There isn't any encoder. I was wrong as per my edited reply. It's just multiplexed input. See above. Should be really easy to use. You just need to switch the red and brown wires high & low depending on which 4 buttons you want to read. Basically the joystick would have done this many times a second, continously sampling each set of buttons. You can do the same from the Arduino. Just switch red HIGH and brown LOW, then read the 4 input wires for buttons 1, 2, 3, & 4. Then switch red LOW and brown HIGH, then read the same wires to read the states of buttons 5, 6, 7, & 8. Do this often enough in your input loop and you'll have access to all 8 buttons. It's very similiar to using a shift register.