I saw part of this film on television long ago; probably in the late 1970s. It was a live-action, action-genre film in which the protagonist participated in an international game like "Assassin"--that is, he advanced in the game by killing another participant before being killed himself. The eventual survivor/winner was to receive a substantial prize. The setting was an overpopulated near future. It seems that the game was mostly accepted as voluntary way of reducing the population, but was condemned by the Vatican. No idea who the lead actor was, but I think he was a typical square-jawed action hero type.
It could be The Tenth Victim (1965), starring Marcello Mastroianni, though I think the game was supposed to prevent war rather than to relieve overpopulation:
In the near future, big wars are avoided by giving individuals with violent tendencies a chance to kill in the Big Hunt. The Hunt is the most popular form of entertainment in the world and also attracts participants who are looking for fame and fortune. It includes ten rounds for each competitor, five as the hunter and five as the victim.
According to this site,
Pains are taken to establish that the Vatican is still in full operation and does not condone 'the Hunt.'