Close your eyes. Apparently your opponent will have an increased tendency to imitate your move increasing the chance of a draw.  At least that is what is reported in this study.  A blindfolded player played RSP against a sighted player and their outcomes were compared to a control treatment in which two blindfolded players played.

A draw was achieved almost exactly 1/3 of the time when the two blindfolded players met, but that rate increased to 36.3% in the blind-sighted treatment, a statistically significant difference.  The authors attribute this to a sub-conscious tendency to imitate the actions of others.  In particular, when the blind player completed his move more than 200 miliseconds prior to the sighted player, the sighted player had an increased tendency to play the same move.

200 miliseconds is too fast for conscious reaction but still within the time necessary for the visual signal to be sent to the brain and an impulsive response signal to be sent to the hand.

If this is true then you should be able to increase your chance of winning in RSP by holding rock until the very last opportunity and then throw paper.  You will sometimes trigger an automatic imitation of your rock and win with your paper.

Are there even more draws when both players have their eyes open?

(Fez float:  Not Exactly Rocket Science.)