In Chapter 1 of To Kill a Mockingbird, what game does Dill invent?

Jem, Scout, and Dill first spend a good deal of time enacting scary dramas about Boo Radley that Jem dreams up. When they have worn this game thin and are sick of it, Dill comes up with idea of getting Boo Radley out of his house so they can see what he looks like. He dares Jem to go up and knock on the front door of the Radley residence and bets him a book called The Gray Ghost against two Tom Swifts that he won't. Jem can't stand to turn down a dare, so he agrees. Jem says to Dill:

“I hope you’ve got it through your head that he’ll kill us each and every one, Dill Harris,” said Jem, when we joined him.

This new game is more dangerous, and therefore ups the excitement value of Boo Radley. Jem sort of goes through with the bet, opening the gate, running into the yard, and slapping the side of the house before running out again. The fascination with seeing Boo will continue.

In Chapter 4 of To Kill a Mockingbird, what new game does Dill suggest after arguing with Scout and Jem?

In Chapter 4, Jem and Scout argue about Jem's retrieving their tire from the Radley yard. Jem gets the tire, pretending not to be afraid, and shortly thereafter announces that they will now play another game, "Something new, something different." He calls the new game "Boo Radley." Scout thinks he made up the game to prove that "he wasn't afraid of Radleys in any shape or form."

In Jem's game, the children staged a little play born of the gossip they had heard about Boo and his family and their own vivid imaginations. Scout was Mrs. Radley, Dill was Old Mr. Radley, and Jem played the role of Boo. As the summer wore on, they "polished and perfected it." Looking back, Scout the narrator recalls it was "a melancholy little drama."

Atticus interrupts them playing their game one day, wanting to know if it had anything to do with the Radley family. Jem denies it, knowing his father would disapprove. After Atticus goes into the house, some discussion ensues among the children about continuing to play. Jem thinks they might, since Atticus might not know what they actually were doing and had not forbidden their playing. Scout disagrees, vehemently. She wanted to quit the game anyway, based on what she had heard when she had rolled into the Radley's yard inside the tire.

What new game do Dill, Scout, and Jem play in To Kill a Mockingbird?

Jem, Scout and Dill play a new game where they act out Boo Radley’s life story.

Jem, Scout and Dill like to playact.  They take stories from movies or books and act them out in the yard.  Dill is so fascinated with the story of the neighborhood bogeyman Boo Radley that they decide to use his life as inspiration for their plays.  Their story is “woven from bits and scraps of gossip and neighborhood legend.”  Scout plays Mrs. Radley, Dill plays Mr. Radley, and Jem hams it up as a crazy young Boo.

As the summer progressed, so did our game. We polished and perfected it, added dialogue and plot until we had manufactured a small play upon which we rang changes every day. (Ch. 4)

Atticus does not approve of the game.  He tells the children to leave the Radleys alone.  Scout is sure that Boo can see them and is at first afraid to take part in the game.   Atticus just feels that the Radleys have been victimized enough.  He does not want his children acting out town gossip for all to see.

Atticus’s arrival was the second reason I wanted to quit the game. The first reason happened the day I rolled into the Radley front yard. Through all the headshaking, quelling of nausea and Jem-yelling, I had heard another sound, so low I could not have heard it from the sidewalk. Someone inside the house was laughing. (Ch. 4)

Clearly, Boo Radley is entertained by the game.  He probably thinks it is funny that someone is taking an interest in him.  He leads a lonely life as a recluse, and he enjoys watching the children.  He comes to be protective of them, carefully leaving the safety of his house to hide presents for them, looking out for them, and finally saving their lives when Bob Ewell attacks them.