You can go with the Zenfone 5. It's armed with Intel dual core atom and it's below $180 in our country. You can play light games. You cannot play graphics-intensive game using such phone.
If you really want the cheapest gaming flagship phone, go with the Nexus 5.
Sony XPeria SP is around $200. Yet it never lags everytime I play with it.
My girlfriend has Xperia C and it also doesn't lag when playing low to mid-graphics games.
Since I don't know where you're from. Do a research. If you want a cheap gaming smartphone, here's a list you want to take note of:
1. CPU: at least dual core. Qualcomm brand, with a clock rate of at least 1.5Ghz.
2. RAM: At least 1gb of RAM
Try asking your tech friends personally.