Addition of even<->odd alphabets positions as they appear in aphanumeric order...
GOLD,7,15,12,4=>19<->19 balanced
LEMONS,12,5,13,15,14,19=>39<->39 balanced
UNCOOKED,21,14,3,15,15,11,5,4=>44<->44 balanced
BORDERLINE,2,15,18,4,5,18,12,9,14,5=>51<->51 balanced
LILLIPUTIANS,12,9,12,12,9,16,21,20,9,1,14,19=>77<->77 balanced
COUNTERBALANCE, ---->balanced
PREDETERMINATION, ---->balanced
so the answer is :
BACKGAMMON
to see how they balence:
BACKGAMMON,2,1,3,11,7,1,13,13,15,14 => 40 <-> 40 Balanced
CHECKERS,3,8,5,3,11,5,18,19 => 37<->35 NOT BALANCED
CRIBBAGE,3,18,9,2,2,1,7,5 => 21<->26 NOT BALANCED
DOMINOES,4,15,9,13,15,5,19 => 47<->31 not balanced
Used following ref :
1 2 3 4 5 6 7 8 9 10 11 12 13
a b c d e f g h i j k l m
z y x w v u t s r q p o n
26 25 24 23 22 21 20 19 18 17 16 15 14