A note on Bruss' stopping problem with random availability

Self Cite

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 have m selection 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.

Section: Introductionsupporting

confidence: 89%

“…We prove only the first part of Proposition 1.1. The monotone selection region for the single selection problem is given by (1) i > 0}, where…”

Section: Multiple Sum-the-odds Theoremmentioning

confidence: 99%

Section: Multiple Sum-the-odds Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Odds Theorem with Multiple Selection Chances

Ano¹,

Kakinuma²,

Miyoshi³

2010

Self Cite

A note on the full-information Poisson arrival selection problem

Kurushima¹,

Ano²

2003

This note studies a Poisson arrival selection problem for the full-information case with an unknown intensity λ which has a Gamma prior density G(r, 1/a), where a>0 and r is a natural number. For the no-information case with the same setting, the problem is monotone and the one-step look-ahead rule is an optimal stopping rule; in contrast, this note proves that the full-information case is not a monotone stopping problem.

Optimal stopping on patterns in strings generated by independent random variables

Bruss

Louchard

2003

Strings are generated by sequences of independent draws from a given alphabet. For a given pattern H of length l and an integer n ≥ l, our goal is to maximize the probability of stopping on the last appearance of the pattern H in a string of length n (if any), given that, if we choose to stop on an occurrence of H, we must do so right away. This contrasts with the goals of several other investigations on patterns in strings such as computing the expected occurrence time and the probability of finding exactly r patterns in a string of fixed length. Several motivations are given for our problem ranging from relatively simple best choice problems to more difficult stopping problems allowing for a variety of interesting applications. We solve this problem completely in the homogeneous case for uncorrelated patterns. However, several of these results extend immediately to the inhomogeneous case. In the homogeneous case, optimal success probabilities are shown to vary, depending on characteristics of the pattern, essentially between the value 1/e and a new asymptotic constant 0.619…. These results demonstrate a considerable difference between the true optimal success probability compared with what a first approximation by heuristic arguments would yield. It is interesting to see that the so-called odds algorithm which gives the optimal rule for l = 1 yields an excellent approximation of the optimal rule for any l. This is important for applications because the odds algorithm is very convenient. We finally give a detailed asymptotic analysis for the homogeneous case.