### Abstract

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we havemselection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

Original language | English |
---|---|

Pages (from-to) | 1093-1104 |

Number of pages | 12 |

Journal | Journal of Applied Probability |

Volume | 47 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2010 Dec |

Externally published | Yes |

### Keywords

- Multiple selection chances
- Optimal stopping
- Selecting the last success

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Applied Probability*,

*47*(4), 1093-1104. https://doi.org/10.1239/jap/1294170522

**Odds theorem with multiple selection chances.** / Ano, Katsunori; Kakinuma, Hideo; Miyoshi, Naoto.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 47, no. 4, pp. 1093-1104. https://doi.org/10.1239/jap/1294170522

}

TY - JOUR

T1 - Odds theorem with multiple selection chances

AU - Ano, Katsunori

AU - Kakinuma, Hideo

AU - Miyoshi, Naoto

PY - 2010/12

Y1 - 2010/12

N2 - We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we havemselection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

AB - We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we havemselection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

KW - Multiple selection chances

KW - Optimal stopping

KW - Selecting the last success

UR - http://www.scopus.com/inward/record.url?scp=80053431619&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80053431619&partnerID=8YFLogxK

U2 - 10.1239/jap/1294170522

DO - 10.1239/jap/1294170522

M3 - Article

VL - 47

SP - 1093

EP - 1104

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 4

ER -