Minimum points required to get top 4 in a group of 9 (DOTA2 TI7 Group Stage ) I would like to know a way to calculate the minimum points required to guarantee Winer's bracket. I.e, how many points can guarantee a team falls into top 4 of a group. 
The rules are as following:
Eighteen teams divided in two groups of nine teams each where they play in a round-robin format.
All matches are played in a Bo2.


*

*A win (2-0) provides 2 points. 

*A draw (1-1) provides 1 point.

*A loss
(0-2) provides 0 points.


Top four teams in each group advance to the Upper Bracket of the Main Event.
Bottom team in each group is eliminated.
Remaining teams advance to the Lower Bracket of the Main Event.
Tiebreaker rules:


*

*If there is a tie along the Upper and Lower Bracket divider, an
additional set of games will be played. For other ties (in order of
importance): Head to head result between the tied teams Results
against lower seeded teams (from directly below till last place) If a
tie can be broken along the way, the process is restarted (back to
head-to-head result) Coin toss


Please refer to DOTA2 TI7 Group Stage page for more information.
 A: How many points are up for grabs? The total number of points awarded per match is constant, and the number of matches for a round robin of $n$ teams is well-known.

 $\left(\array{9\\2}\right)=36$ matches, with $2$ points per match, so $72$ in total.

You want to find the threshold to guarantee being in the top $k$ teams, $k=4$ in this case. Let's say that there are $k+1=5$ good teams, and the rest are bad; this creates the most competitive race for the top 4. The bad teams play some matches between each other, so their points total is non-zero. Work out how many points they must have between them at minimum (i.e. assuming they each lose to each of the top 5), which gives you the total number of points the top 5 can have at maximum.

 The bottom 4 teams play $\left(\array{4\\2}\right)=6$ matches between each other so must have at least $12$ points between them, leaving no more than $60$ points between the top 5.

If you have $x$ points, and the top 5 have $y$ points in total, how do you ensure you're not last out of those 5? Hint: if you're above average, you're not the worst.

 If you have more than $y/5$ points, then the others cannot all have more than or equal to you, because then the total would be more than $5*y/5=y$, but the total is $y$. So we need $x > y/5$, and $y\le60$ so we need $x>12$. As $x$ is an integer, you thus need $13$ points to guarantee a top 4 finish. $12$ points isn't sufficient, because you could have each of the top 5 scoring $12$, and you could lose the tiebreak.

