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Browder Calculates Balance, Overturns Mathematics

Blogs > jdseemoreglass
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jdseemoreglass
Profile Blog Joined July 2010
United States3773 Posts
November 10 2011 22:47 GMT
#1
 
Dustin Browder Calculates Balance, Overturns Mathematics


5x +3y = 20
x = ?


For thousands of years, certain mathematical truths have been considered incontrovertible. One of these supposed truths was that it is impossible to solve for two independent variables from a single dependent variable. In other words, in order to solve a linear system with two unknowns, it was necessary to have at least two equations. However, it seems one of the minds at Blizzard has finally overcome this problem, and overturned the foundations of mathematics as we know them.

The breakthrough occured when Blizzard's balance team for the popular video game Starcraft 2: Wings of Liberty, tried to solve one such equation. Most of the development team immediately dismissed the problem as impossible to solve, citing the long-standing principles of mathematics. However, that didn't dissuade one man...


[image loading]
Dustin Browder has a Bachelor's Degree in English Literature from the prestigious UCLA.


I sat down with the mind behind this incredible feat for in depth look at how he reached the solution to the problem which has stumped mathematicians for millennia.


Browder - "Well, you see, we were trying to come up with an equation to calculate the balance of the races in our game, Starcraft 2. The only data we really had was the win and loss statistics of the millions of players worldwide, and we had to find a way to calculate the balance of the game from these game results.

"Unfortunately, we have a matchmaking system which pits players of equal strength against each other, so the statistics eventually revert to 50%. To overcome this fact, we decided to take into account each player's skill at the game, in order to determine which race was winning a disproportionate amount of games based upon player skill.

"This is where we encountered another problem. We couldn't calculate the relative skill of each race without knowing the balance of the races. The skill of each player was unknown, and the balance of the game was unknown. We couldn't solve for the balance without knowing the skill, and we couldn't solve for the skill without knowing the balance. This means we had two unknowns, and only equation, based on the win ratios for each race. To put it simplistically, the equation we had was something like this:"


Relative Player Skill * Relative Race Balance = Win Ratio

"Now, I was told again and again that it was impossible to solve for two independent variables given only the win/loss statistics, but I was determined to find an answer.

"My solution was rather innovative, genius really... One day, while I was sitting at home, poking a needle into my reaper doll, it suddenly hit me like a ton of bricks! What if we just ignored one variable temporarily, solved the equation, and then substituted the result back into the original equation?"



[image loading]


The solution was elegant in its simplicity, and yet incredibly effective. Here is an example of the process at work, taking the sample equation above:


5x + 3y = 20 ---> 5x = 20 ---> x = 4
5(4) + 3y = 20 ---> 20 + 3y = 20 ---> y = 0


And shazam. Just like that, thousands of years of work by mathematicians and scientists was turned on its head. When the evidence was presented to Dr. Noam Elkies, head of Mathematics at Harvard University, he responded with dismay.


[image loading]
Dr. Noam Elkies - "You trollin' me, dawg?"


Browder - "This is how I solved the problem of figuring out both player skill and game balance from individual game statistics. I ignored balance in the equation and calculated each player's relative skill. Then I plugged skill back into the equation. Incredibly, we discovered that imbalance was practically zero; in other words, the game was nearly perfectly balanced, give or take a few percent.

"When I came to the rest of the team and told them of my solution, they couldn't believe it. They just sat there with their mouths hanging open, staring at me. Eventually they got over the shock of my accomplishment and started getting to work, running the numbers."



Blizzard Developer - "Look, I'm only talking to you on conditions of anonymity. He came to us with this 'solution' to the problem, and well, we just didn't know what to say. I mean, our jobs were on the line... So we got to work trying to mask what was actually being done by making an equation as complicated looking as possible, to confuse the general public."


[image loading]


However, not everyone was convinced by Browder's achievement. A few notable pros have made comments critiquing Blizzard's claims of balance.


[image loading]

Greg 'IdrA' Fields - "He's just retarded. The whole thing is fucking stupid.

"You can't determine skill from game stats. Otherwise I'd have a perfect record against Cruncher. The only way to determine skill is to watch people play the game."



Sen - "You see!?! YOU SEE!?!?!"


From here, Browder feels he has no where to go but up. Recently, he has turned his attention to more advanced mathematical theorems. His next project is to solve the problem of division by zero.


