The concept of an infinite game played on a finite graph is perhaps novel in the context of an rather extensive recent literature in which infinite games are generally played on an infinite game tree. We claim two advantages for our model, which is admittedly more restrictive. First, our games have a more apparent resemblance to ordinary parlor games in spite of their infinite duration. Second, by distinguishing those nodes of the graph that determine the winning and losing of the (...) game , we are able to offer a complexity analysis that is useful in computer science applications. (shrink)