I could swear I read that the EU does not allow this kind of bidding war for tax reduction by companies, but I can't find it online. The US certainly allows it, which I think is a bad idea.
Assuming Amazon is going to create an HQ somewhere in America, consider the two situations: 1) Current situation, they will eventually get a very large tax cut from wherever they select. 2) Hypothetical situation: local tax breaks are not allowed, Amazon selects a location and pays the local taxes that it avoided in situation #1.
You think small governments (states, or cities being 'small') having autonomy over how they tax themselves is a bad thing?
There are more than two situations, or you're over-simplifying the effect of a massive company like Amazon moving to a city.
A 'tax cut' does not mean they are contributing $0 to the local economy. I would guess (I'm not a city planner or anything like that) the overall economic impact of such a move vastly offsets any tax breaks offered (even if that means there are less taxes being given directly to those governing bodies).
Is having high taxes an inherently good thing we want to create incentives to encourage? In private industry, governments try to break up cartels because they're thought to be bad for consumers (residents of the cities in this case).