Browder - "I had this idea, see, that if we define the quantity zero times infinity as one, then the math all seems to work out. Once you divide by zero, the zeroes cancel out."


For now, however, this brilliant mind is still a lead game developer at Blizzard, and the hero of millions around the globe.
 

****
"If you want this forum to be full of half-baked philosophy discussions between pompous faggots like yourself forever, stay the course captain vanilla" - FakeSteve[TPR], 2006
mangomango
Profile Joined September 2009
United States265 Posts
November 10 2011 22:54 GMT
#2
It was only a matter of time. Bow before his greatness; lest ye be destroyed.
Husky: Every drone you lose is like a needle in the eye. Nony: probes win $10k (Earn it! Idra Fighting) :P
Crill
Profile Joined September 2011
Netherlands10 Posts
November 10 2011 22:54 GMT
#3
Haha, brilliant!
Kudos to you my good man
OK, here's the plan
Alejandrisha
Profile Blog Joined July 2010
United States6565 Posts
November 10 2011 22:55 GMT
#4
haha 5/5
get rich or die mining
TL+ Member
ZeromuS
Profile Blog Joined October 2010
Canada13389 Posts
November 10 2011 22:58 GMT
#5
spec-fucking-tacular
StrategyRTS forever | @ZeromuS_plays | www.twitch.tv/Zeromus_
Emporio
Profile Blog Joined December 2010
United States3069 Posts
Last Edited: 2011-11-10 22:59:30
November 10 2011 22:59 GMT
#6
+ Show Spoiler +
[image loading]
How does it feel knowing you wasted another 3 seconds of your life reading this again?
fatum
Profile Blog Joined November 2010
Spain83 Posts
November 10 2011 23:00 GMT
#7
Less trolling and this post should deserve to be featured

It was so fun, except for a few extreme comments
Aim to climb as high as you can dream.
Soleron
Profile Blog Joined September 2010
United Kingdom1324 Posts
November 10 2011 23:01 GMT
#8
The maths looks kind of plausible, integration of the error function (standard deviation of normal distribution) over all space in several dimensions.

It is perfectly possible to probabilistically solve for several unknowns with fewer equations than you need. You end up with an estimate and error bars.
zul
Profile Blog Joined February 2010
Germany5427 Posts
November 10 2011 23:03 GMT
#9
haha. golden. 5²
keep it deep! @zulison
Hyren
Profile Blog Joined June 2010
United States817 Posts
November 10 2011 23:07 GMT
#10
I have no idea what any of this means.
Power-tripping mod for Trump's stream
Conquerer67
Profile Joined May 2011
United States605 Posts
November 10 2011 23:11 GMT
#11
10/5.

+ Show Spoiler +
[image loading]


Actually, this calculation simplifies down into: ((Pwr=100%-(Twr+Zwr))/(13XY*(5Z+2a)))*(3MVP+3NSTea)

If Pwr= Protoss winrate, Twr= Terran winrate, Zwr= Zerg winrate, then it is currently mathematically impossible for a protoss player to win Code-S
I hate when people compare SC2 and rochambeu. One race isn't fucking supposed to counter another one. | Protoss isn't OP. Their units on the other hand....
danl9rm
Profile Blog Joined July 2009
United States3111 Posts
November 10 2011 23:13 GMT
#12
I logged in to give you a 5.
"Science has so well established that the preborn baby in the womb is a living human being that most pro-choice activists have conceded the point. ..since the abortion proponents have lost the science argument, they are now advocating an existential one."
HypernovA
Profile Joined October 2010
Canada556 Posts
November 10 2011 23:17 GMT
#13
5/5
Brilliant post
Probe1
Profile Blog Joined August 2010
United States17920 Posts
November 10 2011 23:26 GMT
#14
For most parts of life, there are words to describe your laughter. For these moments, there is 5/5.
우정호 KT_VIOLET 1988 - 2012 While we are postponing, life speeds by
iGrok
Profile Blog Joined October 2010
United States5142 Posts
November 10 2011 23:33 GMT
#15
JEsus thats is hilarious. His face is perfect.
MOTM | Stim.tv | TL Mafia | Fantasy Fighting! | SNSD
AnachronisticAnarchy
Profile Blog Joined July 2011
United States2957 Posts
November 10 2011 23:37 GMT
#16
This. Is. AWESOME!
"How are you?" "I am fine, because it is not normal to scream in pain."
]343[
Profile Blog Joined May 2008
United States10328 Posts
Last Edited: 2011-11-11 00:00:24
November 10 2011 23:59 GMT
#17
good quotes and pictures, but the math isn't that funny :/

a little amusing, I guess
Writer
jeeeeohn
Profile Blog Joined May 2011
United States1343 Posts
November 11 2011 00:45 GMT
#18
haha, the, "you see?! YOU SEE?!?!" totally slayed me. 5/5

