What characteristics of children in general are revealed through the Boo Radley game? Why doesn't Scout enjoy the games as much as the boys do?
Jem invents the Boo Radley game. It involves acting out episodes from Boo Radley's life—as imagined by Jem, at any rate. Boo is a very fascinating figure to the local kids. Children tend to be drawn towards anything that's strange, weird, or in any way scary—Boo's definitely all of those things, and more.
Scout is not very impressed by Jem's little game. For one thing, she knows that Jem's just acting up to try and make himself look tough in front of her and the other kids:
Jem's head at times was transparent: he had thought that up to make me understand he wasn't afraid of Radleys in any shape or form, to contrast his own fearless heroism with my cowardice.
At the same time, Scout doesn't want to be left out, so she plays along at first. Even though Scout, like her father, has the gift of empathy, she's still a child and, like any child, wants to feel like a part of a group. But ultimately, it's her inherited sense of empathy that prevents Scout from enjoying the Boo Radley game as much as the other children.
When Dill hits upon the bright idea of delivering a note to the Radley residence to try and get Boo to come outside, it's Jem who actually does the dirty work. Maybe he isn't so brave after all, Scout suggests. Perhaps not. But in the world of children, no less than in the adult world, there's always a distinct pecking order, always a leader of the gang. Among other things, this means that someone can initiate a plan while someone else risks their own neck. In this way, the children are unintentionally acting out what will later happen in the adult world.
If we're talking about the game to get Boo Radley to come out of his house, it primarily shows that children are curious and have wild imaginations. It also shows (actually, Atticus shows them) that children are naive and sometimes unknowingly cruel. When they think of Boo, they see a monster. It is only as they get older and get instructions from Atticus that they realize he is a human being, with feelings, who happened to have had a rough life.
If we're talking about the game where Jem plays Boo, while Scout and Dill play Mrs. and Mr. Radley, the reasons are essentially the same. They are curious about this mysterious family and are trying to role play it. At this point (Chapter 4), the children still consider Boo to be a monster or a fictional character. They don't know the truth about Boo, so they make it up based on the hyperbolic things they've heard as well as the gossip from Stephanie Crawford. They have yet to consider his feelings - this is one of Atticus' most enduring lessons in the book. They're still just playing, primarily out of curiosity.
After Atticus warns them not to play the game, Scout gives her reasons why she was tiring of the game:
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 head-shaking, 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.