  This paper answers a long-standing open question concerning the
$1/e$-strategy for the problem of best choice. $N$ candidates for a job arrive
at times independently uniformly distributed in $[0,1]$. The interviewer knows
how each candidate ranks relative to all others seen so far, and must
immediately appoint or reject each candidate as they arrive. The aim is to
choose the best overall. The $1/e$ strategy is to follow the rule: `Do nothing
until time $1/e$, then appoint the first candidate thereafter who is best so
far (if any).'
  The question, first discussed with Larry Shepp in 1983, was to know whether
the $1/e$-strategy is optimal if one has `no information about the total number
of options'. Quite what this might mean is open to various interpretations, but
we shall take the proportional-increment process formulation of \cite{BY}. Such
processes are shown to have a very rigid structure, being time-changed {\em
pure birth processes}, and this allows some precise distributional
calculations, from which we deduce that the $1/e$-strategy is in fact not
optimal.
