This kind of contest is especially dangerous in certain bars. Avoid trying this with the customers at the Snake-Haired Dame and beware the terrible gaze of the bartender at Thulsa's Bar & Doom.
Also, never challenge the Eye of Death!
Don't blink! For a real staredown where you break your opponent's Will, use the Contest of Wills rules from GURPS Martial Arts, p. 130.
Fast Version: Quick Contest between the lower of your HT or Will. Modifiers: up to -3 worth of distractions from either side's cohorts. Starers can try trickery - this is a Quick Contest of IQ. Apply the margin of victory as a penalty to the loser's side. However, the starer initiating the contest must make a Will check to avoid distraction - failure imposes a -2 penalty on the starer.
Slow Version: Regular Contest of the lower of HT or Will. Modifiers: up to -3 worth of distractions, as above. Each starer may use "distraction techniques" - take a -1 to your own roll for a -1 to your opponent's. Risky - if you both fail your rolls, you both lose - but a good way for a dominant competitor to try to end a contest quickly.