Configuration Bias (Unfairness)
in four-player chess

Symmetry and Bias


SYMMETRY
In order for each player to have an equal opportunity of winning in chess (an unbiased chance to win), there  must be adequate symmetry.  In two-player chess, only one axis of symmetry is required.  But four-player chess requires two axis of symmetry, as follows:

Possible fold lines in four-player chess

1)  At least two fold lines perpendicular to each other must also be axes of symmetry for the piece placement.  The possible combinations are the two dashed, purple lines (W-E and N-S), or the two dashed, blue diagonal lines in the depiction above. To be an axis of symmetry, the folded board must have like-pieces overlay like-pieces exactly (pawn on pawn, rook on rook, knight on knight, bishop on bishop, king on king, and queen on queen),

2)  Or, the board when rotated through 90 and 180 degrees must have all like-pieces overlap perfectly (pawn on pawn, rook on rook, knight on knight, bishop on bishop, king on king, and queen on queen).  The rotational axis is a line coming perpendicularly out of the board, depicted by the circular arrow on the depiction above.  This is called a rotational axis of symmetry

In four-player chess, E-W  and N-S axes of symmetry are not possible, because a king will always overlay a queen on either fold line, making symmetry impossible.  That means only diagonal symmetry or rotational symmetry are possible.  Configuration 1 provides rotational symmetry.  Configuration 2 provides diagonal symmetry.  But Configuration 3 is biased (unfair) and thus should not be used.  Click on the links below for a demonstration and further discussion.  

Configuration 1
All queens on white (or all on  black)
No Bias (Fair)
Configuration 2
All queens on their left (or all on  right)
No Bias (Fair)
Configuration 3
queen facing queen
(Biased (Unfair)
    This

2010 Simon Revere Mouer III, PhD, PE, all rights reserved