Please post more. ^.^
If you can't jam with the best, then you have to slam with the rest.
jdseemoreglass
Profile Blog Joined July 2010
United States3773 Posts
November 11 2011 04:01 GMT
#19
Glad you guys enjoyed reading it as much as I enjoyed writing it. As soon as I saw Browder's faces on google search I laughed my ass off
"If you want this forum to be full of half-baked philosophy discussions between pompous faggots like yourself forever, stay the course captain vanilla" - FakeSteve[TPR], 2006
igotmyown
Profile Blog Joined April 2009
United States4291 Posts
Last Edited: 2011-11-11 05:28:45
November 11 2011 05:26 GMT
#20
beta^2+sigma1^2+sigma2^2+sigma3^2 is the "variance" microsoft used for it's true skill rating for multiplayer.

mu1-mu2 is the difference in skill level

gamma[i] is usually the draw coefficient. No idea what psi is

1/sqrt(2 pi sigma^2) e^(-variable/ 2 sigma^2) is obviously the normal distribution/gaussian

It's being integrated across 3 dimensions and then normalized in the denominator (because that adjusted variance won't necessarily integrate to 1).

product of g from 1 to G is either number of players or races, by the context it's probably races.

I'm not sure what it is exactly, but I don't see why any of this is hard to understand to someone with a decent probability and statistics background. I'm also assuming they can look at matchups for balance problems and aggregates all 3 for an independent (but not orthogonal) measure of skill.

The blog is like when a creationist mocks some easy science target/result for looking dumb. Internet glass houses. A lot of this is pretty dirty, like mocking someone's degree then launching into oversimplifications. But I guess it's easier to gain popular support when the average reader can feel smarter than the target?

Edit: I can't vouch for the validity of Blizzard's math either way
arsenic
Profile Joined January 2009
United States163 Posts
November 11 2011 06:00 GMT
#21
Hilarious.
Fuhrmaaj
Profile Joined January 2011
167 Posts
November 11 2011 07:14 GMT
#22
Overall it was pretty funny and 5/5.

As I was telling Dusty at Blizzcon (or D-Brow, as his friends prefer to call him), he actually does have three equations: one for each race.

ax^(1+k) + by = r

a = coefficient of skill (amount of skill that the game should be balanced on)
b = coefficient of balance (amount of skew in game design towards racial balance)
x = player skill
y = racial balance
k = constant which describes how players of higher skill can better exploit their race
r = actual win ratio

He has a y and r for each race, and an x for each player. You can determine relative skill against players of the same race and use ratios to solve for y. In his equation, 0<x<4 and 0<y<20/3 but I'm not sure if I agree with the constants he chose. Hopefully he sees this.

Hey, what was that equation you stuck on? It looks like particle physics or something. I don't know how useful that will be for game balance
Random player
The_Templar
Profile Blog Joined January 2011
your Country52797 Posts
November 11 2011 13:43 GMT
#23
Genius, good sir.


y=5, x=1.
Moderatorshe/her
TL+ Member
Hesmyrr
Profile Blog Joined May 2010
Canada5776 Posts
November 11 2011 14:34 GMT
#24
Earthbreaking blog. 5/5
"If watching the MSL finals makes you a progamer, then anyone in Korea can do it." - Ha Tae Ki
darmousseh
Profile Blog Joined May 2010
United States3437 Posts
Last Edited: 2011-11-11 16:34:44
November 11 2011 16:28 GMT
#25
I've researched this problem and commented a few months ago. Mmr is a function of both mean and deviation and its possible to determine the value of each race if you can isolate certain variables. Send me a pm and ill send you more information on the math.

Edit; its called bayesian appriximation. You can read about it in this link.
http://jmlr.csail.mit.edu/papers/volume12/weng11a/weng11a.pdf
Developer for http://mtgfiddle.com
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