Why and how is push-notification (like GCM) batter

Wanted to understand the fundamental reasons for push-notification like Google Cloud Messaging (earlier called Google Cloud to Device Messaging) being more battery friendly, for cloud <--> device communication ?

In my view, the alternative technologies involve "polling" (over TCP/IP) while keeping the connection in CONNECTED state, using keep-alives. Or is there something better ?

My limited undertanding of GCM is that, it also uses TCP/IP and keepalives, but the client never polls the server for status. Instead the server informs the client about an incoming message, and applications who subscribe to certain type of messages, are notified of the message asynchronously. Also, the common GCM connection, is shared between multiple applications, thus allowing the device electronics to sleep / hibernate at "coordinated" times, without multiple applications keeping the electronics more "ON" (electrically active) than they need to be. Is this the correct understanding ? Or is there more to it ?

Finally, how exactly does this compare to MQTT over TCP/IP with keepalives ? What are the reasons for MQTT being (apparently) less battery efficient than GCM ?

One of the main reasons it's efficient is it scales well. The android device keeps a single connection open to GCM servers to listen for notifications for ALL apps on the device, and then routes messages to the appropriate applications they are intended for. This is much more scalable and efficient than keeping a network connection open for every single application wanting to have some sort of push notifications.

The connection itself is likely a TCP connection that's left in an open state, even when the phone's goes idle. It can wake the device when data is received. I'd imagine there's some sort of heartbeat ping going on too that can have the connection be re-established if necessary.

The socket stuff is probably something you could do yourself, however like I said earlier the main reason for efficiency is the single connection for all apps. Very scalable.