A true hickey is blood that has been drawn into the interstitial space.  It's going to take time for natural breakdown of the heme molecule for it to go away.  Simior to a hematoma or bruise, the amount of blood (size) is somewhat related to time for it to dissappear.

The ice will shut down the blood supply to the area and the heat will increase the blood supply. Also, but to a lesser extent I would guess, ice will slow the chemical reactions that break in down and heat will do the opposite.

So the alternation of ice and warm make no sense to me.  What might make sense with the ice/heat thing is ice right after getting it.  That would discourage more bleeding into the area.  But after a few hours, the natural repair mechanisms would have progressed to the point that this would no longer be effective.  Heat will then increase blood flow to the area and help move the breakdown products from the site.