mathematical PvP analysis

combatx

No doubt, a common excuse in PvP is that party A outnumbers party B, and that is why party B lost. So in this article I will try to answer the question: "how big of an advantage is an extra player, really?"

Now, this is impossible to do without writing an 80-page thesis and taking into account way too many variables, so I will make two huge assumptions:

- Each player in party A and player B have the same amount of experience and 'skill'
- Each player has the same gear/equips

Very unrealistic, right? Thats what happens when you cut corners. But it does prove useful when you want to assume that there is no 'gear' advantage in either direction. Skip to the end if you want to read my findings, otherwise I will justify the process below:

note: If you have taken an intro to calculus, and maybe even an intro to differential equations course everything should be relatively easy to follow

Let f(t) = the number of players standing with respect to time for party 1
Let g(t) = the number of players standing wtih respect to time for party 2

Suppose party 1 has 'u' number of participants
Suppose party 2 has 'v' number of participants

Then the number of participants of party 1 before the battle starts (at time, t = 0) is u. In other words, f(0) = u.
Similarily, the number of participants of party 2 before the battle starts (at time, t = 0) is v. In other words, g(0) = v.

Also, the rate at which party 1 dies is directly related to the amount of participants party 2 has. Let k = average DPS per player. Then we can say that the longevity of parties 1 and 2 can be modelled by the following differential equations:

f'(t) = the rate at which party 1 dies, is
f'(t) = -k*g(t)

Similarily,
g'(t) = -k*f(t)


Furthermore, if we take the second derivative of f(t), we get:

f''(t) = f'(f'(t))
f''(t) = f'(-k*g(t))
f''(t) = -k(-k*f(t))
f''(t) = (k^2)*f(t)

Similarily,

g''(t) = (k^2)*g(t)


We notice that functions f(t) and g(t) behave similar to the function e^x when the derivative is taken. With some critical thinking we can deduce that functions f and g are simply linear combinations of e^x and e^-x.

Since the average dps does not matter as I assumed that both parties have the same average dps, i will exclude them from my calculations.

Thus, we get that the number of players still standing in Party 1 is simply:

f(t) = A*e^t + B*e^-t (for some constants A and
and the derivative f'(t):
f'(t) = A*e^t - B*e^t

We know f'(t) = -g(t)

Therefore, g(t) = B*e^-t - A*e^t


Since we know that g(0) = v, and f(0) = u,

Then from the above equations (e^0 and e^-0 = 1), we get:

u = A + B
v = B - A

If we solve this system of linear equations we get:

A = (u-v)/2
B = (u+v)/2

If we try to find the time at which party 2 is dead, we want to find the t such that g(t) = 0

Therefore, taking all the equations from above:

0 = g(t)
0 = (u+v)(e^-t)/2 - (u-v)(e^t)/2
u + v = (u - v)(e^2t)
thus

e^t = sqrt[(u+v)/(u-v)]

Finally, using this information, we can now solve for the relationship using f(t)

f(t) = (u-v)(e^t)/2 + (u+v)(e^-t)/2

f(t) = (u-v)sqrt[(u+v)/(u-v)]/2 + (u+v)/2sqrt[(u+v)/(u-v)]

f(t) = sqrt(u^2-v^2)/2 + sqrt(u^2-v^2)/2

f(t) = sqrt(u^2 - v^2)

Which is, surprisingly, the very same formula we learned in grade 5 (pythagorean theorem).

TL;DR -- Summary:

The advantage of one party fighting another party can be given by sqrt(u^2 - v^2) where u = the number of players in party 1 and v = the number of players in party 2.

This means if party 1 has 5 players, and party 2 has 4 players, then party 1 will have 3 players left standing when all of party 2 is dead.

Party 1= 13, Party 2 = 12, then Party 1 will win with 5 left standing.

As the number gets bigger, the advantage diminishes (as it should).

The only real application I can see this having, is if you there is a war between two parties that is very evenly matched, supose 4 vs 4. If Party 1 brings in an extra member which has the same relative skill as all the other members, then this relationship starts coming into play.

