Answer to an open question concerning the $1/e$-strategy for best choice under no information

Abstract

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.