Round Robin Scheduling

Advantages:

Disadvanages Terminology

This unit in the course always seems to confuse students due to a lack of consistency in terminology.  Oftentimes, administrators are inconsistent in the way they use certain words and phrases associated with tournament scheduling.  For example, programmers and administrators may use the words, "tournament" or "league" (or may use the words division, pool or others) interchangeably.  For the purposes of this course, the following will serve as consistent definitions for this unit:

Tournament:
We will use the word "tournament" when we are referring to the overall event.   For example, we might program an intramural basketball tournament for 100 teams.  We will always use the word "tournament" when we are referring to the largest unit which we are programming.

League:
We will use the word "league" when we are referring to the different "groups" that we put our entries into.  For example, if we are programming an intramural basketball tournament for 100 teams, we might choose to break down our tournament into 20 leagues with 5 teams in each league.  This means that not all 100 teams will play each other.  Teams will only play other teams within their particular league.

Round Robin Basics -- Total Number of Games

In order to schedule a round robin tournament, it is necessary to understand several basic pieces of information.  Because round robin tournaments generally take a longer amount of time to complete (longer than single elimination tournaments or even double elimination tournaments), you need to figure out how many games it will take to complete your tournament. For example, if you are programming an intramural singles tennis tournament in round robin format and you have 30 people entered, you need to place these people into different leagues.  You have several choices that you can make.  You could offer one big league of 30 teams and have every tennis player play everyone else.   You could offer ten small leagues with 3 teams in each league, where teams will only play the other teams in their particular league.  How do you choose?  What is the difference between one BIG 30 team league and ten small leagues of 3 teams?   You still have the same number of TOTAL teams, you are just formatting them differently.  The major differences between these choices lie in the number of games that EACH TEAM will play, depending on the number of teams in their league, and the number of games that it will take to complete EACH LEAGUE, depending on the number of teams in their league.  We use some simple formulas to arrive at each answer:

Where "n" = the number of teams in a LEAGUE:

	Number of games per team/entry = n - 1
	Number of games per league = n(n-1)/2
	Number of games needed to complete the tournament = n(n-1)/2 * number of leagues

Lets look at some examples:

ex. 1:  Four leagues of 8 teams each

number of games per team:  8 - 1 = 7  each team will play 7 games
number of games per league:  8(8-1)/2 = 28
number of games to complete the tournament = 28 * 4 = 112 games

ex. 2:  Three leagues of 7 teams each and ten leagues of 6 teams each
Hint:  when you have leagues with unequal numbers of teams, treat these as separate problems.  For example:

number of games per team:  7 - 1 = 6 each team will play 6 games in these league
number of games per league:  7(7-1)/2 = 21 games
number of games to complete the tournament:  21 * 3 = 63 games

REMEMBER:  You aren't done....you need to figure out the second half of the problem!

number of games per team:  6 - 1 = 5 each team will play 5 games in these leagues
number of games per league:  6(6-1)/2 = 15 games
number of games to complete the tournament:  15 * 10 = 150 games

REMEMBER:  You STILL aren't done...in order to figure out TOTAL games, add both answers together:

63 + 150 = 213;  in this example, it will take 213 games to complete this tournament!

Lets go back to our tennis tournament that we are programming from above!
If you were to use the formulas, you would find the following information regarding some different scheduling combinations involving 30 total teams:
 

Leagues Offered	Games per team n-1	Games per league n(n-1)/2	Games per tournament n(n-1)/2 * # of leagues
One league of 30 teams
Two leagues of 15 teams
Three leagues of 10 teams
Five leagues of 6 teams
Six leagues of 5 teams
Ten leagues of 3 teams

Do you see that even though the total number of teams in our tournament remains constant (30 teams), the league format that we choose has a big impact on the number of games it will take us to complete our tournament?  It also has a big impact on the number of games that each team will get to play.  If you were on a team in one big 30 team league, your team would get to play 29 games!  However, if you were on a team that was in a 3 team league, your team would only get to play 2 games.  So it becomes apparent that how you format your league will have a big impact on the time it takes to finish your tournament and the number of games that each team in your tournament will get to play.

