Games have become increasingly successful when it comes to modelling a variety of logics and programming languages. These games are typically played on trees - once play has diverged, two strands can never be reunited. This does not agree wit ha description of games via positions, where it is of no importance how a given constellation was arrived at. Furthermore, even for the simples games (eg Hyland's 'Games for Fun', see CLICS Summer School Lecture Notes, Cambridge 1995) the categorical structure is not as simple as one might like - the linear function space of two such simple tree games can become quite complex. Choosing positions as a primitive concept we introduce games played on graphs rather than trees. We keep the standard restrictions, that is there are two players, P and O, who move strictly alternatingly, and O always starts. We show that we can define a linear function space - its positions are elements of the product of the set of positions of the constituent games - allowing us to define a category in the usual way with morphisms being given by strategies on the linear function space of A and B. This category is symmetric monoidal closed and has all products. A full subcatgegory has the property that morphisms can be seen as relations, and composition (originally defined via 'parallel composition and hiding') becomes relational composition. We further define a 'Curien exponential' giving rise to a cartesian closed category in the standard way. This exponential formalises the idea that O should be allowed to explore a given strategy (for P) on A and satisfies the property that it produces a finite games when applied to a finite game, unlike the standard construction for tree games. We conclude in identifying applications of this model.