cost, ignore production cost, etc.). It is given by:

DT = DPS_E * T

where DPS_E is the average damage per second. It is given by the function:

DPS_E (D,AS,ACC,EV) = D*AS*(ACC-EV)

where D = d*Nd is the damage per salvo from Nd weapons with d damage per hit, AS is the attack speed, ACC is the accuracy value in percentage points and EV is the evasion value in percentage points. ACC-EV is bounded from below by 0.1 and from above by 1.0.

We will now consider situations where T is constant (i.e. no change in defensive components etc. and DPS_E is proportional to DT by which we judge the value of a ship). We want to investigate the EV dependence of changing only one of the modifiers ACC, D and AS by calculating the change of DPS_E:

DeltaDPS_E = DPS_E [new] - DPS_E [old]

with:
DPS_E [old] = DPS_E ( D = D_0 , AS = AS_0 , ACC = ACC_0 , EV )
and DPS_E [new] defined according to the argument investigated.

 

Case I) Damage

DPS_E [new] is given by DPS_E ( D = D_0 + DeltaD , AS = 0, ACC = ACC_0 , EV )

therefore we get:

DeltaDPS_E,D = DeltaD*AS_0*(ACC_0-EV)

which is a function of EV.

 

Case II) Attack speed

DPS_E [new] is given by DPS_E ( D = D_0 , AS = AS_0 + DeltaAS , ACC = ACC_0 , EV )

(footnote [1]) therefore we get:

DeltaDPS_E,AS = D*DeltaAS*(ACC_0-EV)

which is a function of EV.

 

Case III) Accuracy modifier

DPS_E [new] is given by DPS_E ( D = D_0 , AS = 0 , ACC = ACC_0 + DeltaACC , EV)

therefore we get:

DeltaDPS_E,ACC = D*DeltaACC

which is not a function of EV.

 

Influence of EV:

High values of EV will reduce both DeltaDPS_E,D and DeltaDPS_E,AS linearily but leave DeltaDPS_E,ACC unchanged.
For given modifiers DeltaD and DeltaACC and (ACC-EV) in the interval [0.1 , 1.0] there is a value of EV_0 at which DeltaDPS_E,ACC = DeltaDPS_E,D, it is given by:

DeltaDPS_E,ACC = DeltaDPS_E,D (EV_0)

D*DeltaACC = DeltaD*AS_0*(ACC_0-EV_0)

EV_0 = ACC_0 - D*DeltaACC/DeltaD/AS_0

Below that value DeltaDPS_E,D > DeltaDPS_E,ACC and above it DeltaDPS_E,D < DeltaDPS_E,ACC.
The argument for AS is analogous.

If one now determines whether when DeltaDPS_E,AS is the preferable choice over DeltaDPS_E,D, one will solve:

DeltaDPS_E,D < DeltaDPS_E,AS

DeltaD*AS_0*(ACC_0 - EV) < DeltaAS*D_0*(ACC - EV)     | use (ACC_0 - EV) =/= 0

DeltaD*AS_0 < DeltaAS*D_0                                         | use AS_0 , D_0 =/= 0

DeltaD/D_0 < DeltaAS/AS_0

Therefore determining whether an AS change or a damage change is preferable has nothing to do with evasion.

Conclusion:
It therefore should not be considered an action you take once you recognize you have to counter evasion. Therefor it is not a counter.

Above calculation does exactly that with N = Nd + n where n is determined by the value of DeltaD = n*d.

 

[1] I am ignoring here, that the actual ingame math actually rather works like AS_0/(1-DeltaAS) then AS_0+DeltaAS, this is unconsequential, since there is an DeltaAS' that corresponds to the change for the real ingame math. The AS values should therefore not be confused with AS modifiers in the game.

