There’s an unstated assumption here that you start with $0. Suppose instead you start with $0.5: then E(U(a))=√1=1 while E(U(b))=23√1.5+13√0.5≈1.05. So if you play game (a) first, you’d then prefer to play game (b) second.
But this doesn’t fully resolve the question, because you’d still prefer (b, b) over (a, b).
There’s an unstated assumption here that you start with $0. Suppose instead you start with $0.5: then E(U(a))=√1=1 while E(U(b))=23√1.5+13√0.5≈1.05. So if you play game (a) first, you’d then prefer to play game (b) second.
But this doesn’t fully resolve the question, because you’d still prefer (b, b) over (a, b).