Bocce league: How many wins to advance to playoffs?

Question

In our bocce division, we have $14$ teams (including us). Everyone plays each other $1$ time during the $13$ week season ($13$ games for all). At the end of the $13$ weeks, the top $6$ teams advance to the playoffs. What is the minimum wins a team must have to land in the top $6$? At this time no one is undefeated so the very best record could be $12-1$.

Thank you.
I’m going around in circles in my head.

Answer 1

This answer concerns the minimum number of wins to be guaranteed a position in the playoff.

Unintuitively, a team can still be eliminated with $8$ wins here is an example: "Team 1 wins against Teams 2,3,4,5,8,9,10,11,12 (9 Wins). Team 2 wins against Teams 3,...,11 (9 Wins). Team 3 wins against Teams 4,...,12 (9 Wins). Team 4 wins against Teams 5,...,13 (9 Wins). Team 5 wins against Teams 6,... 14 (9 Wins). Team 6 Wins against Teams 7,..., 14 AND Team 1 (9 Wins). A team with 8 wins is not guaranteed a place in the playoff."

A similar example can show that a team with $9$ wins can also be eliminated.

Suppose we have a league where there are: $6$ very strong but equal teams, and $8$ garbage teams. The $6$ teams are guaranteed to win against the $8$ garbage teams, and split the remaining wins amongst themselves evenly. There will be $$13+12+11+10+9+8=63$$ games played by these top $6$ teams. Since each of these games will be a win for the strong teams, and they divide them equally, each top team will have $$\frac{63}{6}=10.5$$ wins, or better put, $3$ teams will have $11$ wins and $3$ teams will have $10$ wins.

Thus, if you want to be in a guaranteed position to be in the playoff you must have at least $11$ wins. If you have $10$ wins you are at least in a tie-breaker situation (There is one situation where $7$ teams all have $10$ wins.)

In reality though, you could win a bit more than half your games and be fine. Have fun playing bocce :)

Answer 2

You could get into the playoff any time you tie for sixth place. The minimum score that could do that would come if there are five teams that win all their games against the other nine. They play a tournament among themselves for the first five places. The other nine teams play a tournament for sixth place. Each team plays eight games in this tournament and it is quite possible all teams finish with four wins, so you could get into the playoff with as few as four wins.