Xavier es and Mxrider have the right ideas. They each recommended a type of screw extractor, which is exactly the tool for this job. That, plus a drill to bore a hole in the top of the screw for starters. Xavier es even told you how to get that hole started right. The EZ-Out looks like a tapered shaft with a left-hand spiral coarse "thread", and a square top end to grip. Use the right diameter tool for your screw - they come in sets. You drill a hole in the screw, then turn the tool into it COUNTERCLOCKWISE. The left-hand thread bites into the hole and pulls the tool down in until it jams because of the conical taper. From then on, turning the tool turns the screw in the "unscrewing" direction. Sometimes to help it bite in and jam you can push down on it as you turn slowly.
Completely drilling the screw out to a new hole that you then re-thread with a tap is the last resort option when a screw extrator fails.