Note: I am aware of some of the mathematical problems, such as the rate of damage decreasing as the health drops, and not as people die. Admittedly, this can be fixed by modelling the functions as f'(t) = -k*ceiling(g(t) / avgHP) and the same for g'(t).
But this is much harder to deal with and im not sure this has a nice looking closed form formula :P

More examples: (from a later post)

Consider pythagorean triples: http://en.wikipedia.org/wiki/Pythagorean_triple

More specifically these examples:

Case 1: Party 1: 5 Party 2: 4 Players alive after battle: 3 Advantage ratio: 0.416
Case 2: Party 1: 41 Party 2: 40 Players alive after battle: 9 Advantage ratio: 0.111
Case 3: Party 1: 65 Party 2: 63 Players alive after battle: 16
Case 4: Party 1: 85 Party 2: 85 Players alive after battle: 36

In case one, where an extra player consists of a 25% increase of party numbers, 3 are left standing, which is a gained advantage of 60% (3/5). This means for one extra person, a 60% advantage is gained in the 5 vs 4 scenario. The advantage given/recieved ratio is 25%/60% which is 0.416

In case two, where an extra player consists of a 2.5% increase of party numbers, 9 are left standing, which is a gained advantage of 22.5% (9/40). This means for one extra person, a 22.5% advantage is gained in the 41 vs 40 scenario.The advantage given/received ratio is 2.5%/22.5% which is 0.111.

Etc. Basically the ratio gets lower as the number goes higher.

I use the pythagorean triples because I want integer solutions, not rationals/radicals. Non integer solutions would make less sense

Devoted - Lost City

YEP YOU GO TO WATERLOO.. lol :>

MrKillAlot - Lost City

*grabs you and puts your head in a tiolet and gives you a flushie* "NERD!"

combatx

MrKillAlot - Lost City wrote: »

*grabs you and puts your head in a tiolet and gives you a flushie* "NERD!"

if you are implying i'm smart let me tell u this:

spanish looks like gibberish to me. and its only because i have never taken the time to learn it. with enough time i'm sure i can become fluent in it.

i'm really not smart, i just had the chance to learn.

that being said, im actually not nerdy enough b:sad

LloydAsplund - Sanctuary

I. speechless.

Itachikun - Lost City

Wow... I tried to follow along with all that. And I'm taking Calculus my senior year in high school (this year to be exact) and I still couldn't seem to keep my mind straight around all of that. But I guess it's good work, I don't know why anyone on here would want to try to follow along with that since every pk usually ends in this conversation.

(winning team): "Ya, we totally rock. They are such nubs!"

(loosing team): "Whatever! You guys only won because you brought in more people!"

(winning team): "Uhh.. duh!"

combatx

Itachikun - Lost City wrote: »

Wow... I tried to follow along with all that. And I'm taking Calculus my senior year in high school (this year to be exact) and I still couldn't seem to keep my mind straight around all of that. But I guess it's good work, I don't know why anyone on here would want to try to follow along with that since every pk usually ends in this conversation.

(winning team): "Ya, we totally rock. They are such nubs!"

(loosing team): "Whatever! You guys only won because you brought in more people!"

(winning team): "Uhh.. duh!"

In canada, the calculus course is divided into a half course. It's named calculus and vectors, with one half being a very crude intro to calculus, and the second half being a very crude intro to linear algebra.

In high school calculus we learned limits and differentiability... at the time I didnt really know what I was doing besides maybe some speed, velocity, trajectory, and acceleration problems (which was probably because of grade 12 physics anyways).

In my university intro to calc course we did a lot more limits and differentiation... actually learning what the derivative meant, and its applications such as optimization. Learned a bunch of theorems, some which are very helpful (l'hopitals rule) and some which are very obvious (mean value theorem). And then we ended off with the reimann sum, the definition of the integral, and some antiderivatives.

What i used here was nothing more than in this course (although i used some later terminology)

In Calc 2, you do a lot of integration techniques, linear and seperable differential equations, . Sequences, series, taylor polynomials, and parametric curves.

