How does Oedipus Rex define leadership?
Great question! I have never thought of this play as offering a commentary on leadership. However, one way to approach this question would be to consider the ways that Oedipus is presented as a good leader. Certainly what strikes me is that he is a concerned leader desperate to try to find out the cause of the plague and to discover the murderer of Laius so that he can save his people. These are surely good leadership qualities! Although we could arguably say at times he displays arrogance with his dealings with Creon and Teiresias, this is partly because of his intense frustration at not being able to get to the truth that already he seems to be able to half-grasp.
I think therefore we can take Oedipus as a good but tragic model of leadership. Note how he responds to the pleas of the Priest at the beginning of the play:
Poor children! You may be sure I know
All that you longed for in your coming here.
I know that you are deathly sick; and yet,
Sick as you are, not one is as sick as I.
Each of you suffers in himself alone
His anguish, not another's: but my spirit
Groans for the city, for myself, for you...
We might read this as self-serving rhetoric, but I actually believe Oedipus here--he gives us no reason to doubt what he says and throughout the rest of the play sets himself on a course to discover what has happened and how Thebes can be healed--whatever the cost.