Against $9 rock, X always chooses $1. Consider the problem “symmetrical ultimatum game against X”. By symmetry, X on average can get at most $5. But $9 rock always gets $9. So $9 rock is more rational than X.
I don’t like the implied requirement “to be rational you must play at least as good as the opponent” instead of “to be rational you must play at least as good as any other agent in your place”. $9 rock gets $0 if it plays against $9 rock.
(No objection to overall no-free-lunch conclusion, though)
I don’t like the implied requirement “to be rational you must play at least as good as the opponent” instead of “to be rational you must play at least as good as any other agent in your place”. $9 rock gets $0 if it plays against $9 rock.
(No objection to overall no-free-lunch conclusion, though)
In the first case, the problem is “symmetrical ultimatum game against X”, in which $9 rock does get $9.
In the second case you are correct, in the problem “symmetrical ultimatum game against $9 rock” $9 rock gets $0.