I'm actually enrolled in calc 3 right now and we basically do all of calc 1 and calc 2 again, but this time with multiple variables, multiple integrals, etc.

Devoted - Lost City

brb stealing for masters thesis

LloydAsplund - Sanctuary

Devoted - Lost City wrote: »

YEP YOU GO TO WATERLOO.. lol :>

Ill be the same in 2-3 years. (hopefully).

combatx wrote: »

In canada, the calculus course is divided into a half course. It's named calculus and vectors, with one half being a very crude intro to calculus, and the second half being a very crude intro to linear algebra.

Hey, which do you think is harder (Im assuming in Ontario), calculus or vectors in the calc/vect course? Im in 3rd year high school, almost finish with the calculus section of the course. Heard vectors is a lot harder than the calculus portion.

Aadi - Lost City

We're just at the end of working with taylor and maclaurin polynomials in my calculus course, also senior year in high school calculus. But it's AP Calculus BC, which is supposed to be two semesters of college calculus, so I guess that's about right?

On-Topic though, interesting, but the question is, at what point does one extra person become not as important? 5 v 6? 20 v 21? I'm too tired to understand all the math and I'm on Spring Break x_x

BattleFairy - Lost City

from a mathematics point of view that works perfectly.

from a game point of view, not so much. Its hard to isolate people and bring them down if you are in the losing group. My experience is that its pretty dang obvious USUALLY which party will win and that party usually loses very few amounts of players if they are communicating correctly.

Devoted - Lost City

LloydAsplund - Sanctuary wrote: »

Ill be the same in 2-3 years. (hopefully).



Hey, which do you think is harder (Im assuming in Ontario), calculus or vectors in the calc/vect course? Im in 3rd year high school, almost finish with the calculus section of the course. Heard vectors is a lot harder than the calculus portion.

Nothing in high school is hard. High school grades are simply an indication of your ability to digest/process knowledge. Obviously the concepts are the building blocks of post secondary studies but what you learn in high school is a very very vague look into future endeavors.

As for calculus and vectors, vectors is easily the harder part of the course for most people. From what I remember, people thought vectors was hard due to it not being as straight forward as previous high school math topics were. The only reason I remember doing a bit worse in vectors than calculus was because it was the last 2 months and I already had my acceptance letter :>

combatx

Aadi - Lost City wrote: »

We're just at the end of working with taylor and maclaurin polynomials in my calculus course, also senior year in high school calculus. But it's AP Calculus BC, which is supposed to be two semesters of college calculus, so I guess that's about right?

On-Topic though, interesting, but the question is, at what point does one extra person become not as important? 5 v 6? 20 v 21? I'm too tired to understand all the math and I'm on Spring Break x_x

Taylor/Maclaurin series are my fave :P

Anyways, consider pythagorean triples: http://en.wikipedia.org/wiki/Pythagorean_triple

More specifically these examples:

Case 1: Party 1: 5 Party 2: 4 Players alive after battle: 3 Advantage ratio: 0.416
Case 2: Party 1: 41 Party 2: 40 Players alive after battle: 9 Advantage ratio: 0.111
Case 3: Party 1: 65 Party 2: 63 Players alive after battle: 16
Case 4: Party 1: 85 Party 2: 85 Players alive after battle: 36

In case one, where an extra player consists of a 25% increase of party numbers, 3 are left standing, which is a gained advantage of 60% (3/5). This means for one extra person, a 60% advantage is gained in the 5 vs 4 scenario. The advantage given/recieved ratio is 25%/60% which is 0.416

In case two, where an extra player consists of a 2.5% increase of party numbers, 9 are left standing, which is a gained advantage of 22.5% (9/40). This means for one extra person, a 22.5% advantage is gained in the 41 vs 40 scenario.The advantage given/received ratio is 2.5%/22.5% which is 0.111.

Etc. Basically the ratio gets lower as the number goes higher.

I use the pythagorean triples because I want integer solutions, not rationals/radicals. Non integer solutions would make less sense

LloydAsplund - Sanctuary wrote: »

Ill be the same in 2-3 years. (hopefully).



