In addition to what Coderguy said, all good stuff. I would look at if you would like something that is portable or not. Meaning do you want one that you could take into bedroom, or unplug it and put it in den, easily and effortlessly.
Also, some of the 4 he mentioned have internal speakers, some don't. Some have 20 watt's (10x2) and some have a lot less or NONE. In a pinch, all you would need would be xbox 360 and PJ with the built in speaker PJ's. All of the PJ's Coderguy listed you can plug into your sound system, so don't worry about that.
Also important is the Warranty on the different models , if you are spending 1,000 dollars, some of these PJ's only have 1 year warranty, or 2 year warranty or 3 year warranty. How would you feel if you bought a 1 year model and something went wrong 13 months later. Expensive to fix.
Also, Bulb life. Some of those 4 models coderguy suggested have 2,000 hours stated bulb life, some have 6,000 hours bulb life. That would be something that would interest me. Also how much to replace bulb? Also, some of the models he suggested have reported problems with early bulb pop, so do you want a PJ with a KNOWN problem.
Now Rainbow Effect is something to take into account, some are sensitive to that, some not. Coderguy himself sold his hc4000 because he couldn't take the Rainbow Effect, and bought the pro8200 because it didn't bother him as much or at all. Again something to think about.
Now as to gaming, I think all 4 are good at gaming and TV viewing. Coderguy was going at it from just Picture viewpoint, which he does, because that is the most important thing for him. I look at it from complete package, that is why I list some of things I look for in PJ's. We are both right, it comes down to priority. The order of importance of how we make the decision to buy one or the other PJ.
Now, most PRO reviews don't go into how many problems USERS have with the PJ's they review, most PRO reviews just view the PJ for a few days to test it, so go into forum rooms to see the problems people are having the the 4 PJ's you are looking at to see, which is the most reliable. Not to say everyone will have a problem, no, but is there a pattern. If a number of people claim to have exactly SAME problem. It could be production run gone bad, don't know. Again, the PJ with the longest warranty does give you more PEACE of MIND, because if you have 3 years to have problem show up, so you can send it in to repair center to have it repaired or replaced, free. Short warranty means your window is shorter. And on a 1,000 dollar investment, how do you feel about that?
Now of the 4 PJ's coderguy listed, the Pro8200 is the cheapest by 300 dollars or more, has the longest warranty of 3 years, has the longest bulb life of 6,000 hours, has the lowest rainbow effect, has the best internal speakers 20watts (10x2). So look at the whole picture. But check the 4 models out, because what is important to me might not be important to you.