Subject:
2-stage Side Events with Group Qualifications and KO Finals
Date:
Wed, 03 May 2000 22:24:32 +0200
From:
Robert Jasiek
Newsgroups:
rec.games.go, fj.rec.games.go, japan.games.go
A popular system for side events has been to have group
qualifications and knockout finals. So far organizers
had trouble to handle group sizes, rest players, round
numbers, or seedings. After an example I give a complete
basic theory that can be applied by event organizers.
*******
EXAMPLE
*******
Suppose you want to held a lightning tournament, e.g., on
two evenings. On the first evening you have 5 hours for the
qualifications and on the second evening you have 4 hours
for the knockout finals. 10 minutes thinking time. You
estimate the break between two games as 3 minutes. This
makes roughly 23 minutes per game. So you can play at
most 13 games on the first evening and 10 games on the
second evening. You get 431 registered players. For the
groups you set placement criteria within each group as
points-direct_comparison-SODOS, e.g.
In the qualifications you want to let everybody play
every round. Thus you choose the groups to have an
even number of players and the number of rounds one less.
This gives 13 rounds and groups of size 14. You need 31
groups. 30 groups have 14 players, the 31st group has 11
players. You fill it up to 13 players by taking 1 player
each from groups 29 and 30. Groups 29 to 31 then all have
13 players and 1 Bye.
(Filling groups with players by mixed strengths, e.g., is
well known.) In each group fix player 1 or the Bye and let
rotate all other players after each game. Thereby you need
exactly 13 rounds, not more.
The groups provide places for all players and you will seed
the top places first and so on until the finals' players
are determined. (The latest seeded players with a low place
are chosen randomly among all players with that place in
some group.)
At most 10 games for the finals would allow you to seed
2^10 = 1024 players. Oops, you don't have so many. So you
play fewer rounds. How many? Well, qualifications are meant
to eliminate some players, let's say slightly more than half
the players. 431/2 = 215.5. This is a first hint but not good
enough. We need a power of 2. Thus you seed 128 players and
have a knockout with 7 rounds. That's it.
******
THEORY
******
Initial Parameters
------------------
n = # registered players
R = max # rounds in group qualification stage
S = max # rounds in KO finals
For the numbers of rounds you consider your available time
and the desired thinking time.
Even Groups
-----------
g = even group size
r = played group rounds
R odd => g := R+1, r := R
R even => g := R, r := R-1
This always gives even groups and one round less to play
than the group size.
Number of Groups Z and Handling of Rest Players Y
-------------------------------------------------
Z := round_up(n/g)
If Z <> n/g, then a group of
Y := n mod g
rest players must be handled.
("mod" gives the integer rest of a division.)
This rest group is filled up to g - 1 players.
For this take 1 player each out of
g - 1 - (n mod g)
randomly chosen other groups.
Then g - (n mod g) groups have one Bye each.
Player-Group-Distribution
-------------------------
Use either the mix strategy or the homogenous strategy.
Mix: Let sink the highest ranked player into one group
at a time.
Homogenous: Fill a single group with the highest ranked
players left.
Round by round within each even group fix player 1 and
let all others move one place clockwise. In an even group
with a Bye fix it instead.
KO Rounds s
-----------
s := min { S, round_down(ld(n/2)) }
Normally, you take s := S. However, sometimes you could
play more rounds than are needed for a KO. Then the
second parameter is an emergency measure ensuring
that the maximal power of 2 less than half of n is taken.
("ld" is the logarithm in base 2.)
KO Participants k
-----------------
k := 2^s
They will have to win their way through a binary knockout
tree.
Seeding
-------
Pi; i=1..n is the place due to the placement conditions
(like points-direct_comparison-SODOS) after round r of
player i in his group. The same place within a group
may be reached by more than one player. Places are
numbered due to the order 1,2,..,g.
Q := { Pi | i=1..n } is the qualification pool of all
players' places.
Example:
All Pi (the places of the players):
In group 1 with 6 players: 1 2 2 2 5 6 (here 3 players
tie for place 2 even by SODOS).
In group 2 with 5 players and 1 Bye: 1 2 3 4 5.
Then you combine all groups and get the places (and their
corresponding players): 1 1 2 2 2 2 3 4 5 5 6.
If you want to seed 4 players, then the two first place
players are seeded and two of the four second place players
are randomly drawn.
End Example.
// seeding procedure
// all players of the remaining top places are seeded
// ,however, for the last free of all k finals' places
// random selection is done among equal qualification
// place getters
c := 0;
for (j:=1; j<=g; j++)
{
M := { q in Q | q = j };
c += |M|;
IF (c<=k) {seed M; IF (c=k) break;}
ELSE {randomly_select_and_seed k-c+|M| of M; break;}
}
--
robert jasiek
http://www.snafu.de/jasiek/rules.html