Why was the Boo Radley game invented in To Kill a Mockingbird?

In To Kill a Mockingbird, Jem invents the Boo Radley game to prove his bravery to Scout and Dill.

On page 35, Dill says that he’s able to tell if someone is about to die just by smelling them, teasing Scout that she would die in three days. Jem scoffs at this conversation and says:

“Yawl hush,” growled Jem, “you act like you believe in Hot Steams.”

Dill is unfamiliar with Hot Steams, and Jem begins describing the breath-sucking ghosts to him. Scout interrupts Jem, telling Dill not to believe him, then suggests the three of them roll in a tire to help escape the day’s boredom, as the children have all grown tired of acting out their usual scenes. Apparently upset at being snubbed by his younger sister, Jem pushes Scout, the first to roll, with much vigor, causing her to roll all the way into the Radley’s yard. Disoriented and scared, she runs back to the house without the tire, telling Jem to get it himself. A prideful boy, Jem cannot not refuse and runs to get the tire, “scowling triumphantly” when he returns. After a lemonade break, Jem suggests they play Boo Radley, wherein they each portray individuals from that family.

"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." - Scout, page 37

Scout sees through his attempt to show off. He invents the game as a power move, to prove once and for all that he is above questioning when it comes to his fears about the Radleys, himself playing the part of Boo in their performances over the course of the summer.