Sc2 is imbalanced and the math to prove it

General Discussion
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Hello.

Introduction.
For awhile now Blizzard has been claiming that Starcraft 2 is (for the most part) balanced. They have not provided any proof and have expected it's client-base to assume these claims to be legitimate. Considering the number of unsatisfied players, how Blizzard is making vast changes to HOTS and the lack of proof of Sc2's balance, it is... questionable whether Sc2 really is balanced as Blizzard claims it to be.

In this article I provide a second opinion of Sc2's balance. I provide statistics, analysis, charts, solid and easy to follow logic, and most importantly: an deliberately unbiased summary of the results.

Measuring the Balance.
Today I sat down to put Blizzard's claims of Sc2 being balanced to the test. As I worked the math, with a dummy set of statistics, I came to the realization that win-rates are not needed to measure balance. In fact, they are the one thing we want to avoid. What we want to focus on instead is the percentage of each race per-league. This is based off of the mathematical proof that race distribution across leagues directly corresponds to balance in the special case of equally used races. Its a mathematical fact, lets do the logic:

Battle.net's ranking system assures that players get dang-near 50/50 win/loss ratios by ranking players against those with similar skill levels. Because of this measuring the win-rates (which would be only 50/50) will tell us just about nothing. But, it shifts the real information about balance from the win-rates to the Race Percentage Per-League (or RPL). The RPL is merely the percentage of players in a specific league that use a individual race.

Follow the logic with me. Because of Battle.net's ranking system all players' win-rates will be 50/50. But, players doing better will end up in higher leagues. So if a higher league is filled with a majority of a single race, it means that for whatever reason that single race is doing better than other races.

Take the Master league for example. If 90% of players in the Master league where Zerg, then it suggests that Zerg is over-powered. Likewise if 90% of the players in Bronze are Terran, then that says Terran is underpowered.

For the game to be truly balanced the RPL must be as close to 33.33% as possible for each race in each league. Any deviation from this suggests a imbalanced aspect of the game.

Statistics.
Using Sc2Ranks.com I was able to find the required statistics. They provided me with the RPL for each race for each league on the American server.

Here is the report in PDF format: http://dl.dropbox.com/u/12096020/Sc2%20Balance%20Stats.pdf

Analysis.
The top chart shows the RPL for each race organized by league. The average column is the average of all the leagues. The green line/bar is used to show what it would be if it where balanced (33.33%), and the other colors symbolize the various races. Anything above the balanced line suggests it is overpowered, anything bellow underpowered. As you can see, there are considerable deviations from what would be considered balanced. Do note that these charts are slightly biased because they don't account for Random players. Not a biggie though.

To get a closer look at whats up we need a better look at the deviations. The second chart down (titled "Distance From Balanced Per League (% of players)") plots the distance from the balanced line from chart 1, and helps magnify whats going on to understand it easier.

Much to my surprise I found that Terran has the greatest deviations, followed by Zerg, then by Protoss. This chart strongly suggests that the game is very unbalanced.

The Terran line is far bellow balanced in every league except for Bronze, where it spikes above, and bottoms out in the Diamond/Platinum range. This suggests that Terran is underpowered in Master, Diamond, Platinum, Gold, and Silver Leagues, but is overpowered in the Bronze league. Looking at the average, which is far bellow balanced, suggests for the most part that Terran is very underpowered.

The Zerg line is slightly better than the Terran line. It is practically balanced in Master, overpowered in Diamond & Platinum, but underpowered in Gold, Silver and Bronze. The running average for Zerg is bellow the balance line, and suggests zerg is slightly underpowered.

The Protoss line is very impressive. It practically hugs the balanced line, only taking a slight dip in Diamond/Platinum range but making up for it in a slight spike in Grand Master range. The overall average suggests Protoss is right on the money, nice and balanced.

