Strangers On a Train (1951)

Guy Haines is a well-known amateur tennis player who wants to go into politics in his post-tennis life. He is equally well-known in the society papers as a man who wants to divorce his wife in order to marry Anne Morton, the refined daughter of a state senator. Bruno Antony is a charming, gregarious man who lives off his father's wealth. He's also a sociopath who loves his mentally-childlike mother, but detests his father, who wants to have him restrained. Guy and Bruno meet one day on a train. Their social chit-chat, directed by Bruno, leads to a discussion of what Bruno considers the perfect murders: each of two strangers kill the other's "problem" person, as each killer would have no motive or ties to the person murdered. As Guy and Bruno part, Guy believes Bruno's talk was just social chit-chat, but Bruno believes they have a pact and goes ahead with his murder. Guy knows Bruno committed murder but doesn't feel he can go to the police with the fantastical story, which be thinks will implicate himself more. The situation gets worse for Guy as Bruno starts inserting himself in Guy's life to pressure Guy into keeping to his supposed end of the bargain. Guy must figure out what to do to free himself, not kill anyone, and keep those around him safe. Bruno has Guy's personalized cigarette lighter which he is threatening to use to implicate Guy if Guy doesn't comply with his part of the deal.

IMDb: https://www.imdb.com/title/tt0044079