COLOUR BALANCE

ROUND ROBIN TOURNAMENT

The colour balance after an even number of games,
is elementary according to the rules of the swiss system,
but that is not the case with the FIDE Round Robin tables
with an even number players.

Example:
7-8 players (Berger)
 1    2    3    4    5    6    7
1-8  8-5  2-8  8-6  3-8  8-7  4-8
2-7  6-4  3-1  7-5  4-2  1-6  5-3
3-6  7-3  4-7  1-4  5-1  2-5  6-2
4-5  1-2  5-6  2-3  6-7  3-4  7-1

For example, it is not fair after 6 rounds,
(before the last round)
for some players to have 4 white and 2 black while others
have 2 white and 4 black.

                                 colours per round
GRENKE Baden-Baden 2015               2    4    6
SNo.   Name               Rtg FED    W B  W B  W B
 1  GM Caruana Fabiano   2811 ITA    2 0  3 1  4 2
 2  GM Bacrot Etienne    2711 FRA    1 1  3 1  4 2
 3  GM Aronian Levon     2777 ARM    1 1  2 2  4 2
 4  GM Adams Michael     2738 ENG    1 1  2 2  3 3
 5  GM Naiditsch Arkadij 2706 GER    0 2  1 3  2 4
 6  GM Carlsen Magnus    2865 NOR    1 1  1 3  2 4
 7  GM Baramidze David   2594 GER    1 1  2 2  2 4
 8  GM Anand Viswanathan 2797 IND    1 1  2 2  3 3

The Double Round Robin tournament consists
of a reversing order of the last two rounds of the first phase (cycle).

7-8 players (Berger)
 1    2   3    4    5   6/7  7/6
1-8  8-5  2-8  8-6  3-8 4-8  8-7
2-7  6-4  3-1  7-5  4-2 5-3  1-6
3-6  7-3  4-7  1-4  5-1 6-2  2-5
4-5  1-2  5-6  2-3  6-7 7-1  3-4
 
 8    9   10   11   12   13   14
8-1  5-8  8-2  6-8  8-3 8-4  7-8
7-2  4-6  1-3  5-7  2-4 3-5  6-1
6-3  3-7  7-4  4-1  1-5 2-6  5-2
5-4  2-1  6-5  3-2  7-6 1-7  4-3

This prevents three consecutive games of the same colour,
but after rounds 2, 4, 6 some players have no colour balance. 

                                   colours by round
FIDE Candidates 2016                   2    4    6
SNo.   Name               Rtg FED     W B  W B  W B
 1  GM Karjakin Sergey   2760 RUS     2 0  3 1  3 3
 2  GM Nakamura Hikaru   2790 USA     1 1  3 1  3 3
 3  GM Giri Anish        2793 NED     1 1  2 2  3 3
 4  GM Anand Viswanathan 2762 IND     1 1  2 2  4 2
 5  GM Topalov Veselin   2780 BUL     0 2  1 3  3 3
 6  GM Aronian Levon     2786 ARM     1 1  1 3  3 3
 7  GM Caruana Fabiano   2794 USA     1 1  2 2  3 3
 8  GM Svidler Peter     2757 RUS     1 1  2 2  2 4

Different options
only for tables for even number of players
which is the most common.

Example:
7-8 players (Berger)
 1    2    3    4    5    6    7
1-8  8-5  2-8  8-6  3-8  8-7  4-8
2-7  6-4  3-1  7-5  4-2  1-6  5-3
3-6  7-3  4-7  1-4  5-1  2-5  6-2
4-5  1-2  5-6  2-3  6-7  3-4  7-1

There are several options:

A) All the colours of the last player change in the Berger tables.

Swap colours in the first row:

   7-8 players (Berger)
    1    2    3    4    5    6    7
   8-1  5-8  8-2  6-8  8-3  7-8  8-4 : swap colours
   2-7  6-4  3-1  7-5  4-2  1-6  5-3
   3-6  7-3  4-7  1-4  5-1  2-5  6-2
   4-5  1-2  5-6  2-3  6-7  3-4  7-1

Optionally, in order for the first round to stay the same,
swap 1 with the last number.

   7-8 players (Berger)
    1    2    3    4    5    6    7
   1-8  5-1  1-2  6-1  1-3  7-1  1-4
   2-7  6-4  3-8  7-5  4-2  8-6  5-3
   3-6  7-3  4-7  8-4  5-8  2-5  6-2
   4-5  8-2  5-6  2-3  6-7  3-4  7-8

In Cyclic tables swap all the colours of the first player with the number 1.
Optionally, in order for the first round to stay the same,
swap 1 with the last number.

In Double Round Robin tournament the rounds of the two phases (cycles) are played in reverse colours and in reverse order.

B) The tournament starts from round 2,
the first round is played last (Berger or Cyclic).

