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@BernardoRecamánSantos On Ed Pegg's FB page (in a comment to the relevant post) he says "I didn't find it. There's a chinese website devoted to the orchard problem, and the above was found after a multi-year search." I'm not sure what the right attribution would be, but apparently it doesn't originate with Ed himself.
@justhalf Yes, if both players agree to play uniform distribution, then the average score per turn is 4/9. Even better, if both players agree to alternate between the outcomes (Scissors, Rock) and (Rock, Scissors) then the average score per turn is 1! So the players can profit by collaborating. But neither situation gives a Nash equilibrium. If you do play uniform distribution, then a rational opponent will play Rock against you every time (giving them 2/3 per turn and you 1/3 per turn).