On semi-restricted Rock, Paper, Scissors

Spiro, Surya and Zeng (Electron. J. Combin. 2023; arXiv:2207.11272) 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 Θ(√(n)); they conjecture, based on numerical evidence, that the expectation is ≈ 1.46√(n). We analyse the result of the strategy further and show that the average is ∼ c √(n) with c=3√(3)/2√(π)=1.466, verifying the conjecture. We also find the asymptotic distribution of the score, and compute its variance.