The Free On-line Dictionary of Computing (30 December 2018):

play by electronic mail
PBEM

    A kind of game where the players use electronic mail
   to communicate.  This may be done via a human moderator or
   an automatic mailing list exploder on some central machine
   or it may be fully distributed with each player just
   addressing his mail to all other players.

   This is a natural extension of "play by mail" games conducted
   via snail mail.

   (http://fermi.clas.virginia.edu/~gl8f/pbm.html).  Usenet
   newsgroup: news:rec.games.pbm.

   (1994-10-27)