the talk will describe some problems which arise in finding optimal quantum strategies for games.  in such problems one has a (noncommutative) algebra a which encodes quantum mechanical laws and a noncommutative polynomial b which corresponds to a particular game and tells its score.  the goal of the talk will be to give an idea of some  of the structure, methods and our results which arise in maximizing b.  to get more warning of what is in the talk see the last few years of my postings on arxiv with collaborators: adam bene watts, igor klep and vern paulsen etal.