Well, for what I know the game in Italy is called "Il gioco dell'impiccato" and is not part of the Italian culture (maybe it has been imported from the USA). The name refers more to the "hanged" (impiccato, from the verb 'impiccare', to hang)than to the hangman itself (in italian the hangman could be translated as "l'impiccatore" but more likely "il boia", the executioner).

Ghiro