Your decision to go with ten leagues of 3 rather than two leagues of 15 will be largely dependent __________________ that you have available in which to play.  If you only have a total of 40 games available to you on your facility, and you have 30 teams that have entered to play, you are limited in the choices that you can make.  Your only choice would be to program your tournament in ten leagues of 3 teams, because the total number of games it will take to play under this format (30) is less than the total number of games available to you (40).  If you choose any other combination, you will not be able to play all of your games.

Facility Availability

One of the FIRST items of concern for a recreational sports programmer is to get an idea of the availability of your potential playing site.  Each setting and each organization will have different ways of reserving facility space.  In some organizations, you will have unlimited access to facilities for your programs' use.   In other settings, you will have to request to use certain facilities and you better have a good idea of what you will need, because if you do not utilize the facilities which have been reserved for you, you will undoubtedly be impacting the ability of others to run other programs.  For example, if you are programming an intramural soccer league and you reserve 4 soccer fields for play at 5:00 p.m., 6:00 p.m., 7:00 p.m. and 8:00 p.m., but you only end up playing games at 5:00 p.m. and 6:00 p.m., you will be wasting the game times available to you at 7:00 p.m. and 8:00 p.m.  The next time you decide to run a soccer league, you can bet that the facility coordinator will remember that you didn't maximize the use of your facility!  Count on getting less facility availability the next time around!

Example:

Lets go back to our example of the intramural tennis league that we are going to program.  The first thing that you should do is reserve your tennis facility.  Lets say that our facility coordinator has allowed you to play matches Monday through Friday, on 4 courts, between the hours of 4:00 p.m. and 7:00 p.m. during a 1 week period of time.  Now you have a decision to make.  Because you only have limited use of your tennis courts, you must consider putting a time limit on each of the matches that will be played.  How long are your tennis matches going to last?  Are you going to play matches on the hour?  If so, you have to modify your tennis rules so that your matches will be complete in one hour.  The decision that you make will have an impact on the number of teams that you will be able to allow to enter into your league.  This is your decision, and is based on a number of factors, including the philosophy of your program, the number of participants you want to allow play, the number of matches you want to give each participant, etc.

Okay, lets say you've decided to play matches consisting of one set.   The winner of the set will win the match, and you feel confident that participants should be able to play a set in approximately 45 minutes.  Allowing time for injuries, check-in and other possible delays, you decide to schedule matches every 1 hour.  Your next step is to develop a Master Facility Schedule showing the times you have available to play.   Develop your Master Schedule for every day that the league is going to operate.

Monday:
 

Time/Field	1	2	3	4
4:00 p.m.
5:00 p.m.
6:00 p.m.

Tuesday
 

Time/Field	1	2	3	4
4:00 p.m.
5:00 p.m.
6:00 p.m.

Wednesday:
 

Time/Field	1	2	3	4
4:00 p.m.
5:00 p.m.
6:00 p.m.

Thursday:
 

Time/Field	1	2	3	4
4:00 p.m.
5:00 p.m.
6:00 p.m.

Friday
 

Time/Field	1	2	3	4
4:00 p.m.
5:00 p.m.
6:00 p.m.

Now that you have your Master Facility Schedule drawn, you can figure out how many total matches are possibly available to you.  In this case, you see that you are able to play 12 matches in a day.  This would allow you to play 60 matches in one week.   Since you have only have 1 week reserved, you can play a total of 60 possible matches.  This information is the starting point to help you begin creating your round robin schedule!

Who Plays Who?

Now that you know what your facility availability looks like and now that you know the total number of matches you POSSIBLY could play, it is time to start the scheduling process.  Lets go back to the example we have been using.  Lets say that we have 30 participants signed up to play in our tennis tournament.  After we develop our tournament Master Schedule, we need to make a decision on league and tournament format.  How many leagues are we going to offer?  How many participants are going to be in each league?  Now is the time to start playing with possible team/league combinations, and plugging numbers into the formulas we learned to see how many games it is going to take us to play our tournament.  You can go ahead and try various combinations that add up to your total of 30 participants (one league of 30, two leagues of 15, etc.)!   (Unfortunately you won't have this opportunity on the exam....you'll have to do everything longhand!).

