Nash Equilibrium Brainteaser

We play a modified game of rock, paper, scissors. We each put up two hands (for example Rock and Scissors). We see what each other’s hands.

Then, simultaneously, we both pull one hand back, and play the hands that are still out.

Consider a scenario where Player 1 puts up Rock and Paper. Player 2 puts up Rock and Scissors. What is the optimal play here, which hands does each player pull back?

There does not appear to be a Nash equilibrium here.

On the one hand, Player 1 should favor Rock, as he either ties if Player 2 puts up Rock, or wins if Player 2 puts up Scissors. If we use the same logic, Player 2 should favor Scissors, as he then either wins if 1 puts up Paper, or loses if he puts up Rock. The sample outcomes for Player 2 are worse if he puts up Rock (either tie or loses). However, if Player 2 knows Player 1 is more likely to play Rock, he surely will not play Scissors.

There seems to be a constant flipping of what each player should play, when the two players factor in what the other player should ‘optimally’ do. What is your approach to this? Should both players just play Rock and tie to minimize variance? Although this would be bad of Player 1 as he theoretically has the edge…