Find minimum and maximum wins required in a $8$ team tournament

Question

In a tournament, there are $8$ teams in total and playing against each other $2$ times. We need to find

(-)What is the minimum no of wins required to qualify for the next round?

(-)What is the maximum no of wins required,wining which still a team does not qualify for the next round?

Conditions

(-)Wining gives $2$ points to the teams.

(-)Every game has a result, no draws. no ties.

(-)On the basis of wining matches, teams having most points qualify for next round. 4 teams move to next round.

(-)In case of teams having same points, teams which is having advantage (.e.g more goals scored qualify for next round)

Answer 1

Maximum number of wins with which a team may not qualify: $10$

The idea here is to list all the teams in order of their number of wins, and maximize the $5$th best team's wins.

Hence we want as many total wins as possible for the top $5$ teams - meaning we want the least total possible wins for the bottom three teams. What is this least number? Well each of these three teams must play each other in a total of $6$ games, so together they account for a minimum of $6$ wins. The total number of games is $56$, so the top $5$ teams get to distribute $56-6=50$ wins. Now the best that the $5$th best team can therefore achieve is $\frac{50}{5}=10$ wins. Note that this is possible if each of the top $5$ teams always beats the bottom $3$ teams ($6$ wins) and then wins exactly one match against the other top teams ($4$ wins) - giving a total of $10$ wins to the $5$-th best team. Hence $10$ is the most number of wins which does not guarantee qualification (though this is an extremely rare case).

Minimum number of wins to qualify: $4$

The idea here is to apply the same procedure and look at the performance of the $4$-th best team. Each of the bottom $5$ teams must play each other - giving them $20$ total victories to distribute. Hence the $4$-th best person must have at minimum $\frac{20}{5}=4$ victories. This is again achievable - each of the bottom $5$ teams wins exactly once against each other and looses all other matches.

Answer 2

My guess on minimum wins required.

There are 56 total matches (8 teams playing 14 matches each).

Let's say the best team goes 14-0, the next team goes 12-2, and the third team goes 10-4. That is 36 wins accounted for, leaving 20 for the other 5 teams. Let's say they all have the same record of 4-10. That would mean the 4th best team would have 4 wins and advance on tie breakers.

I assume you could reverse it all and find the best record required for advancement.