I have an experiment for you to perform. Assume you have an incomplete design which, with the parts currently on it, can destroy a given target in some number of shots (say, 10) and fires at some given rate (e.g. 1 shot per second). You can complete this design by increasing damage per shot by 25% (10 shots to kill => 8 shots to kill), increasing rate of fire by 25% (1 shot per second => 1.25 shots per second), or by increasing effective hit chance by 25% (e.g. 40% chance to hit => 50% chance to hit). I think you will agree that in all three cases we have a constant expected time-average DPS for all three cases, at least until we start looking at hit chances over 80% for the high-damage and high-RoF weapons. Plot kill probability against total engagement time for each of these three scenarios for a variety of hit chances. Does an increase in rate of fire look more like an increase in accuracy or an increase in damage per shot? As hit chance decreases, does increasing rate of fire look more like increasing hit probability by the same relative magnitude, or more like increasing damage per shot by the same relative magnitude?

If, after you do this, you still disagree with me, then I can only say that we have irreconcilable viewpoints on this matter.

So you are giving me homework asignments now? Thank you so much! [/sarcasm]

If you knew the answer why didn't you just upload your plots and explain what you did? If I didn't believe you I could have followed your derivation. (Yeah I did it, but since you didnt bother to upload an imgage, I won't either).

Interesting, but irrelevant?

Look, the AS distribution converging against the Acc distribution at low p for relative changes is nice and all... and the difference between the damage and AS distribution is the biggest then, but that is still just a marginal difference. Heck you could even argue, to prefer damage boost over AS bosst based on that picture, when your ships have short life times.

But your ingame decision will allways be (based on effective dps):

Damage: absolute change in dps*effective hit chance (ACC-EV)
Attack speed: absolute change in dps*effective hit chance
Accuracy: absolute change to effective hit chance

The decision of damage vs AS is trivial in 99% of cases (maximize expected dps). Sure at the point where adding one weapon and adding AS, you will have different kill time distributions, but that simply means that you get a higher spread in results for more weapons. The expectation value remains the same.

The decision between more dps (damage or AS, depending on what maximizes your dps) and accuracy however, is intereting. You want to ask what the enemies evasion is. And that is why it is a counter, because your answer will be different depending on how much evasion the enemy has. Because at high evasion your absolute accuracy bonus translates to a relative change in hit chance of 100% and more, while at low evasion values it will have comparable results will be lower relatve changes in hit chance. At these low evasion values you profit more from changing your dps.

I don't know how else to explain this.

Let us say you have two options which both produce the same expected to kill a given target, one of which has a high variance in expected time to kill and the other of which has a low variance in expected time to kill. Do you take the high variance option, or the low variance option? If you're smart, you'll take the low variance option in most circumstances, because consistently acceptable performance is safer than occasionally spectacular but also occasionally dreadful performance, especially when you're dealing with large numbers of events; you may well get lucky some of the time, and not be too unlucky some of the rest of the time, but it is highly improbable that you will be lucky all of the time. Consistent performance is better than variable performance unless you're so badly outclassed that you need to get lucky in order to win or you so badly outclass your opponent that it would take enormously bad luck for you to lose.

A thing which reduces the impact of a random variable on outcome can be said to be a counter to the cause of that random variable. Rate of fire reduces the impact of hit rate on the variance in expected time to kill, which reduces the impact of evasion. It may not be as good a counter to evasion as increasing accuracy, but it is a counter.

This is a common mistake.

flip a quarter and the odds are 50/50 that you get heads or tails.

flip a quarter twice and the odds are 50/50 each flip

flip a quarter 10 times and the odds are still 50/50 each flip

the odds of getting a given result on a coin flip are not influenced by the result of the prior coin flip.... you could get any combination between 5h:5t to 10h:0t or 0h:10t

 

There is a term for it, but I don't recall the name

Can we quantify this somehow?

I used the example above, reduced the number of kill shots to 5 and 4 for damage boost (to make the effect of the variance more prominent) and calculated the probability of killing the target before the expected kill time. Below hit chances of 0.6 these probabilities are within 1% or 2% (absolute probability) of each other for damage and as and below 0.3 hitchance they acc is also in that range.

Interestingly enough, acc is allways the variable with highest the highest probability to kill before the expected kill time (but with no chance to for early kills, since its minimum kill time is 25% higher).

If you take the probability at 75% expected kill time, the damage boost usually has the highest probability (by a few %).