The usual definition of Nash equilibrium requires only ≤, not <, so Defect-Defect, Cooperate-Defect and Defect-Cooperate are Nash equilibria (and pure) but not "strong" Nash equilibria. You want this definition, because games need not have strong Nash equilibria, even if you allow mixed strategies.
(Apparently the game is called "weak prisoner's dilemma" in the literature).
Oops, didn't realize that.
I found this to be a very interesting method of dealing with a modified Prisoner's Dilemma. In this situation, if both players cooperate they split a cash prize, but if one defects he gets the entire prize. The difference from the normal prisoner's dilemma is that if both defect, neither gets anything, so a player gains nothing by defecting if he knows his opponent will defect; he merely has the option to hurt him out of spite. Watch and see how one player deals with this.
http://www.youtube.com/watch?v=S0qjK3TWZE8