With the current trend, purchase wise being towards higher resolution [4k], bargains can be found on Ebay regarding getting a gaming projection system up and running, as those with a penchant for home Theater, tend to be early adoptees, and upgrade their projectors relatively early, so a lot of suitable PJ's are coming on the market as their owners upgrade to 4k, you just need to know which projectors are the ones to get.
To be honest any PJ made for HT duties made from around 2007 onwards will make a good gaming PJ, it's the latency that can be a issue, and usually this only is a problem with the high end PJ's, as these were designed to just play movies basically, so input lag was not a priority in their design brief, as it's not needed when used purely for movies .....you can tweak some PJ's to respond better, but that involves getting into the service menu, and that's getting a bit beyond the scope of this post.
Also, the different between 1080i /1080p and 4k isn't huge, resolution wise, actually most people would be hard pressed to notice the differences unless they were pointed out to them, and a lot depends on viewing distance from the screen if it is noticeable at all.
Where 4k wins is with the expanded colour gamut, it makes gradients much smoother, where large expanses of graduated colours can look pixilated on the lower resolution screens.......I'm still not convinced I need to upgrade to 4k just yet, as unlike most people, I also have my own video calibration gear [I've been a HiFi nut since the mid 70's- and became a videophile over the last 20 years], so my aging Sony PJ is still looking better than most peoples new 4K PJ's, well at least in colour accuracy.
Look on Ebay for Sony PJ's VPL HS51a as they were called in the USA, or VPL HS60 as they were called here in OZ, and I think in the UK, they are 1080i units [Playstations auto detect & output to the display resolution-so no stuffing about needed changing settings], you should be able to pick one up for around Aus$ 350 -£220 [A LOT cheaper than when I bought my unit new for Aus $3500 ]...I see replacement lamps are around £80-90.....that will give you around 3000 hours of game play.....great gaming PJ's....try playing Wipeout HD/Fury/2048 in 1st person cockpit view on one of these.
My original HS60 died on me just as I replaced the lamp...... I left it for a while, thinking it would cost me a arm and a leg to get Sony to repair it, and I might be better served saving the money towards a new 4K PJ.......but curiosity go the better of me, and one day a few months ago I found another used unit on Ebay, complete with spare Lamp [that in itself is over Aus$500 here] It didn't sell at first, then it got re posted, so I cheekily sent the seller a message saying I'd pay the buy it now price of Aus$350 if they dropped the postage costs, they said yes....turned out that it had come straight from a Sony service centre here in OZ and had been recently reconditioned by them, but the owner never turned up to pay for it......you never now your luck until you test it.