Here is a fun chart:
https://i.imgur.com/iRH4z9F.png
These are the elo rankings of “pro” players active in 2020 with at least 100 games played. The rankings are standardized: the ranking of each player and for each matchup is subtracted from the mean of that matchup and divided by the standard deviation of that matchup. This allows for comparisons between different distributions shapes/means. That is, if a player is 2 standard deviations from the mean in PvP, he should be roughly 2 standard deviations from the mean in PvZ and PvT.
Since the distributions are centered around the mean, the “b” constant in the linear regression should be basically 0 (which it is for each equation). What we are left with is how a race’s performance scales with skill on average for all players.
- Terran: 0.9 in TvP and 0.916 in TvZ.
- Protoss: 0.9 in PvT and 0.921 in PvZ.
- Zerg: 0.912 in ZvT and 0.909 in ZvP.
- Protoss players are rewarded with (0.921 - 0.909) = 1.2% higher Elo on average compared to equally skilled Zergs.
- Terrans are rewarded equally to equally skilled Protoss (0.9 vs 0.9).
- Zergs are rewarded (0.912 - 0.916) = -0.4% compared to equally skilled Terrans.
Thus the pro-level stats confirm the ladder stats: P>T>Z.
Another thing to note is that since all asymmetrical matchups are less than the symmetrical ones on average (by about ~10%), it means the asymmetrical matchups have a tighter variance than the symmetrical ones, which basically proves that balance problems do exist and the asymmetry of the matchups is the cause. The asymmetrical matchups have more consistent performance than the symmetrical ones and that happens when there is a bias that reduces the randomness of the outcomes.
Think of it this way. If we flip a coin we’d expect a perfectly random 50/50 split of heads & tails if heads & tails are perfectly symmetrical. But, if the coin is heavier on the heads side (and is thus asymmetrical), it will produce heads consistently more than tails, so the variability/randomness goes down.
The same is true for ZvZ/PvP/TvT. There is a certain element of randomness that we’d expect but for some reason ZvT/PvT/PvZ have less randomness (meaning one race is favored which reduces the randomness of the outcomes).