The 16-year-old victim, Dalton Williams, was found on Tuesday in the state’s capital, Pierre, lying in the shooter’s house with a wound in his chest, the arrest affidavit said on Wednesday.
The high school student was transferred to a nearby hospital, but later died of the gunshot wound.
Pierre Police Chief Bob Grandpre said that the shooter, also 16, was arrested at his home over a first-degree murder charge and was held in Hughes County on Wednesday.
A witness, who was with the boys inside the house, said Williams and the shooter wrestled playfully after they argued over the game.
The shooter then retrieved a shotgun and pointed it at the witness, the affidavit said. The shooter then took a shell from a drawer, loaded the gun and pointed it again at the witness.
The witness said he wanted to escape the scene through a sliding glass door, but Williams suddenly stepped in between the two schoolboys.
The shooter fired the gun and Williams fell to the ground. The witness left the house and called the police, the affidavit added.
The shooting came amid a new hike in the US gun violence. On December 14, a 20-year-old man, identified as Adam Lanza, killed twenty children and six adults at an elementary school in Newtown, Connecticut.
On December 15, a gunman opened fire at a hospital in the state of Alabama, injuring a police officer and two other people before a policeman shot and killed him.
On December 16, in a shooting in the state of Kansas, a gunman killed two police officers outside a grocery store in the city of Topeka.
The United States has experienced several mass killings so far in 2012, including the Colorado cinema shooting in July that killed 12 people and injured nearly 60 others. In August, a killing spree at a Sikh temple left seven people dead.