How to Override Google Play's Compatibility Check I own a mid-range Android phone with pretty decent specs,yet I find many a times that certain games and services are shown to be in-compatible with the phone.However when I download the apk from the internet and run it,the application runs with no snags.

Of course I could continue with my MO,but I was just wondering if there was a method to circumvent the compatibility check that takes place,even if it involves rooting.
I don't see a way to do so without massive system modifications (rooting and more) because the "compatibility check" is done on Google (server) side. 

When using your phone with Google PlayStore your phone registers and identifies itself with a certain brand, name and version at Google.

Furthermore a list of available system libraries, device features and device characteristics (screen size and resolution, ..) is sent to Google.

Last but not least the country and mobile network operator is also sent as some apps country and/or MNO restricted. 

If you now search for an app or try to access one specific app Google compares the app specification with your device spec and then decides if you get access to that app or not.

Note that the app developer can (optionally) manually select the devices the app is compatible with. Google has a large list of each and every Google Android device on the market. 

Hence if you want to bypass the compatibility check your phone would have to pretend to be a different device and model that it actually is. This can get complicated as Google has a list of every Google Android device with every firmware version ever available. Hence forging the data to pretend to be a different phone model can be pretty complicated as Google exactly knows how what data to be expected.

Furthermore there is the problem that sometimes you don't know why the compatibility check fails. Some years ago Google was "so friendly" to exactly send back to the device why your device is incompatible (e.g screen too small, system library xyz not present, feature xyz missing). But that changed at some point and now you only get back the info "hardware incompatible".