Do you buy him a new drink, or do you try to outstare him to make him back off?
From the sciencedaily.com
Mair's lion-like face took on the aspect of the king of beasts trying to outstare its tamer.
From the telegraph.co.uk
He can dance like a bear, sing like a frog and outstare an owl.
From the time.com
He was a hero because he seemed to outstare history, reversing trends and forces that had seemed irrevocable.
From the time.com
And in Baghdad, Saddam Hussein was planning to outstare his opponents and deny access to international inspectors who might spot his plans for germ weapons on a far vaster scale.
From the newscientist.com
I can outlook a panther and outstare a flash of lightning, tote a steamboat on my back and play at rough and tumble with a lion, and an occasional kick from a zebra.
From the en.wikipedia.org
More examples
Stare down: overcome or cause to waver or submit by (or as if by) staring; "He simply stared down his opponent"