# How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which contains the top ~$200$ players in a region.

The matchmaking system will, most of the time, match players from the GM league with other players from the GM league. There are exceptions however, like if no one else from GM is playing, or if someone from a lower league has a very high MMR (Match Making Rating) but is not in GM for whatever reason. The algorithm is quite complex, and as far as I know not all details are even public.

For the purpose of this question, let's say that whenever a GM player is matched with someone from a lower league, the GM player wins that match.

These are the current standings in the GM league for the American region: http://www.rankedftw.com/ladder/lotv/1v1/win-rate/?f=am,grandmaster

You can see that everyone's win/loss ratio is higher than 1 (more than $50\%$ won), so everyone wins more than they lose in GM. The standings change often, but it's rare to see anyone with more losses than wins. Wins and losses are counted from when you start playing, not only from when you entered GM. However, stats are reset a few times per year, at the start of each season. So I would expect this not to influence things too much.

This is rather weird for me to see: I would expect the worse GM players to be, in general, easy pickings for the better ones, and their win/loss to be below 1.

One explanation that I can think of is what I call low transitivity (if there's a proper term for it let me know): if $A$ consistently defeats $B$ and $B$ consistently defeats $C$, then it rarely holds that $A$ also consistently defeats $C$. In such a case, all 3 players $A$, $B$, $C$ can hold similar win/loss ratios, but I still don't see how all 3 can hold them above 1.

Under the assumption I mentioned above, that a GM player will always defeat a lower league player, it's possible that they are all above 1, but it still seems highly unlikely, since inter-league matchups are quite rare.

What is a possible explanation for this phenomenon? Given those win/loss ratios, what is an approximation of the number of games a GM member will play with lower league players?

Without the assumption that a GM player will always defeat a non-GM player, can we say something about the probability of that happening?

• The number of wins refers only to the wins only after the player entered the GM league, or from his first day in the game? This would explain it perhaps (if the second case is true) – Jimmy R. Dec 28 '15 at 11:19
• @Stef from his first day in the game. That might help explain it indeed, but I'd still expect it to normalize after enough time in GM. Also, they are reset a few times a year (about each season). – IVlad Dec 28 '15 at 11:21
• Given the paucity of well-formed assumptions on which to base "a possible explanation", I'm going to suggest this would be more appropriate for the Gaming SE community also known as ArQAde. – hardmath Dec 28 '15 at 11:39
• @hardmath I thought about asking on a gaming site, but the type of answer I have in mind would contain quite a bit of math reasoning and formalism, which I thought would make the question a better fit for here. I can provide more information if necessary. A generic answer that only considers what I did provide is also welcome. – IVlad Dec 28 '15 at 11:44
• While your interest in this is understandable, your current formulation contains statements like "not all details are even public", "I would expect this not to influence things too much", and "it's possible that X, but it still seems highly unlikely". Perhaps an explanation may lie in a censored data set due to weaker GM players (often new ones?) leaving the field. In any case it would seem to call for a data intensive study. – hardmath Dec 28 '15 at 11:56