Gun Points and Impact Armor

[noble_resonance]

TLDR
Large guns are more effective at targeting destroyers due to the chance of a good roll, while large heavily armored ships are best targeted by large batteries of small-to-medium guns.  You can get a decent idea of how much firepower a gun has by adding the crew and gun dice together to get a gun point score and this will be an accurate measure, even if the dice are different (1d8+1d4 is roughly equivalent to 2d6).

Abstract
Due to the fact that different combinations of two dice with the same sum of their sides have similar odds of rolling equal to or greater than a given value we can examine weapon attacks using a single value.  By examining the probabilities of dice sums against a range of target values we arrive at a discernible doctrine for the use of different weapons against different kinds of ships.  Namely, large guns are more effective at attacking destroyers, which require fewer hits but are harder to strike; while batteries of small guns are more effective at attacking larger ships which are easy to strike but require lots of hits.

Segregation of Roles

In the discussion on effective armor the general equation for an attack was stated as

D_g+D_c+(D_p1+D_p2) >= B_n

where D_g is the result of Gun Die, D_c is the result of the crew die, B_n is the breach number, and D_p1 and D_p2 are the results of the profile dice.

Two ships contribute to each attack roll.  In order to better understand the interactions between ships, it is convenient to separate the terms by which ship is referenced when considering the effects of different choices on the result of the attack roll.  This turns the relation into:

D_g+D_c >= B_n-(D_p1+D_p2)

The left hand side of this is referred to as the attack term.  The right hand side of this is the defense term.

Dice Odds: Achieving a Value or More on (A)d(Y)

Both sides of our relation involve the rolling of one or two dice.  Since we do not know the result of any roll before the roll occurs what is of real interest to the player is the odds of the condition being true or false.  Hence a short discussion on dice probability.

I will be using standard dice notation for this discussion.
http://en.wikipedia.org/wiki/Dice_notation

The odds of rolling a value z where z is greater than y on a die is impossible, so probability 0.

The odds of rolling a value z or greater on a die with y sides where z<=y is equal to (y-z)/y.

Example: The odds of rolling 3 or more on 1d8 is 5/8 (or 62.5%).

The odds of rolling a value z or greater on a  2 dice with sides y₁ and y₂ is more complicated, as it involves counting all the different possible combinations of those two dice and dividing by the number of combinations that exceed z by the total number of combinations.

Example: There is only one combination on 2d6 that results in a 12 (both dice come up 6) out of total of 36 possible results.  The odds are therefore 1/36, or 2.8%.

None of this is particularly surprising to anyone who is vaguely familiar with dice probabilities, but the next point is the important one.

The probability of two dice rolling a value z or greater is directly proportional to the sum of the sides.  In other words, 1d4+1d8 have a similar chance of rolling 7 or more as 2d6.

This correlation is not perfect, there is about a 1% or 2% difference between combinations of dice with the same sum of sides, with slightly higher probabilities being assigned to pairs of dice whose sums are similar (2d6 is about 1% to 2% higher than 1d8+1d4).  However, this relation can be used as a basis for an analysis of different gun and crew combinations.

Gun Points and Impact Armor: Definitions

Earlier, the right hand side of the attack equation was defined as the defense term.  Impact Armor is simply the specific result of the defense term as it applies to any given gun role.  Imagine that instead of having the attacking player roll all the dice as per the rules, the defending player roles the profile dice and the attacking player roles the gun and crew dice.  Then they compare to determine the result.  Impact Armor is simply the result of the defenders roll.

Gun Points is also simple; an attack has a number of gun points equal to the sum of the dice sides involved.  This is useful because when we want to know the chance of a particular attack beating a certain impact armor, we need only know how many gun points the attacker has.

Gun Points and Impact Armor: Implications

The following table shows the chance for an attack with a certain number of gun points to breach  impact armors between 5 and 14, a fairly representative range for the ships currently published.  Note that the current maximum of 18 gun points is derived from the potentially incorrect assumption that the best attack in the game is a d6 crew firing a d12 gun.

Gun points are horizontal, impact armors are vertical.

	8	10	12	14	16	18
5	0.63	0.75	0.83	0.88	0.9	0.92
6	0.38	0.58	0.72	0.79	0.83	0.86
7	0.19	0.42	0.58	0.69	0.75	0.79
8	0.06	0.25	0.42	0.56	0.65	0.71
9	0	0.13	0.28	0.44	0.55	0.63
10	0	0.04	0.17	0.31	0.45	0.54
11	0	0	0.08	0.21	0.35	0.46
12	0	0	0.03	0.13	0.25	0.38
13	0	0	0	0.06	0.17	0.29
14	0	0	0	0.02	0.1	0.21

There are some important observations about this table.  The first is that the odds of penetrating an impact armor higher than 10 is fairly low even for high gun point attacks, topping out around 50% .
Another is that there are diminishing returns to attacking a low impact armor with a high gun point attack: For an impact armor of 5 going from 8 to 10 gun points is a 12% increase, while going from 10 to 12 gun points is only an 8% increase.

For reference most ships have an effective armor of around 7.
http://forums.monstersinthesky.com/index.php/topic,194.0.html

Weapon Selection and Role

Given the above table, the next step is to examine what kind of assaults would be most successful at destroying or disable different kinds of vessels.  Since impact armors will follow a distribution based on the target this is the driving force behind what a weapon's role is. 

The chance of a breach drops significantly as impact armor rises so the idea that one can "punch through" heavy impact armor with a high gun point attack is a risky tactic at best.  If your opponent rolls well on his profile dice (i.e. scores 10 or more impact armor), even high power 14-18 gun point attacks stands at best a 50/50 chance of penetration.  On the inverse, even low gun point attacks stand a decent chance at penetrating low impact armors of 6 or lower.  On the upside, only  very lucky armored cruisers or light cruisers have any chance of realistically maintaining impact armors that high.  Still, a good roll by your opponent on his profile dice has a high chance of deflecting any shot.

All is not lost for the heavy hitters, however.   As breach numbers start to fall, the chance of defeating the armor of a target using just the attack term starts to rise.  This gives large guns a chance to score a critical success against targets that have low breach numbers.

If we work off the assumption that larger guns are more valuable than smaller guns and that we have a choice between the option of mounting fewer larger guns or more smaller guns we have a clear dynamic.  Use large numbers of smaller guns to eliminate ships with high breach numbers but low effective armor while deploying large caliber guns to eliminate ships with high effective armor but low breach numbers.

Firepower Score: Point Balancing using Gun Points and Firing Arcs

The most useful aspect of this discussion is that it provides an easy way to determine the relative value of a vessel's attack.  For any given firing arc total the number of gun points that vessel has mounted in that arc.  Then total the number of gun points that can project into that arc from an adjacent arc and multiply by some constant between 0 and 1.  Repeat this process for all four arcs to arrive at relatively accurate value for the firepower of a vessel.

Formally, the firepower score of a left/right symmetric vessel would look like:

F₀+F₁+F₂+F₃

where F_x is calculated as

G_x+C_t*(G_(x-1)t+G_(x+1)t)

if x=0 is fore, x=1 is starboard, and so on.  G_x is all guns mounted in an arc, and G_xt is all turreted guns mounted in an arc.  C_t is some constant between 0 and 1, with 0.25 a likely candidate to compensate for situations where a vessel cannot fire a turreted gun into multiple arcs simultaneously.  This is very similar to what is already being done in the Alpha version of the points document, although the values given in that document make larger guns more point efficient.

[Author's Note: I know this doesn't take into account range.  I'm still working on this.]