The 1/e-strategy is sub-optimal for the problem of best choice under no information

In Memory of Professor Larry Shepp
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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 Bruss and Yor (2012). Such processes are shown to have a very rigid structure, being time-changed pure birth processes, and this allows some precise distributional calculations, from which we deduce that the 1/e-strategy is in fact not optimal.

Section snippets

Dedication and background

At the evening of Professor Larry Shepp’s talk “Reflecting Brownian Motion” at Cornell University on July 11, 1983 (13th Conference on Stochastic Processes and Applications), Professor Shepp and Thomas Bruss ran into each other in front of the Ezra Cornell statue. Thomas was honoured to meet Prof. Shepp in person, but Larry replied “What are you working on?” And so Larry was the very first person with whom Thomas could discuss the 1/e-law of best choice resulting from the Unified Approach [2]

The unified approach

The so-called 1/e-law of best choice is a result obtained in the Unified Approach-model of Bruss [2]. The model is as follows:

Unified Approach: Suppose N>0 points are IID U[0,1]. Points are marked with qualities which are supposed to be uniquely rankable from 1 (best) to N (worst). The goal is to maximize the probability of stopping online, and without recall on a preceding observation, on rank 1.

This model was suggested for the best choice problem (secretary problem) for an unknown number N

Analysis of the open question

Our analysis starts with the following little result, whose proof is immediate from the statement.

Proposition 3.1

Suppose that (N) is a p.i.-counting process. If we define Ñ(u)=N(eu) for u(,0], and t1=logT1, then M(eu)=N(euT1)N(T1)T1euT1Nssds=Ñ(ut1)Ñ(t1)t1ut1Ñ(s)ds is a martingale in its own filtration, so (Ñ) is a pure birth process, started with one individual at time t1.

So the requirement that (Nt) be a p.i.- counting process is not in fact very general — apart from the choice of the time (

The value of a fixed threshold rule

Suppose we use a fixed threshold rule, that is, we do nothing until ub and then we take the first record thereafter. The 1/e rule corresponds to the special case b=1. What is the value of this?

If Ñb=n, then the distribution of the number Y of further observations is known, and is a negative binomial distribution: P[Y=y]=qypnn+y1y(y0),where p=exp(b). Given that Y=y, the probability that the best comes after the first n observations is y/(n+y), and the probability that the first record after

The Hamilton–Jacobi–Bellman equations

If Vn(u) denotes the value of being at time u0 with n events already observed, none of them at time u, then the HJB equations of optimal control for the Vn are 0=Vṅ(u)+n{nn+1Vn+1(u)+1n+1max{Vn+1(u),π̃n+1(u)}Vn(u)}=Vṅ+n(Vn+1Vn)+nn+1(π̃n+1Vn+1)+, together with the boundary conditions Vn(0)=0. The solution is then the value function Vn(u).

Now, if the answer to the open question is affirmative, then for all n, Vn=Vn, where V is the value function for using the 1/e strategy, which we

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank very warmly Professor Philip Ernst for his precious indirect and direct contributions to this article. When Philip organized in 2018 the most memorable conference at Rice University in honour of Larry Shepp, he motivated many of us to look at harder problems. And so this old open problem turned up again and attracted attention. Philip’s interest in this problem and his numerous interesting comments on, and discussions of, a former version of this paper, were very

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Université Libre de Bruxelles, Département de Mathématique, CP 210, B-1050 Brussels, Belgium.

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