It can be done like this:
ping-pong darts billiards cornhole sit out
(18 11 10 3) (19 5 8 15) (1 17 12 7) (20 14 2 6) ( 9 16 13 4)
(19 12 6 4) (20 1 9 11) (2 18 13 8) (16 15 3 7) (10 17 14 5)
(20 13 7 5) (16 2 10 12) (3 19 14 9) (17 11 4 8) ( 6 18 15 1)
(16 14 8 1) (17 3 6 13) (4 20 15 10) (18 12 5 9) ( 7 19 11 2)
(17 15 9 2) (18 4 7 14) (5 16 11 6) (19 13 1 10) ( 8 20 12 3)
note that if you divide the 20 people in to teams of 5 as follows:
team 1= 1-5
team 2=6-10
team 3=11-15
team 4=16-20
then each game has one member from each team.