Is Rock Paper Scissors with two players but three contestants a fair game?

In a game of Rock Paper Scissors with two players, there are $$3$$ outcomes every round each with multiplicity $$3$$.

• Player 1 can win.
• Player 2 can win.
• Both players can draw.

If we were to assign a winning result to a third "Player" when Player 1 and 2 draw, does this become a fair 3-player game?

• Player 1 wins.
• Player 2 wins.
• Both players draw, so Player 3 wins.

Player 1 and 2 have no incentive to cooperate to reduce Player 3's chances because the only way to do so is for one to forgo their own chance at winning.

Player 3 cannot negatively or positively reduce the chances of either Player 1 or Player 2 because they make no moves in this game.

This looks to create a Nash Equilibrium and a fair game.

Am I missing something? I'm worried that the fact that Player 3 does not participate at all may be causing some bias I'm overlooking.

• I wonder if there have been investigations published in the literature on the use and bias of spectator agents in general. Jul 3 '21 at 16:02
• I would advise you to not accept an answer so early, as it discourages other people from answering.
– Joe
Jul 3 '21 at 16:12