We start with giving an axiomatic characterization of the Nash equilib
rium (NE) correspondence in terms of consistency, converse consistency
, and one-person rationality. Then axiomatizations are given of the st
rong NE correspondence, the coalition-proof NE correspondence and the
semi-strong NE. In all these characterizations consistency and suitabl
e variants of converse consistency play a role. Finally, the dominant
NE correspondence is characterized. We also indicate how to generalize
our results to Bayesian and extensive games.