Yes, more math. This time: Basic statistics!

After I created my formula to predict the outcomes of battles, I decided it was much more practical to come up with a way to predict the efficiency of weapons.

I decided the best way to do this was to begin by ranking the weapons by the things that make the weapon efficient: battery usage, damage, velocity, rate of fire, its weight, its auto-aim, and its accuracy. For the following rankings, I’ll disregard the auto-aim ranking and only rank weapons that are accurate, non-unique (Neut III is not unique) small port, energy blasters (such as the flechette, rail, and gatlings). While this seems limiting, it actually encompasses most of the weapons used.

The percentage listed next to the actual statistic is the percentage of that statistic of the average for the statistic group. That way I can find whether a weapon is above average or below average, and by how much. When I have calculated the percentages for all the weapons, I will find the average percentage (hehe, funny term) and rank the weapons overall. If a lower number is better, it is calculated stat/avg. If a higher number is better, it is the reciprocal.

Small Weapons

Blaster Ranking:

Battery Usage:

Average- 11.12222

1. Phase Blaster MKII (7.5)(67.4%)
2. Phase Blaster (8)(71.9%)
3. Ion Blaster MKIII (9.8)(88.1%)
4. Ion Blaster MKII (9.8)(88.1)
5. Ion Blaster (10)(89.9%)
6. Neutron Blaster MKII (11)(98.9%)
7. Neutron Blaster (12)(107.9%)
8. Neutron Blaster MKIII (14)(125.9%)
9. Positron Blaster (18)(161.8%)

Damage:

Average- 492.222222

1. Positron Blaster (800)(61.5%)
2. Neutron Blaster MKIII (600)(82.0%)
3. Neutron Blaster MKII (600)(82.0%)
4. Neutron Blaster (600)(82.0%)
5. Ion Blaster MKIII (380)(129.5%)
6. Phase Blaster MKII (380)(129.5%)
7. Ion Blaster MKII (370)(133.0%)
8. Ion Blaster (350)(140.6%)
9. Phase Blaster (350)(140.6%)

Velocity:

Average- 168.888888

1. Neutron Blaster MKIII (215)(78.6%)
2. Neutron Blaster MKII (185)(91.3%)
3. Neutron Blaster (180)(93.8%)
4. Positron Blaster (180)(93.8%)
5. Phase Blaster MKII (165)(102.4%)
6. Phase Blaster (160)(105.6%)
7. Ion Blaster MKIII(150)(112.6%)
8. Ion Blaster MKII (145)(116.5%)
9. Ion Blaster (140)(120.6%)

Delay:

Average- .1677777

1. Neutron Blaster MKIII (.14)(83.4%)
2. Neutron Blaster MKII (.14)(83.4%)
3. Ion Blaster MKIII (.14)(83.4%)
4. Ion Blaster MKII (.14)(83.4%)
5. Ion Blaster (.15)(89.4%)
6. Neutron Blaster (.15)(89.4%)
7. Phase Blaster MKII (.2)(119.2%)
8. Phase Blaster (.2)(119.2%)
9. Positron Blaster (.25)(149.0%)

Mass:

Average- 288.88888

1. Phase Blaster MKII (100)(34.6%)
2. Phase Blaster (100)(34.6%)
3. Ion Blaster MKIII (100)(34.6%)
4. Ion Blaster MKII (100)(34.6%)
5. Ion Blaster (100)(34.6%)
6. Neutron Blaster MKIII (400)(138.5%)
7. Positron Blaster (500)(173.0%)
8. Neutron Blaster MKII (600)(207.7%)
9. Neutron Blaster (600)(207.7%)

Overall (Brute Ranking):

1. Ion Blaster MKIII (89.64)
2. Phase Blaster MKII (90.62)
3. Ion Blaster MKII (91.12)
4. Phase Blaster (94.38)
5. Ion Blaster (95.02)
6. Neutron Blaster MKIII (101.68)
7. Neutron Blaster MKII (112.52)
8. Neutron Blaster (116.16)
9. Positron Blaster (127.82)

That’s right! In the brute ranking, meaning no weighting any of the statistics, the Ion Blaster MKIII tops the list followed by all the blasters that you used to bot as a n00b. However, we all know that this cannot be the fact in-game (or do we?). This is because certain statistics are more important when fighting than others. For instance, battery drain to a certain point does not matter. Also, most of the changes in the high velocities are not important. I will follow up this post with another post in which I will present the weighted equation to find the ranking of a weapon.

