in a rock paper scissors game, there are no pure Nash equilibrium and only a completely mixed Nash equilibrium.
in prisoner's dilemma, there is no mixed Nash equilibriumand only a pure Nash equilibrium.
Is it established in game theory how you can know if a game only has completely mixed Nash equilibrium or only pure Nash equilibrium?
For example, games with only mixed nash have a cyclical payoff structure: there are no fixed points in the players pure best responses.
Whereas for a game to have no mixed strategy, the pure strategy must have a strict dominance property: if it is obviously the best strategy that the players can use, then there is no need for the players to mix. But I am not certain if any game with a strict dominating pure strategy will necessarily not have a mixed strategy.
Has someone thought about these before?