On Semi-Restricted Rock, Paper, Scissors

Spiro, Surya and Zeng (Electron. J. Combin., 2023) recently studied a semi-restricted variant of the well-known game Rock, Paper, Scissors; in this variant the game is played for $3n$ rounds, but one of the two players is restricted and has to use each of the three moves exactly $n$ times. They find the optimal strategy, and they show that it results in an expected score for the unrestricted player $\Theta(\sqrt{n})$; they conjecture, based on numerical evidence, that the expectation is $\approx 1.46\sqrt{n}$. We analyse the result of the strategy further and show that the average is $\sim c \sqrt{n}$ with $c=3\sqrt{3}/2\sqrt{\pi}=1.466$, verifying the conjecture. We also find the asymptotic distribution of the score, and compute its variance.