Well, we think the answer to this is that we are looking at the 'problem' from a Newtonian perspective, when we should be looking at it from a relativistic one. The Newtonian picture is that of a fireworks display with each ember being a galaxy speeding away from us, or a cloud of gas expanding into space. This is very intuitive, and it is the picture we all prefer to think about when we ask about the expanding universe. If you are not exactly at the center of this expanding cloud, there will be faster particles trying to overtake you and slower ones that you will be plowing into, so you would see both red and blue shifts.
But this is not the correct picture to use based on the observational evidence we have.
In the relativistic picture of Big Bang cosmology, there is no pre-existing space into which the cloud is expanding, and quite literally, space is expanding or 'stretching' between each galaxy in a precisely defined way. In this picture, there would be NO blue shifts for distant galaxies, only redshifts. This is exactly what astronomers observe. Since you cannot argue with the evidence, you are forced to accept the relativistic picture of what the expansion of the universe looks like.
Newton is dead, long live Einstein!