Summary.
This is just one way of looking at the balance of Sc2, and is in no way conclusive. The only way we can achieve a truly accurate analysis is to compare & contrast several different analyses from multiple sources, each taking a slightly different view that focuses on other ways of measuring balance.

But one thing is for sure, Blizzard is totaling bs-ing about Sc2 being balanced.

Thank you.
Terrible analysis. Absolutely terrible.

For the game to be truly balanced the RPL must be as close to 33.33% as possible for each race in each league. Any deviation from this suggests a imbalanced aspect of the game.


This is not the definition of balance.
what if more noobs play terran because of campaign? What if one race is just as good, but harder at say NA master level?
The main problem with your logic is this.

People will prefer a race over another. It's impossible to get 33.33 players to play as zerg, terran and toss because you forgot random. Another thing is if a player is really good at his race more so than the other player, why should the first player get a nerf bat for showing skill?

If 45% of a league uses terran, 35% use Toss and 20% use zerg, how does that cry imbalance? Just means that people in that league prefer not to play as zerg. It's not imbalanced if those races are equally matched in said league. The ZvT, ZvP and TvP are pretty equal percentage wise in the statistics. The truth is however, Terran is gonna win the most match ups cuz there is more playing but also, lose the most matches as well.

The problem with balancing with three races is, at most, you can get two match ups balanced but the third one will be off set.
The main problem with your logic is this.

People will prefer a race over another. It's impossible to get 33.33 players to play as zerg, terran and toss because you forgot random. Another thing is if a player is really good at his race more so than the other player, why should the first player get a nerf bat for showing skill?

If 45% of a league uses terran, 35% use Toss and 20% use zerg, how does that cry imbalance? Just means that people in that league prefer not to play as zerg. It's not imbalanced if those races are equally matched in said league. The ZvT, ZvP and TvP are pretty equal percentage wise in the statistics. The truth is however, Terran is gonna win the most match ups cuz there is more playing but also, lose the most matches as well.

The problem with balancing with three races is, at most, you can get two match ups balanced but the third one will be off set.


Random only holds a slight bias, and I do mention that I omitted it in the article, but its so small that it makes little difference. If you take a look at the Sc2Ranks you will see that pretty close to 33.33% of players play each race, which means that if the game where balanced each league would have equal shares of each race. So your claims of "more people prefer one race over another" are invalid.
The main problem with your logic is this.

People will prefer a race over another. It's impossible to get 33.33 players to play as zerg, terran and toss because you forgot random. Another thing is if a player is really good at his race more so than the other player, why should the first player get a nerf bat for showing skill?

If 45% of a league uses terran, 35% use Toss and 20% use zerg, how does that cry imbalance? Just means that people in that league prefer not to play as zerg. It's not imbalanced if those races are equally matched in said league. The ZvT, ZvP and TvP are pretty equal percentage wise in the statistics. The truth is however, Terran is gonna win the most match ups cuz there is more playing but also, lose the most matches as well.

The problem with balancing with three races is, at most, you can get two match ups balanced but the third one will be off set.


I don't know if it's likely that a league will have a significant preference for a specific race. I mean, this is like saying professional polls like Gallup don't work because "what if you, by luck, happen to only call _____ supporters and don't get any of the opponents? what then?"
Balanced isn't based on "which race is stronger in bronze" or "which race is stronger in diamond". It's based on "which race is stronger when played to its absolute fullest potential". Your "RPL" doesn't have anything to do with actual balance.
03/12/2012 03:47 PMPosted by Yaksha
Your "RPL" doesn't have anything to do with actual balance.


Yes it does, you simply misunderstand how the bnet ranking system works. Lets use a extreme test-case scenario: If 100% of Terran players are in Master, and 100% of Zerg are in Bronze, then terran is waaay overpowered and zerg is way underpowered. Their winrates will all be 50/50 because of Bnet's ranking system, but all the Zerg players are shoved into Bronze because its underpowered and can't win vs Terran, while Terran gets shoved into higher leagues because its overpowered and always wins vs zerg.

