Why do self-driving cars use a radar? The slideshow 10 astonishing technologies that power google’s 
self-driving cars documents some of the technologies used in Google's self-driving car. It mentions a radar.

Why does Google use radar? Doesn't LIDAR do everything radar can do? In particular, are there technical advantages with radar regarding object detection and tracking?

To clarify the relationship with AI: how do radar sensors contribute to self-driving algorithms in ways that LIDAR sensors do not?

The premise is AI algorithms are influenced by inputs, which are governed by sensors. For instance, if self-driving cars relied solely on cameras, this constraint would alter their AI algorithms and performance.
LIDARs, especially cheap LIDARs, have problems with reflective surfaces (like metallic paint on cars), strong lights like car headlights, weather(rain, snow, hail, fog), and have a considerably shorter range than comparable in price radars. Of course, they have much better precision, so some hardware stacks for cars are using both.