To buy a cheap HD projector that will be compatible with consoles such as Xbox 360 or Ps3

There isn't really a short answer to this question... it depends on what you want from the projector and what limitation you have in your home theater or room you'll be using your projector in. If you have a limited space and you still want to get a large picture, you should consider a projector with a a short throw distance. also if it's not possible for you to put your projector in the center of your room, you should consider a projector with not only vertical lens shift, but also with a horizontal lens shift. Brightness is another issue, if your room is dark enough then you can just pick up any of the projectors without worrying about your projected image brightness. if you have some windows and some light that you can't eliminate, then you should consider a projector with no less than 1600 lumens of brightness, the higher the better. also you should do a research about the differences between a LCD projector and a DLP projectors and see what suits you well. connectivity shouldn't be a big issue since most 720p projectors will have 1 HDMI port at least and a couple of component video input, so it will be no problem to hook up your PS3 and xbox360 at the same time. The cheapest projector that you can buy and hook up to your HD gaming console and gives you a very good picture needs to be at least 720p compatible..... since you're looking for a cheap HD projector I'll just exclude the 1080p choice.... You can find 720p projectors for as low as $660 from amazon also you should consider visiting www.projectorpeople.com because some of the items you'll find there for lower price than what you can find in amazon.com The cheapest HD projector you'll find online is the Mitsubishi HC1600 and I think it's the best cheapest 720p projector available on the market.