... Hence, you can measure balance by RPL.
Take the Master league for example. If 90% of players in the Master league where Zerg, then it suggests that Zerg is over-powered.


That just means that Zerg is the most popular. If Zergs are dominating with 75% win ratio versus Terran or Protoss races, THEN it's imbalanced.



In the lower leagues Terran is most represented. Does that mean Terran is OP? No.
Another major issue is that you can't balance a game at all levels.

Zerg will inherently be unbalanced in bronze league because the mechanics (larva, creep spread, injects and the like) are something that takes more skill than most bronze (heck, silver and probably gold) players are going to have. So trying to balance that problem in will make Zerg overpowered in higher leagues where the players will have the skill to utilize those mechanics.

That's just the most obvious one; I'm sure similar statements can be made for Protoss and Terran as well. All-in-all, I think they've done a pretty good job of balancing SC2. Some tweaks here or there would be nice (carrier build time, I'm looking at you), but overall, they've done well with a tripod system.
For the game to be truly balanced the RPL must be as close to 33.33% as possible for each race in each league. Any deviation from this suggests a imbalanced aspect of the game.


Assuming the same number of players play each race... (also forgetting random)
03/12/2012 03:54 PMPosted by Ruin
Assuming the same number of players play each race... (also forgetting random)


Ruin: I have already covered both these claims in multiple responses and in the original article. Please read it carefully rather than skimming through it, as I am not going to explain it again.



03/12/2012 03:52 PMPosted by rccars
That just means that Zerg is the most popular. If Zergs are dominating with 75% win ratio versus Terran or Protoss races, THEN it's imbalanced.


Yes, and what happens when they dominate with a 75% winrate? All the zergs go up into higher leagues, so their RPL is becomes biased. You are simply describing the RPL in a different way, but saying the same exact thing.
If you yanked out the Koreans (the majority of which play Terran), that Terran line would be wayyyyyyy lower. Which tells us that even these results are skewed.
Thank you for wasting your time and telling us something we've already known for along time.
Saying the game isn't balanced until each race is being played at 33.33% is pointless. I chose zerg not because of units but simple because I love the evil races in a game. In warcraft 3 I always picked Undead. That has no affect on balance. Sorry but your thread is pretty much pointless.
03/12/2012 04:00 PMPosted by xXDefianceXx
Saying the game isn't balanced until each race is being played at 33.33% is pointless. I chose zerg not because of units but simple because I love the evil races in a game. In warcraft 3 I always picked Undead. That has no affect on balance. Sorry but your thread is pretty much pointless.

You're another one of those guys that just skimmed through and didn't understand it. When the number of players for each race is close to equal (which it is) it opens a special-case where the balance of the game can be measured by the number of players playing each race in each league. Its a mathematical fact, so if you think otherwise you are merely misunderstanding =).

I have covered the exact point you are pointing out in the original article and in multiple responses. Please actually read the article rather than skimming through it, and if you would like clarification on how my theorem measures game balance I would be happy to explain it further.
So let me get this straight... Terran is "very underpowered" Zerg is "slightly underpowered" and protoss is "nice and balanced". Do you really not see the lack of logic in that?

This has to be a trolling post... please don't tell me the American education system is this bad... or tell me you're European or something. This post has me depressed about the future of my country.
So let me get this straight... Terran is "very underpowered" Zerg is "slightly underpowered" and protoss is "nice and balanced". Do you really not see the lack of logic in that?

Another person complaining about the "random bias" that I mentioned in the original article. Seriously people, read it and understand it, or do I have to spell everything out?
03/12/2012 03:54 PMPosted by Ruin
For the game to be truly balanced the RPL must be as close to 33.33% as possible for each race in each league. Any deviation from this suggests a imbalanced aspect of the game.


Assuming the same number of players play each race... (also forgetting random)


And how exactly would the random choice affect this?... either you're meaning something different from what you said, or you fundamentally misunderstand statistics.

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