Learning Video Game Design on the Tabletop shows the reader how game AIs can be created in the form of finite state machines.
For a real-world example of a finite state machine in Splinter Cell: Chaos Theory, download and Clint Hocking’s 2006 Game Developers Conference talk “Designing to Promote Intentional Play” (35MB).

The World’s Most Boring Tower Defense Game is a web-based “game” that I use to demonstrate to my students how a video game’s “real time” action actually takes place over a series of discrete turns (what programmers call “ticks”). The game is also a useful example of what a digital prototype might entail.