Expository notes on computability and complexity in (arithmetical) games

In this paper an attempt is made to resurrect the pioneering work of Michael Rabin on a class of games that are arithmetically defined and recursion theoretically analysed for effective playability and computational and diophantine complexity. Mathematical tools and concepts that were unavailable at the time of Rabin's original contribution simplify proofs and generalise results in arithmetical games. They are explained and used in an introductory and expository way. Rabin's effectivization of a Gale-Stewart game is also explained. Background motivation is provided in terms of a standard barriers to entry problem in industrial organisations theory and a parlour game.