The 10th Victim

The 10th Victim (La decima vittima) is an Italian cult sf film directed by Elio Petri in 1965. It is based on Robert Sheckley's 1953 short story "Seventh Victim". He published a novelization of the film in 1966.
The film begins with a man chasing a woman thru the streets whilst shooting at her. He's stopped by a policeman for questioning. The man shows his licence to kill. The policeman allows him to continue. Scenes switch between the pursuit & a narrator explaining the rules & justification of the game.
In the near future, big wars are avoided by giving violence-prone individuals a chance to kill in the Big Hunt. The Hunt is the world's most popular form of entertainment, attracting participants seeking fame & fortune. It includes ten rounds for each competitor, five as hunter & five as victim. The survivor of the ten rounds becomes the ultimate champion.
Caroline Meredith is the huntress armed with a high caliber Bosch Shotgun looking for her 10th victim. Marcello Poletti is the victim. He's reluctant to kill Meredith as he's unsure she's his hunter. Killing the wrong person would get him locked up in prison for 30 years. To maximize her financial gain, Meredith wants to get a perfect kill in front of the cameras as she's negotiated a major sponsor from the Ming Tea Company.