Upon reviewing your choices, which combination of leagues and teams makes the most sense?  Remember, we only have 60 possible matches available to us.  Knowing this, we can forget about one big 30 participant league (takes 435 matches), two leagues of 15 participants (takes 210 matches), three leagues of 10 participants (takes 135 matches), or five leagues of 6 participants (takes 75 matches).  So what's left?  It looks like we could choose, 1) six leagues of 5 participants (60 matches) or, 2) ten leagues of 3 participants (30 matches).  In this case, we want to maximize the use of our facility and waste as few matches as possible.  If we go with choice #1, we will not waste any matches...we will use every available match time (60 minus 60).  If we choose choice #2, we would be wasting 34 matches (64 minus 30).  So lets go with choice #1.

So how is our schedule going to look?  Who plays who?  In some settings, your work is practically done.  Now that you made the tough decisions, you may have the luxury of plugging all of this information into a computer scheduling program and the computer will do all of the scheduling work for you.  Other settings aren't so fortunate, though, and you will be stuck doing schedules the old way.  Since we chose to go with six leagues of 5 participants, we should come up with a basic scheduling template.

Scheduling Template

The first step to accomplish is to figure out how many rounds are going to be played in your tournament.  A round is just one completed set of games played in the entire round robin tournament.  To figure out how many rounds make up your tournament, you can use these formulas:

where "n" = the number of teams in the league:

For even number entries:  n - 1
For odd number entries:  n

In our case, we have six leagues of 5 teams.  Since "5" is an odd number, it will take 5 rounds to complete one of our 5 team leagues.
 

Rounds	1	2	3	4	5

The second step to accomplish is to figure out how many games (or matches) are going to be played in each round.  You can accomplish this by using these formulas:

where "n" = the number of teams in the league

For even number entries:  n/2
For odd number entries:  n-1/2

In our case, we are going to play 2 games per round.  This is because 5 is an odd number; so we use the formula n-1/2 to yield our number of games.  Since n-1 = 4, and 4/2 = 2, we will play 2 games per round.
 

Games/Rounds	1	2	3	4	5
Bye
1
2

You can see that in our example, we placed the word "Bye" in the top slot.   Whenever you have an odd number of entries in a league, you will always have one team that will not be able to play during that round.  The "bye" rotates, so that every team eventually will sit out for one round.  Even though we don't consider the "bye" slot to be a game, we still account for it because it makes our scheduling process easier.

In order to figure out who plays who in each round (for odd number entries), begin in the first round.  Always place a "B" in the upper left had corner, followed by a sequence of numbers starting with "1" for each game in the round.   When you hit the last game listed, come back up the column, beginning from the bottom to the top.  The "v" between the numbers stands for "versus".  For example:
 

Rounds	1	2	3	4	5
Bye
1
2

For subsequent rounds, again begin with the "B" in the upper left hand corner.  The other numbers will rotate counterclockwise, beginning with the "bye" team.  The "B" always remains constant.  Do this for all subsequent rounds.  For example:
 

Rounds	1	2	3	4	5
Bye
1
2

You now have a scheduling template for a 5 team league.  Notice that by the end of 5 rounds, every team has played every other team, and each team has taken a turn as the "bye" team.

To create a scheduling template for a league with an even number of teams, use the same principle, except the number "1" will be your constant.  Place the "1" in the upper left-hand corner and rotate all other numbers around the 1 in a counterclockwise pattern.  For example, lets look at a 4 team league:

Number of rounds:  use n - 1; we have 4 -1 = 3 rounds
Number of games per round:  use n/2; we have 4/2 = 2 games per round
 

Games/Rounds	1	2	3
1
2

Lets recap...we have been working on putting together a round robin schedule for a tennis tournament consisting of 30 teams.  In order to accomplish this feat, we have gone through the following steps: Created a Master Facility Schedule based on facility availability

Determined the total number of possible games available to us to play

Figured out how many leagues we would offer and how many teams would be in each league

Determined the number of rounds and number of games played in each round for each of our leagues

Created a scheduling template for each league, determining the opponents for each round of the tournament

Now its time to place our participants in their appropriate leagues and schedule our tournament.

