• WordNet 3.6
    • v outstare overcome or cause to waver or submit by (or as if by) staring "He simply stared down his opponent"
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Webster's Revised Unabridged Dictionary
    • v. t Outstare To excel or overcome in staring; to face down. "I would outstare the sternest eyes that look."
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Century Dictionary and Cyclopedia
    • outstare To stare out of countenance; face down; browbeat; outface.
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Chambers's Twentieth Century Dictionary
    • v.t Outstare owt′stār (Shak.) to stare down or abash with effrontery.
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In literature:

We were less than a skiff's length apart, each trying to outpose and outstare the other.
"Roof and Meadow" by Dallas Lore Sharp
If you have the stoniest of British stares he will outstare you.
"Brothers of Pity and Other Tales of Beasts and Men" by Juliana Horatia Gatty Ewing
On the rock of ages he first opened his eyes to the sun, in noble instinct affronting and outstaring the light.
"Recreations of Christopher North, Volume 2" by John Wilson
This, then, becomes a matter of who can outstare whom.
"A Practical Guide to Self-Hypnosis" by Melvin Powers
Solomon Owl did not appear to mind that, but calmly outstared Grumpy Weasel without once blinking.
"The Tale of Grumpy Weasel" by Arthur Scott Bailey
She was thin and fair, with a faded skin and unhappy eyes, outstared by a blaze of jewels.
"Barnaby" by R. Ramsay