They use the game to determine who has to perform the unpleasant task of for evidence—specifically the paper and rags that a triple homicide weapon was wrapped in.

The "paper" you are putting together refers to this evidence-gathering scene. Here is the context of the episode:

Chicken, Dynamite, Chainsaw - Chicago P.D. (Season 2, Episode 4)

: Burgess and Roman are assigned to a gun buyback program but discover a weapon linked to a triple homicide.

: They track the gun to a woman named "Miss Birdy," who informs them she threw away the materials the gun was wrapped in.

: The "Chicken, Dynamite, Chainsaw" game decides who has to go into the dumpster to retrieve the discarded items.