The 1/e-strategy is sub-optimal for the problem of best choice under no information
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 -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:
This model was suggested for the best choice problem (secretary problem) for an unknown numberUnified Approach: Suppose points are IID . Points are marked with qualities which are supposed to be uniquely rankable from (best) to (worst). The goal is to maximize the probability of stopping online, and without recall on a preceding observation, on rank 1.
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 is a p.i.-counting process. If we define for , and , then is a martingale in its own filtration, so is a pure birth process, started with one individual at time .
So the requirement that 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 and then we take the first record thereafter. The rule corresponds to the special case . What is the value of this?
If , then the distribution of the number of further observations is known, and is a negative binomial distribution: where . Given that , the probability that the best comes after the first observations is , and the probability that the first record after
The Hamilton–Jacobi–Bellman equations
If denotes the value of being at time with events already observed, none of them at time , then the HJB equations of optimal control for the are together with the boundary conditions . The solution is then the value function .
Now, if the answer to the open question is affirmative, then for all , , where is the value function for using the 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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Cited by (1)
A NOTE ON THE CONSTANT 1/e
2023, Palestine Journal of Mathematics
- 1
Université Libre de Bruxelles, Département de Mathématique, CP 210, B-1050 Brussels, Belgium.