Hey, which do you think is harder (Im assuming in Ontario), calculus or vectors in the calc/vect course? Im in 3rd year high school, almost finish with the calculus section of the course. Heard vectors is a lot harder than the calculus portion.

Everyone in high school who wasnt in grade 12 yet always looked at the calculus course as the hardest thing in the world, and was afraid of it. Sure its hard, but its very interesting (maybe not in high school) but learning how functions are developed, and then modelling real life problems as curves to find their solutions, or even looking at a parametric curve of two functions to find a relationship can become amazingly useful. Financial mathematics revolves around calculus, among any profession where a math background is required. When you learn more and more, you will see recurring numbers, and think to yourself how does "e" and the natural logarithm find a place in everything. e^i*pi = 1. Most people dont care, but im sure you will learn to appreciate that. I can't count the amount of times i will evaluate the limit of a nasty function and realize that it converges to pi. Or take the integral of a very messy function and realize it's just e. Maybe you'll be amazed as I was at taylor/maclaurin series, when almost every function from sin, arctan, ln, e, any binomial etc. can be modelled by their derivatives.

Vectors is almost completely new, but if you can wrap your head around some of the counter-intuitive concepts (like imagining a line in the 4th dimension) the actual arithmetic is very very easy. Most of what you do is find the intersection between vectors, cross and dot product and their applications, and then they mix in planes, plane intersections, distance from vector to plane etc. This is all re-covered within 2 weeks of Linear Algebra 1, and 1 week = 3 hours of class.

If youre good at picturing things in your head, itll be very easy for you. And even though its an introduction to linear algebra, you don't do any matrix problems, (and hence the determinant, basis, vector space, eigenvector etc. are all left out of the course)

personally, i did a lot better in the calculus portion, probably because by the time I hit vectors I already got accepted into university, and didnt care about marks anymore

P.S. good luck getting in. I'm a bioinformatics student, so my math knowledge is very limited, but i heard math gets very very interesting the further you go into, so if youre going into math, you couldnt have picked a better school (in canada)

BattleFairy - Lost City wrote: »

from a mathematics point of view that works perfectly.

from a game point of view, not so much. Its hard to isolate people and bring them down if you are in the losing group. My experience is that its pretty dang obvious USUALLY which party will win and that party usually loses very few amounts of players if they are communicating correctly.

Yeah this formula does assume that players are taken down one at a time (and focused). But then again, both parties use this assumption. So therefore its covered by the remark that both parties are of equal skill. But yeah, this formula does not model the PvP system, its just for fun :P Besides, i thought it was cool how it can be represented by a right angled triangle. Huge coincidence? Recurring things in math are always interesting to me, like the number e and the golden ratio.

If i did model the situation as i described in the footnote, I wouldnt be able to find such a nice formula, and it would most likely only exist as some sort of series approximation. At least to me, maybe a combinatorics major could puzzle it out and find a closed form formula. Too much work though. Would require hard thinking and then an inductive proof.

ssilly

combatx wrote: »

No doubt, a common excuse in PvP is that party A outnumbers party B, and that is why party B lost. So in this article I will try to answer the question: "how big of an advantage is an extra player, really?"

Now, this is impossible to do without writing an 80-page thesis and taking into account way too many variables, so I will make two huge assumptions:

- Each player in party A and player B have the same amount of experience and 'skill'
- Each player has the same gear/equips

Very unrealistic, right? Thats what happens when you cut corners. But it does prove useful when you want to assume that there is no 'gear' advantage in either direction. Skip to the end if you want to read my findings, otherwise I will justify the process below:

note: If you have taken an intro to calculus, and maybe even an intro to differential equations course everything should be relatively easy to follow



TL;DR -- Summary:

The advantage of one party fighting another party can be given by sqrt(u^2 - v^2) where u = the number of players in party 1 and v = the number of players in party 2.

This means if party 1 has 5 players, and party 2 has 4 players, then party 1 will have 3 players left standing when all of party 2 is dead.

Party 1= 13, Party 2 = 12, then Party 1 will win with 5 left standing.

