someone asked this question recently, struggling to find the answer. Can anyone help here…
In a tournament 8 teams will play with each other (1 home , 1 away) total 14 matches each and total 56 matches.
Point system:
Win - 3 points
Tie - 1 Point each
Loss - 0
And only top 4 teams will qualify for playoffs, how many minimum points that team need to have a guaranteed seat in playoffs.
I can’t solve it mathematically for you, but I think I can get the answer with common sense. I determined the highest total you could get without making the playoffs, which is achieved when a few teams are terrible and the others are grouped as closely as possible.
So if three teams are winless (0-14) and the other teams are all 10-4, for a total of 30 points, there is a 5-way tie for the playoffs. Therefore, a total of 31 points should guarantee you’re in the top 4.
How are ties in the standings resolved? If, for instance, the home team always wins, each team will end up with the same number of win-points.
56 games in total.
Maximum total points available is when no games are tied, i.e. 3 points per game.
56 x 3 = 168
168/5 ~= 33
Therefore if you have more than 33 points, you must be in the top four.
You could have four teams at 11-3 and one at 12-2 so even 33 points will not guarantee you would be in the top four, but 34 points ought to do it.
Gack, you’re right. I somehow missed 6 games in my tally.
Umm, I don’t think that is correct.
While there are 56 games in total, 6 of those game would be between the three losing teams.
That only leaves 50 games to share between the top 5 tied teams - 10 wins each = 30 points.
So that would make 30 the highest total where it is possible to finish fifth (on goal difference or whatever would be used to break the tie)
I reckon any team with 31 or better must be in the top 4.
As Max Power notes, this is incorrect. The other way to look at it is that the top five teams play a total of 20 games against each other, so there must be at least 20 losses spread among the top five teams.
My bad! Of course you’re right.
I’ve even done this sort of thing before. I watch the curling tournaments. Typically, they are 12 team round robins, followed by a playoff involving the top 4. The rule is that ties among the top 4 are resolved by tie-breakers (the playoffs are structured in such a way that the first two teams have an enormous advantage over 3 and 4), but no team can be eliminated merely by a tie-breaker but ties involving 4th place must be broken by pre-playoff games. The worst case would be a scenario in which 3 teams were 10-1 and the other 9 were all 4-7.
Thanks everyone…so looks like around 30 points will do the trick.
Around 30 would be a good ballpark.
But if we’re assuming that 14 games for each of the 8 teams means a double round robin, it’s impossible for more than one team to go 0-14 for the season. The second and third lowest teams would each have to be 3-11 or 2-10-2.
Some back of the envelope calculations showed that 31’s probably the magic number. Troutman was correct though his initial numbers were off.
I did a single round robin.
Team 1 went 0-7
Teams 2 and 3 went 1-5-1
Teams 4-7 went 5-2 for a 5 way tie at 15 points apiece.
Which means if you double that to
0-14
2-10-2
10-4 for a 5 way tie at 30 points.
Basically what this means is this. If you score 30 points it is highly unlikely though still possible to not be assured a place in the playoffs.
If your team scores 31 points it is mathematically impossible for 4 other teams to tie or beat your score. You must be in the playoffs.
I showed this thread to a colleague who works in Operations Research. In a few minutes, he was able to put together a computer model for the situation. The computer was able to confirm that 30 points was the best possible 5th place showing, where the top 5 teams all tie.
As a digression, I’m amazed by the efficiency of the model. A brute force attack would require analyzing 2^56 different scenarios, ignoring the symmetry, which would decrease this muber somewhat. 2^56 is approximately 72 quadrillion. His model solved the problem in less than 0.01 seconds.
Yeah, it’s actually not that hard to cast this as a network flow problem. Kleinberg & Tardos has a nice exposition of the details.
29 points will get you into the playoffs. Here’s my reasoning.
To contradict your friend in OR there is no scenario with 5 teams finishing with 30 points. In order to score 30 points you must win 8 games or more. To maximize the fifth place finisher’s total you would have a 5-way tie for the top teams. The top five teams can each garner 6 wins from the sad sacks in the bottom three. The other two wins would need to be at the expense of the other teams in the top five. Since a 5-way tie ensures the highest point total for the fifth place team each of the top 5 would have 2 losses resulting from the 5 team round-robin that they play amongst themselves. This only leaves 4 ties for each team in the top 5 and a maximum total of 28 points for the teams in a 5-way tie for the lead. Ergo, 29 points will beat the optimum 5th place finisher.
(N.B. spelling corrected in quote.)
Small correction: there are eight losses to be distributed among the top five teams. The top two teams could get away with one loss each and 29 points, but the best finish for the fifth place is still 28 points.
Top five teams beat the bottom three teams home and away. That’s six wins each. Then they play each other, four home games, four away. Say the home team wins every time. That’s ten wins for each of the top five teams, or 30 points.
I believe I stand corrected. Thank you.