  7-8 players (Berger)
    2    3    4    5    6    7    1
   8-5  2-8  8-6  3-8  8-7  4-8  1-8
   6-4  3-1  7-5  4-2  1-6  5-3  2-7
   7-3  4-7  1-4  5-1  2-5  6-2  3-6
   1-2  5-6  2-3  6-7  3-4  7-1  4-5

In Double Round Robin tournament the rounds of the two phases (cycles) are played in reverse colours and in reverse order.

C) The rounds are played from the last round to the first round:
(7,6,5,4,3,2,1) (Berger or Cyclic).

   7-8 players (Berger)
    1    2    3    4    5    6    7
   4-8  8-7  3-8  8-6  2-8  8-5  1-8
   5-3  1-6  4-2  7-5  3-1  6-4  2-7
   6-2  2-5  5-1  1-4  4-7  7-3  3-6
   7-1  3-4  6-7  2-3  5-6  1-2  4-5

In Double Round Robin tournaments
the first cycle is played with reverse colours and
the rounds in reverse order:

   7-8 players (Berger)
    1    2    3    4    5    6    7 
   8-4  7-8  8-3  6-8  8-2  5-8  8-1
   3-5  6-1  2-4  5-7  1-3  4-6  7-2
   2-6  5-2  1-5  4-1  7-4  3-7  6-3
   1-7  4-3  7-6  3-2  6-5  2-1  5-4

then in the second cycle the FIDE table:

    8    9    10   11   12   13   14
   1-8  8-5  2-8  8-6  3-8  8-7  4-8
   2-7  6-4  3-1  7-5  4-2  1-6  5-3
   3-6  7-3  4-7  1-4  5-1  2-5  6-2
   4-5  1-2  5-6  2-3  6-7  3-4  7-1

Links:
calmapalma
devenezia

TWO TEAMS TOURNAMENT (SCHEVENINGEN)

(with equal number of players)

The conditions of the system in order of priority:

1. Each player on one team plays each player on the other team.

2. Each player plays
an equal number of games with white and black.

If the number of players in the team is an odd number, then in all games:

a) One of the teams has one more colour than the other.

b) For half number of players (rounded to the nearest integer) one colour is increased by 1 and for the other players decreased by 1.

3. No player plays the same colour three times in a row.

4. Each team in each round plays
an equal number of games with white and black.

This does not apply if the players in each team are 2.

If the number of players in the team is an odd number,
then one colour is increased by 1,
while in the next round this colour is decreased by 1.

5. In the n-th round the player A1 plays with the player Bn.
Player A1 changes color in the next round.
In the first round the player An plays with the player Bn.

The problem is not easily solved
if the number (6, 10, 14, 18 ...) of players in the team
divided by 4 gives a remainder 2.

One solution precedes another
if the conditions in the order of priority are met:

A. After all rounds

1. the number of players changing colour in the next round is the maximum.

2. the absolute difference of the number of players in a team
who change colour in the next round
minus the same of the other team is minimal.

B. After all even number rounds

1. the number of players with balanced colours is the maximum.

2. the absolute difference of the number of players in a team
which have balanced colours
minus the same of the other team is minimal.

Example:
Match on 6 Boards
Round 1 A1-B1 B2-A2 B3-A3 A4-B4 B5-A5 A6-B6
Round 2 B2-A1 A2-B3 A3-B5 B6-A4 A5-B4 B1-A6
Round 3 A1-B3 B5-A2 B1-A3 A4-B2 A5-B6 B4-A6
Round 4 B4-A1 B6-A2 A3-B2 A4-B1 B3-A5 A6-B5
Round 5 A1-B5 A2-B4 A3-B6 B3-A4 B1-A5 B2-A6
Round 6 B6-A1 A2-B1 B4-A3 B5-A4 A5-B2 A6-B3
Players with alternation color:
Team A: 22 times,
Team B: 22 times
Players with color balance in even number round:
Team A: 16 times,
Team B: 16 times
 
FIDE table
Round 1 B1-A1 B5-A2 A3-B4 A4-B2 A5-B3 B6-A6
Round 2 B2-A1 A2-B1 B3-A3 B4-A4 A5-B6 A6-B5
Round 3 A1-B3 A2-B2 B1-A3 B6-A4 B5-A5 A6-B4
Round 4 A1-B4 B6-A2 A3-B5 A4-B1 B2-A5 B3-A6
Round 5 B5-A1 B4-A2 A3-B6 B3-A4 A5-B1 A6-B2
Round 6 A1-B6 A2-B3 B2-A3 A4-B5 B4-A5 B1-A6
Players with alternation color:
Team A: 20 times,
Team B: 24 times
Players with color balance in even number round:
Team A: 16 times,
Team B: 16 times

Links:
calmapalma
wikipedia

Bibliography: books.google

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