A note on bounds for the odds theorem of optimal stopping. (English) Zbl 1059.60056

The odds theorem gives a unified answer to a class of stopping problems on sequences of independent indicator functions. It can be applied to natural stopping problems such as, for example, the secretary problem, the group-interview problem, the last-peak problem, but also to many other simple problems of games, betting or investment. The success probability of the optimal rule is known to be larger than \(R e^{-R}\), where \(R\) defined in the theorem satisfies \(R \geq 1\) in the more interesting case. This result is strenghened by showing that \(1/e\) is then a lower bound. This best possible uniform lower bound extends to the general setting of the odds theorem.


60G40 Stopping times; optimal stopping problems; gambling theory
Full Text: DOI Euclid


[1] Bruss, F. T. (2000). Sum the odds to one and stop. Ann. Probab. 28 1384–1391. · Zbl 1005.60055
[2] Hill, T. P. and Krengel, U. (1992). A prophet inequality related to the secretary problem. In Contemporary Mathematics (F. T. Bruss, T. S. Ferguson and S. M. Samuels, eds.) 209–215. Amer. Math. Soc., Providence, RI. · Zbl 0760.60046
[3] Hsiau, S. R. and Yang, J. R. (2000). A natural variation of the standard secretary problem. Statist. Sinica 10 639–646. · Zbl 0963.62076
[4] Samuels, S. M. (1992). Secretary problems as a source of benchmark bounds. Stochastic Inequalities 371–387. IMS, Hayward, CA. · Zbl 1400.60057
[5] Tamaki, M. (2001). Optimal stopping on trajectories and the ballot problem. J. Appl. Probab. 38 946–959. · Zbl 1002.60035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.