As the number gets bigger, the advantage diminishes (as it should).

The only real application I can see this having, is if you there is a war between two parties that is very evenly matched, supose 4 vs 4. If Party 1 brings in an extra member which has the same relative skill as all the other members, then this relationship starts coming into play.

Err.. LOL wut ? b:sad

PandaXpresss - Harshlands

f''(t) = (k^2)*f(t)

Similarily,

g''(t) = (k^2)*g(t)

*copied from your formula

I could be wrong but I believe it's sin(x) and cos(x) is the solution to the differential equation. Or exp(ix).

Though admittedly I could be wrong, kinda late and prime's don't show up too well with my font and screen.

Anyways that aside, it's a nice idea to prove it mathematically, and it does make sense however, there's way too many variables that you neglected that actually are very important. For example, even though DPS is considered constant, stuns, buffs, kiting all lower opposing DPS, while debuffs all raise allied DPS. Charms and clerics make the hp rise and fall depending on rates and that lucky crit that no one expected suddenly stops that.

As someone said before, works out on paper but in real life (or in this case a game) not so much.

Wonder how your calculations change with the use of a different function though.'

p.s. I agree completely with the conclusion you came to that when the scale of players gets larger an extra hand does not help nearly as much as at lower scales.

_Rosaly_ - Lost City

This is too long to copy and paste in the chat. I'll just stick to calling people noobs. b:victory

combatx

f''(t) = (k^2)f(t)

If we use a trigonometric for f(t)

(-k^2)cos(kt) = (k^2)f(t)

not equal, similar for the sine funcction. So instead I used the exp function.

e^x hit me instantly, finding the second one was a pain. i was experimenting with many trigonometrics, then i realized that i can use e^-x :P


And of course its very unreliable, not even taylor himself could model human thought and this many variables xD

Ahira - Lost City

lol why all that math (which I can follow incidently,) to highlight what intuitively everyone could tell you quite easily? b:chuckle

Nakhimov - Lost City

Good math but this is only good as a theoretical application for two reasons:

1. Tracking DPS in this game is pretty difficult (wtb Recount b:surrender)
2. Perhaps a nitpick, but HPS isn't included in the equation. If you consider clerics and charms, which not every group will have equally, everything goes to hell.

In any case, interesting read.

Nasume - Lost City

b:shutup let me summarize this

• PvP doesnt win by outgeared Players, but by the Numbers, Numbers alone

combatx

Ahira - Lost City wrote: »

lol why all that math (which I can follow incidently,) to highlight what intuitively everyone could tell you quite easily? b:chuckle

so youre not surprised that the relationship is that of the pythagorean theorem? Was surprising to me, and not intuitive at all. The general premise is beyond obvious, if thats what you mean, then i agree with you.

Its also beyond obvious that in order to fill a box to the most amount of spheres, there has to be a tetrahedral space in between each sphere. ie, a pyramid of oranges.

But this wasn't proven until 2003, and even then proof is still informal. "A formal proof would take 20 years of work" http://en.wikipedia.org/wiki/Kepler_conjecture

Nakhimov - Lost City wrote: »

Good math but this is only good as a theoretical application for two reasons:

1. Tracking DPS in this game is pretty difficult (wtb Recount b:surrender)
2. Perhaps a nitpick, but HPS isn't included in the equation. If you consider clerics and charms, which not every group will have equally, everything goes to hell.

In any case, interesting read.

Well the idea was to find out how big of an advantage an extra player is, from an equal fight. That is to say, if there is a 4 vs 4 where both sides seem to be equal, i want to find how big of an advantage an extra player for one side would be.

Yes, HP does fluctuate, and coming up with a conjecture that defines whether or not a player who needs healing actually gets it, along with cutting the DPS from the cleric who healed instead, is way beyond my mathematical knowledge.

As of right now, the formula assumes that everyone is killed within 10 seconds (after all, they are focused). I think its safe to say as long as u or v >= 2, then the heiro won't tick. Further, we must assume that if u or v < 1, it can take an infinite amount of time to kill the other player, which then creates a lot of problems

Crackdealer - Harshlands

And that, Charlie Brown, is what Wikipedia is all about.

