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.'
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