In Chapter 1, Jem and Scout become friends with Dill, who is spending the summer with his aunt, Rachel Haverford. Scout describes Dill as a talented, eccentric individual who could play any character during the children's dramas. Scout mentions that by the end of August they were becoming bored with their productions. Fortunately, Dill gave them the idea of making Boo Radley come out of his house. After hearing many fascinating stories about the reclusive neighbor, Dill urges Jem and Scout to help him get a look at Boo Radley. Shortly after, Dill bets Jem that he wouldn't run into the Radleys' yard. Although Dill does not invent a "game" in the first chapter, he does encourage the children to try to get a look at Boo Radley. His fascination with Boo spurs Jem and Scout's interest in their neighbor, and Jem eventually invents a game called "One Man's Family" in Chapter 4.
In Chapter One of To Kill a Mockingbird, Dill, a newcomer to Maycomb, is introduced to the people in Scout's world--Jem, Atticus, Calpurnia, and most importantly, the Radelys. Dill takes a fascination with the Radley family and their reclusive ways. He is escpecially interesting in Arthur "Boo" Radely, the youngest son of the family (who is now approximately 33) who has been locked in the house for over about 15 years.
To break up the monotony of summer, Dill suggests that he, Jem and Scout try to get Boo Radley to come out of his house. The following chapters will explore their different methods, Scout's reluctance to do so, and Atticus' anger at them bothering the family.
Later than chapter 1, actually chapter 4, Dill invents the Boo Radley game. This game shows the kids acting out the neighborhood legend about Boo, his mother, and father.
Jem plays the part of Boo and hides under the porch while Dill and Scout play the roles of Mr. and Mrs. Radley. This is the only 'game' he is truly credited with inventing and it occurs in chapter 4, not 1.
Otherwise, chapter one does position Dill as a contributor to the plays that Jem and Scout have been playing for years.