Storytelling game - Wikipedia
https://en.wikipedia.org/wiki/Storytelling_gameA storytelling game is a game where multiple players collaborate on telling a spontaneous story. Usually, each player takes care of one or more characters in the developing story. Some games in the tradition of role-playing games require one participant to take the roles of the various supporting characters, as well as introducing non-character forces (for example, a flood), but other systems dispense with this figure and distribute this function among all players.