I've been asked by my professor to come up with a two-player game in which both players have three moves, there is no Nash equilibrium when they move simultaneously, and there is also no first-mover or second-mover advantage. I have a feeling this might be a trick question, as intuition tells me that any game with no mover advantage must have some kind of Nash equilibrium, but I'm not super confident in that and I might just be wrong. Can someone better at game theory than me confirm or deconfirm my suspicions?