Balance Discussion Math(Best of N format analysis)

OP = overpowered.
UP = underpowered.
Best of 1 = 1 win is required to beat your opponent.
Best of 3 Series= 2 wins are required to beat your opponent.
Best of 5 Series = 3 wins are required to beat your opponent.
N = {1,3,5} Could be higher odd numbers.
x = win rate of OP race. 0.500 < x < 1.0000
1-x = win rate of UP race. 0.0000 < 1-x < 0.5000
f(x) = win rate of OP race in best of 1 as a function of x
g(x) = win rate of OP race in best of 3 as a function of x
h(x) = win rate of OP race in best of 5 as a function of x
Race A = OP race. It wins more than 50% vs Race B. This thread assumes OP race exists. Statistical probability is the only attribute it has, not player skill/map/meta game etc.
Race B = UP race. It wins less than 50% vs Race A.
Race C = the other race. Basically ignored so that 1 race vs 1 race calculation becomes clear.
OPlevel = x in percentage. 50% < OPlevel < 100%

Visual Representation of Best of 3



Visual Representation of Best of 5



Best of 1 win rate is f(x) = x for obvious reason. All of these 3 functions are defined between 50% and 100%.

As you can see, f(x) < g(x) < h(x) at all points. That is, best of 3 win rate is always higher than that of best of 1 at the same OPlevel. Best of 5 win rate is even higher than that of best of 3 as well. At 60.00% OPlevel, Race A wins 64.80% in best of 3 series, and 68.26% in best of 5 series. This means that best of non-1 series favors Race A even more, when Race A is already OP in the first place. The more wins you need, the better deal Race A gets just from the format.

This section assumes ideal tournaments. One Race A player plays against Race B players twice. At a separate setting, one race B player plays against Race A players twice. For the first Race A player, it is as if winning a 4 player tournament with ABBB race distribution. For the latter Race B player at a separate setting, it is as if wining a 4 player tournament with BAAA race distribution. Since there are 4 players involved, both have 1/4 = 25.00% chances to win the separate tournaments if 2 races are totally balanced at 50-50. Take a look at how OPlevel affects the win rate.



Extending the idea, ABBBBBBB & BAAAAAAA 8 player tournaments with 1/8 = 12.50% standard chance to win.



These tournaments don’t happen in reality due to mirror matches and existence of Race C. Nevertheless, these graphs essentially show how easier it is for OP race to place high in a tournament compared to UP race even with relatively small OPlevel like 52.50% or 55.00%. 70% OPlevel is not required to “look like” 70% OP from tournament placements. Since almost all tournaments have multiple rounds of Best of 3 or 5, the tournament format itself emphasizes OP&UP states in this way.

Whenever you see someone argue balance with statements like “There are 5 Race A players in Round of 8” or “7 tournaments are won by Race A players in the last 10,” you have to take these with a grain of salt. They are trying to impose the image that Race A is super OP with 70-80% OPlevel, when in fact; Race A is likely to have only 55% or so OPlevel. Although one cannot deny the fact that Race A is OP, it is not as much as people make it to be from those results. Along with player skill and other factors, the tournament format factor plays a role here as I presented above. Citing recent tournament results as a source of balance discussion could be misleading because OPlevel always gets exaggerated. You always have to examine individual map-by-map results instead of how many of which race placed high in a tournament. Racial distribution section of tournament liquipedia is not a good source to start your balance discussion.

Sadly, we only spectate tournament matches. Therefore, our image of a race tends to be exaggerated by seeing Race A players more often due to them advancing further, partially thanks to the format that makes Race A look more OP than its actual OPlevel unlike 1-match-only ladder games. This may be one of the reasons why actual overall win rate of UP race is not as bad as some think it is. Tournaments with multiple rounds of best of N series are destined to be biased in favor of OP race. We have no choice, but accept it. Just know that it is a biased format.

Race A could be any of the 3 races. Which race is Race A is 100% irrelevant to my point. It could be one race today, but it could be another race 1 year from now. My point still stands no matter which race is widely considered Race A at that particular point of time. Note that I already acknowledged that Race A is OP. Nowhere in this thread did I deny that fact. “OP race is OP anyways” goes nowhere and is missing the point.

Some might have thought this was another cool picture thread from me, but it’s not. I don’t think this thread was interesting for majority or readers, but I hope at least some found it intriguiging.