Having Arduino read a video game screen
Most Arduino platforms contain a very low power embedded processor.  As such it is very unlikely such a device will decode an NTSC or PAL signal let alone modern digitally compressed video.

Instead consider using a light sensor attached to your monitor.  This is more in line with what, for instance, the Arduino Uno is capable of.  If you can identify a spot on the monitor that will only change to a certain brightness at the conclusion of a game, you could look for that brightness level using a light sensor attached to the monitor and connected to an Arduino.

Here is a tutorial about using a light sensor with an Arduino.  Most of the circuit is for the display of which you need not bother with.