This is a finding, but I don't like it. Please someone prove my math wrong!

The velocity pretty much determines hit rate, and that's why phase blasters are usually not used in PvP... for botting, if you can get in a good flow, phase blasters are the best ever.
TPG sparrow phase and orion XGX are even better, and are the most efficient weapons, but even at 170 speed of orion, it's not easy to use.

I think you need some refinements to your model. Here's what I came up with.

Let body A be player A's ship and body B be player B's ship. Let d be the distance between A and B. Player A shoots a projectile P with velocity v towards the center of B, compensating for lateral velocity differences (auto-aim).

Player B immediately begins to accelerate his ship in order to dodge. We will call the acceleration orthogonal to the shot's velocity the dodging acceleration, a. I submit that a will equal the engine thrust times the player's dodging skill divided by the ship's mass but there might be other factors involved, like the reaction delay. Nevertheless, a is the resultant acceleration.

The position of B now differs from the anticipated position of B by ½ a t².

Now let t (time, obviously) be equal to the time the shot was predicted to hit B, which is obviously d/v (technically, (d-r)/v where r is the radius of the ship along the axis of A-B, but r is taken to be small enough to not matter overly much). Thus the position of B differs by ½ a d²/v² at ground zero once the projectile arrives.

Now we want to find a probability of a hit. Let's use a modified normal distribution as an approximation. The standard deviation, σ, shall be the cross-sectional length of body B along the axis of dodging action taken. This is roughly equal to the length of the ship, probably times some balancing constant. Let's say L is the length of the ship.

Let N(x,σ) be the modified normal distribution (unit normal distribution?)
N(x,σ) = e^(-x² / (2σ)²)

Thus the probability of a hit is N(a d²/2v², γ L) where γ is the aforementioned balancing constant. You can do the substitution from there if you want.

Now if we want to simply gauge weapon effectiveness, we can just plug in typical values for d, a, and σ and it's obvious that the velocity factor is extremely important in determining whether there was a hit and it's not just a simple multiplier - twice as much acceleration is much better than twice as effective.

Edit: I've changed 'D' to 'd' and 'p' to 'P' as suggested and added γ as a balancing constant which can be fit to data.

Ok, how the hell did you get them fancy greek symbols into the textbox? Jellous.

For consistency, it should probably be projectile P, and distance d. I would also submit that network delay is a much more significant factor for a than reaction time (poor modem users). It also affects the (preceived) position of B at t0 when A fires its projectile.

Another interesting feature is the dmg/sec considering the first shot to be free:

for example,

in the first 0.4 secs, the N3 does 2400 dmg
while the Gauss does 2200 dmg, making them roughly equal

This is useful, since it shows that certain weps rely heavily on how long you can keep the stream on them. Which is why gauss can be so good.

@a1k0n- Wow, great model. I was planning on proceeding to something like that, but since you posted it, I guess I won't =p. But I'll try to use your formula to come up with the importance of all weapon stats, (as you did at the end of your post) and formulate them into a simple weighted equation for weapon 'goodness'.

@andreas- you're right, the network delay is a problem. However, there's no way to factor in network delay (i don't think), as it varies so often. The lag in a sector also depends on things such as the amount of 'stuff' in the sector.

@shape- the dmg/sec is an important stat, but can't be used alone to determine the 'goodness' of weapons. In you're example, it won't matter as much if you miss two n3 shots instead of two Gauss shots. Also, gauss shots have such a low velocity and a slow delay, they are much easier to dodge, and it's much harder to land a string of guass shots than to land a string of N3 shots.

