my observations thus far:

with this scoring format, picking randomly is a bad strategy because that can be exploited by picking scissors very often (scissors vs random would have a positive expectation of 2/3 pt per turn, rock and paper vs random has exp. of -1/3 pt per turn each).
So generally against people with a losing or breakeven record, I just try to outplay what I expect them to be thinking on the first level; against winning opponents I'm either trying to outplay their 2nd/3rd level thinking on each individual turn or trying to make them fire paper too often on the whole.

In 1-1-1 RPS it becomes basically a crapshoot between top players thinking at the umpteenth level because picking randomly is an unexploitable breakeven strategy. However, in 1-2-3 RPS I think there is a ton of strategy involved and playing optimally would have to involve degrees of both good reading of opponent and some randomization (in a non 1:1:1 ratio)