Cdmtcs Research Report Series Clusters of Two Player Games and Restricted Determinacy Theorem Clusters of Two Player Games and Restricted Determinacy Theorem 1

We introduce a new notion of a cluster of in nite two player games between players 0 and 1. This is an in nite collection of games whose game trees can be composed into a graph which is similar to a tree except that the graph might not have the initial node. For each node of the graph there is an ancesstor node. We call this graph the arena of the cluster. For a game cluster we introduce a notion of a winner for the whole cluster. This notion is weaker than the requirement to win every game of the cluster. Any two player game can be viewed as a game cluster consisting of all its residual games [3, 18]. We extend the restricted memory determinacy (RMD) theorem of Gurevich-Harrington (GH), [3] to game clusters. We think that the notion of a game cluster improves the modeling power of two player games used to give semantics for concurrent processes [10, 11].