It doesn't work, or so I've heard. Drivers in those cities learn to expect the safety margin of the delay, and statistically they become more willing to run a red light right after it changes.
Back in the day they tried the overlapping red thing in my city. Once everyone got used to the new way of doing things the time when both light were red became a sort of anarchy zone. Everyone assumed that everyone else would strictly observe the red so they didn't have to.
I find myself reminded of an article about a German town that stripped out their lights and signs and just had everyone drive by basic priority rules. Apparently this sharply reduced incidents.
Some time ago I drove through that town without knowing anything about it. I didn't realize that they had such a concept, it just felt really stressful, because your normal expectations and intuitions don't apply.
This basic principle also explains why roundabouts (traffic circles) are so much safer than traffic lights. They feel unsafe to drivers, so they slow down and pay attention, instead of blindly trusting the lights.
In the UK, the lights go [red] → [red+amber] → [green], so that drivers are ready to move as soon as the light turns green. People don't run red lights.
