If you search for a cheap phone and are familiar with unlocking the bootloader to install a custom rom (original chinese miui is horrible) its a nice phone (though you need to charge it each evening).
Mi5 is quite cheap for what you get and for gaming very nice i think. Also, its not too big.
If you don't care about the size of the Phone, leeco le max is powerful.
If you really search for a budget phone, take a look at the redmi series or buy an old one and update it with custom rom (If you don't play heavy games, you dont really need a new phone.) I Used my HOX untill this summer with cm 12 and next kernel, battery is very nice (around 1.5 days with whatsapp, sometimes browser and little gba emulation).
Hope i could help u choosing a phone
