We propose an equilibrium concept, the Robust Nash equilibrium (RNE), that combines the best-reply rationality and the first mover invariance condition. The single-stage 2x2 symmetric information game G is transformed into sequential two-stage games with two sub-trees: STA has the row player starting and STB has the column player starting. A profile in G is robust if it is the strict SPNE of the two branches; it is ephemeral if it is not the SPNE of any branch. We show that every strict dominant strategy equilibrium of G is robust but not every strict Nash equilibrium of G is. We show further that every robust profile of G is always a strict Nash equilibrium of G. A Robust Nash equilibrium (RNE) of G is any robust profile of G. The RNE of G is unique. We show in particular that the payoff dominant strict Nash equilibrium of a coordination game G is RNE while the strictly payoff-dominated Nash equilibrium of G is ephemeral. The original Harsanyi-Selten preference for payoff dominance over risk dominance is supported by robustness without invoking collective rationality.