Inspired by recent algorithms for electing a leader in a distributed
system, we study the following game in a directed graph: each vertex
selects one of its outgoing arcs (if any) and eliminates the other
endpoint of this arc; the remaining vertices play on until no arcs
remain. We call a directed graph {\em lethal} if the game must end
with all vertices eliminated and {\em mortal} if it is possible that
the game ends with all vertices eliminated. We show that lethal
graphs are precisely collections of vertex-disjoint cycles, and that
the problem of deciding whether or not a given directed graph is
mortal is NP-complete (and hence it is likely that no ``nice''
characterization of mortal graphs exists).