Based on pigeon anatomy, the bobbing head during walking quite likely is just a matter of keeping balance.

Suggestions that it relates to vision - as made here - don't appear to be well founded, although other head motions may be associated with vision. The head-bob while walking is consistent with control of center of gravity alone.

It might be of interest to pose the alternative question:

          Why do frogs jump?

Most animals have small involunary motions of the eyeball, called saccadic motion, that cause the eyeball to change position fairly rapidly even when staring fixedly at a single point. When an image is imposed on the retina, the strength of the neural signal "fades" fairly rapidly, and moving the eye moves the image to a new set of neurons to keep the image "fresh."

It is reported that a frog has no capability for saccadic motion of the eye, so the image can be kept refreshed only if the frog moves - or if some part of the image moves.

The claim is that the only way a frog can see stationary or slow moving parts of the surrounding scene is to move - i.e. jump. It also means that while the frog sits still, the stationary scene fades out of view, and the only thing the frog can see is the bug that flies through its field of view. This makes for incredible accuracy in catching the bug, but the frog has to remember where the next lily pad was if he wants to jump to a different one, since he (probably) can't see the pad until he jumps.

John