Determining which participants are placed in which leagues is totally at the discretion of the programmer.  This is your chance to place participants into leagues based on skill level, time preferences, gender, housing units or other units of participation.  You can obtain this information from your participants when they register for your tournament.  Have the participants note on their registration form the information that you would want in order to place them in a league.  Lets keep our example simple....in our tournament, we have 30 total participants.  We have 15 men and 15 women.  Of the 15 men, 5 are considered "advanced" and 10 are considered "intermediate".  Of the women, 5 are considered "advanced", 5 are considered "intermediate" and 5 are considered "beginners".  We will assume that all participants will be available to play at the times we schedule them...we will not have any participant time conflicts.

Our participants are:
Arnold, Alice, Bob, Barbara, Chuck, Candy, Doug, Darla, Ed, Emily, Frank, Feona, Greg, Glenda, Harry, Helen, Jerry, Jackie, Kurt, Karla, Larry, Laura, Mike, Millie, Nick, Nora, Pete, Paula, Rich, Rachael

Remember, we decided to program six leagues of 5 participants each.
 

League/Participant	Men's Advanced	Men's Intermediate 1	Men's Intermediate 2	Women's Advanced	Women's Intermediate	Women's Beginner
1
2
3
4
5

We now need to place our league matches on our Master Facility Schedule.   Lets go ahead and begin with the first round of our tournament.  There are several rules of thumb that we want to use when we are determining our league schedules.   Remember, we want to be fair to everyone.  When possible, try to follow these basic rules:
 

Lets go ahead and start.  We need to first determine how many match slots one of our leagues will occupy in order to complete one round.  We can get this information by looking at the number of games that we will play in each round.   Remember, we can use the formula n-1/2 to figure this out.  In this case, we are playing in 5 team leagues.  Therefore, 5-1 = 4 and 4/2 = 2.  So each league will require us to play 2 matches to complete one round.  This means that each league will require the use of 2 courts to play their first round matches.

Lets use the following abbreviations for our leagues:

Men's Advanced = MA
Men's Intermediate 1 = MI-1
Men's Intermediate 2 = MI-2
Women's Advanced = WA
Women's Intermediate = WI
Women's Beginner = WB

Begin plugging in leagues into available time slots, making sure you use 2 courts for each league.  Where you place the leagues on the schedule is up to you!

Monday:
 

Time/Field	1	2	3	4
4:00 p.m.
5:00 p.m.
6:00 p.m.

Remember our scheduling template?  This is going to decide the schedule for Monday's matchups.  In a 5-team league, we determined that our first round would be:

Bye - 5
1 v 4
2 v 3

So our first round matchups in the Men's Advanced League will be:

Court 1 at 4:00 p.m.  Arnold v. Doug
Court 2 at 4:00 p.m.  Bob v. Chuck
(Ed has the Bye!)

Go ahead and repeat this scheduling process for all of your leagues, and you have scheduled the first round of your tournament!

In order to schedule the subsequent rounds of your tournament, repeat this process.  Remember to try and keep match times consistent, when possible.  If the Men's Advanced league is playing at 4:00 p.m. on Mondays, try to have them play at 4:00 p.m. on Tuesdays as well.  Lets try this again:

Tuesday:
 

Time/Field	1	2	3	4
4:00 p.m.
5:00 p.m.
6:00 p.m.

Notice that our league times are able to stay the same.  The only thing that will change, here, is going to be our scheduling matchups.  For example, you will now use the scheduling template for round 2 of a 5 team round robin to determine Tuesday's matches.

Our second round template shows that:

Bye - 4
5 v 3
1 v 2

So our second round matchups in the Men's Advanced league will be:

Court 1 at 4:00 p.m. Ed vs. Chuck
Court 2 at 4:00 p.m. Arnold vs. Bob
(Doug now has the Bye!)