What I believe shapes point was: There is no "stream" of gauss shots, there are only short bursts, due to the nature of the weapon. And then the dps measure is far less of an issue.

other thing that needs to be though of when you measure "difficulty" of landing a stream is the mass of the weapon. Give a Gauss the same mass as the Neutron III and you'll have far more gauss ingame and far more deaths to it. And it will be harder to dodge because the pilot -shooting- is more manouverable.

Andreas, fancy greek symbols (funny, I first read that as "fancy geek symbols") are done with ampersand codes.
http://www.cs.tut.fi/~jkorpela/html/guide/entities.html is the most complete reference I've found.

ΑΒΓΔαβγδ

True, it should be projectile P and distance d. It doesn't matter what the projectile is called since I never use it. I'll correct the 'D'.

Network delay and such also just kind of factors into reaction time - there is still a net resultant acceleration but I didn't bother to allow for a time offset, but it would be pretty easy to add to the model (½a (t-α)²).

Also, after posting I realized that it'd be better to come up with a typical dummy ship to plug in for a, d, and σ instead of using 1s.

Your point about the perceived position of B is good, though (not that your other points aren't!). I was doing this purely from a weapon effectiveness point of view assuming perfect aim and skill on the part of player A, but the model could be fixed to allow for uncertainty in B's position (which would lower the peak probability and spread out the distribution a bit).

In order to get a number for weapon 'goodness', we need to use the derived probability of a hit and then apply it toward hitting people with it repeatedly. Damage per second is important, but only when taken together with probability of a hit. Gauss has a far lower probability of hit (although it tends to be higher in practice due to auto-aim radius, which I didn't consider here - I consider the weapon to always be autoaiming in the probability model).

So let p = probability of hit (defined in a previous post), H = hitpoints of damage done per hit, T = period of firing (AKA delay between shots). The damage done to a player per unit time is thus p H/T.

H/T may be roughly equal for a Gauss Mk II and a Neut Mk III, but the p is vastly better for the Neut Mk III. Anyone want to run some numbers?

Edit: thinking some more, assuming perfect aim is only half of the problem of weapon goodness. Factors like mass, autoaim radius, and lag in perceived position of B will also need to affect the probability of a hit and need to be considered. Well, at least it's a start. I'll post something later if I think of a good way to include those factors.

/vote mute "a1k0n" =D

I always thought the best reference was the char map viewer, you know the one =D Gnome has one, WinBlows has one, as far as i know MAC has one, even KDE has one =D

I think i agree with myself when i say this thread should be locked =D

a1k0n, yah it the N3 catches up in DMG/sec, my only point was that these weapons dmg calculations are less like a straight rate, and more like:

let
Damage Due to 1 shot = BaseDMG
firing time=t
delay=d

then dmg dealt is going to be:

Damage= BaseDMG+ ((t/d)*BaseDMG)

(where, technically, we should round t/d down)

In the limit t->inf

DMG per sec-> BaseDMG/d

otherwise

DMG per sec-> (BaseDMG/t)+(BaseDMG/d)

Ah, I see what you're saying and I follow your math. I'm comfortable with taking t → ∞ for now, but when we factor in ship B's hit points, then we will definitely need your updated damage/time equation. Plus the energy usage becomes a factor.

I'm also not wedded to the idea of the normal distribution over the cross-sectional length of B's ship, but we can use it for the uncertainty in B's position instead. But assuming perfect information about B's position: if you're autoaiming and your target is simply unable to maneuver due to his high mass/low thrust, then the probability that you hit him is 1, and if he is able to maneuver and he is skilled enough to do so, then the probability is 0.

Also of note is that a skilled pilots A and B can vary the apparent velocity v of the projectile by accelerating toward or away from one another.

My head is spinning too, but it is fun to read this anyway :)

I have one question about auto aim. Is the autoaim carried out by the server or the client? I am wondering if lag/bandwith problems are eliminated or by autoaim?

@Othmaar: autoaim is in the client

Other things to consider is that the probabiltiy of a hit goes down the more hits you've had. both due to evasive manouvers of your target, And the fact that you yourself have to keep manouvering to track your target/dodge.

This isn't a big problem in a light ship, but heavier ones get far more problems due to their own relatively uncertain position.