Regenbogen - Lost City

to much much math...b:bye

but since pvp in pwi is independent from a players skill your formula should work everytime...b:chuckle

LaTrappe - Lost City

I shall summarize this thread for the little people of PWI.

combatx is Sorbik and Sorbik is a pretentious **** with delusions of Intellect.

I have saved you countless hours of back and forth on a thread created by a toolbag just so he could argue semantics and pretend he knows more than the rest of us.

/thread

combatx

Regenbogen - Lost City wrote: »

to much much math...b:bye

but since pvp in pwi is independent from a players skill your formula should work everytime...b:chuckle

haha,
ohh no u didnt

LaTrappe - Lost City wrote: »

I shall summarize this thread for the little people of PWI.

combatx is Sorbik and Sorbik is a pretentious **** with delusions of Intellect.

I have saved you countless hours of back and forth on a thread created by a toolbag just so he could argue semantics and pretend he knows more than the rest of us.

/thread

at first i was confused, and then i thought you were joking,
and then i looked at some of my replies and i see what you mean

i guess i can come off like that, but its not my intention to come across as i do. i did not make this thread to assert my intelligence. i guess one way i can justify this is that im posting under an alias. i didn't have an in-game face to anyone here until you gave me one

i dont even know how anyone knows who i am, i only ever told one person

i guess im just that nerdy to find it as genuinely interesting as i do. and of course the formula doesnt 'work', its margin of error is huge, but the general solutions are quite close to intuition and astounding (to me)

its just something i did for fun, and then found a result which formed a perfect right angle triangle. nothing more

if youre going to make a conclusion about me, the best one you can make is that im socially inept, and dont realize half the stuff which i actively do

p.s. i dont believe in intelligence

Crackdealer - Harshlands

Sorbik lost to me when I was 67 and he was 83.

Hellish - Lost City

combatx wrote: »

No doubt, a common excuse in PvP is that party A outnumbers party B, and that is why party B lost. So in this article I will try to answer the question: "how big of an advantage is an extra player, really?"

Now, this is impossible to do without writing an 80-page thesis and taking into account way too many variables, so I will make two huge assumptions:

- Each player in party A and player B have the same amount of experience and 'skill'
- Each player has the same gear/equips

Very unrealistic, right? Thats what happens when you cut corners. But it does prove useful when you want to assume that there is no 'gear' advantage in either direction. Skip to the end if you want to read my findings, otherwise I will justify the process below:

note: If you have taken an intro to calculus, and maybe even an intro to differential equations course everything should be relatively easy to follow



TL;DR -- Summary:

The advantage of one party fighting another party can be given by sqrt(u^2 - v^2) where u = the number of players in party 1 and v = the number of players in party 2.

This means if party 1 has 5 players, and party 2 has 4 players, then party 1 will have 3 players left standing when all of party 2 is dead.

Party 1= 13, Party 2 = 12, then Party 1 will win with 5 left standing.

As the number gets bigger, the advantage diminishes (as it should).

The only real application I can see this having, is if you there is a war between two parties that is very evenly matched, supose 4 vs 4. If Party 1 brings in an extra member which has the same relative skill as all the other members, then this relationship starts coming into play.



More examples: (from a later post)

God Sorbik ur such a dork

Hellish - Lost City

LaTrappe - Lost City wrote: »

I shall summarize this thread for the little people of PWI.

combatx is Sorbik and Sorbik is a pretentious **** with delusions of Intellect.

I have saved you countless hours of back and forth on a thread created by a toolbag just so he could argue semantics and pretend he knows more than the rest of us.

/thread

+1 b:chuckle

Zanryu - Dreamweaver

So... what happens if one squad has a sin?

EDIT: Woops, necro >.> Woke up like 10 minutes ago so wasn't paying attention to dates, my bad b:surrender

avenged911221

can all you spec kami and essence idiots read this thread thnx

Adroit - Lost City

avenged911221 wrote: »

can all you spec kami and essence idiots read this thread thnx

I thought you quit... *sigh*