# Odds Theorem with Multiple Selection Chances

@article{Ano2010OddsTW, title={Odds Theorem with Multiple Selection Chances}, author={Katsunori Ano and Hideo Kakinuma and Naoto Miyoshi}, journal={Journal of Applied Probability}, year={2010}, volume={47}, pages={1093 - 1104} }

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…

## 17 Citations

The Sum-the-Odds Theorem with Application to a Stopping Game of Sakaguchi

- Mathematics
- 2016

The optimal stopping problem of maximizing the probability of stopping on the last success of a finite sequence of independent Bernoulli trials has been studied by Hill and Krengel (1992), Hsiau and…

Lower Bounds for Bruss' Odds Problem with Multiple Stoppings

- Mathematics, Computer ScienceMath. Oper. Res.
- 2016

This paper gives a nontrivial lower bound of the probability of win (obtaining the last success) for the problem with m -stoppings and proves a conjecture on the classical secretary problem, which gives a connection between the probabilities of win and the threshold values of the optimal stopping strategy.

Maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon

- MathematicsAdvances in Applied Probability
- 2011

We consider the problem of maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon of length N, where m is a predetermined integer. A…

Weber's optimal stopping problem and generalizations

- Mathematics
- 2013

One way to interpret the classical secretary problem (CSP) is to consider it as a special case of the following problem. We observe n independent indicator variables I1,I2,…,In sequentially and we…

Multiple stopping odds problem in Bernoulli Trials with random number of observations

- Mathematics
- 2016

This paper studies an optimal multiple stopping problem, in which the objective is to maximize the probability of selecting the "last success" on Bernoulli trials with random number of observations…

Sequential selection of the k best out of nrankable objects

- Mathematics, Computer ScienceDiscret. Math. Theor. Comput. Sci.
- 2016

The objective of this paper is to find in a setting of n sequential observations of objects a good online policy to select the k best of these n uniquely rankable objects to investigate threshold functions which maintain all present information.

Concerning an adversarial version of the Last-Success-Problem

- Mathematics
- 2018

There are $n$ independent Bernoulli random variables with parameters $p_i$ that are observed sequentially. Two players, A and B, act in turns starting with player A. Each player has the possibility…

Finding the best out of n rankable objects. A consecutive thresholds Algorithm

- Mathematics
- 2015

The objective of this paper is to find in a setting of n sequential observations of objects a good online policy to select the κ best of these n uniquely rankable objects. This focus is motivated by…

2-E-7 A Typical Lower Bound for Odds Problem in Markov-dependent Trials

- Mathematics
- 2012

$\alpha_{0}:=P(X_{1}=1)=1-\beta_{0}$ . Each $\alpha_{i}$ and $\beta_{i}$ are given. We assume $0<\alpha_{i},$ $\beta_{i}<1$ for all $i$ . We observe these $X_{i}$ ’s sequentially and claim that the…

Dynamic Programming and Linear Programming for Odds Problem

- Mathematics
- 2021

This paper discusses the odds problem, proposed by Bruss in 2000, and its variants. A recurrence relation called a dynamic programming (DP) equation is used to find an optimal stopping policy of the…

## References

SHOWING 1-10 OF 21 REFERENCES

Selecting a sequence of last successes in independent trials

- Mathematics
- 2000

Let I1, I2, . . . , In be a sequence of independent indicator functions de- fined on a probability space (Ω, A, P ). We say that index k is a success time if Ik = 1. The sequence I1, I2, . . . , In…

A note on bounds for the odds theorem of optimal stopping

- Mathematics
- 2003

The odds theorem gives a unified answer to a class of stopping problems on sequences of independent indicator functions. The success probability of the optimal rule is known to be larger than Re -R ,…

Sum the odds to one and stop

- Mathematics
- 2000

The objective of this paper is to present two theorems which are directly applicable to optimal stopping problems involving independent indicator functions. The proofs are elementary. One implication…

Recognizing the Maximum of a Sequence

- Mathematics
- 1966

Abstract The classical dowry, secretary, or beauty contest problem is extended in several directions. In trying to find sequentially the maximum of a random sequence of fixed length, the chooser can…

Strategies for Sequential Search and Selection in Real Time

- Political Science, Computer Science
- 1992

This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Strategies for Sequential Search and Selection in Real Time, held in June 1990 at the University of…

Selecting the last success in Markov-dependent trials

- MathematicsJournal of Applied Probability
- 2002

In a sequence of Markov-dependent trials, the optimal strategy which maximizes the probability of stopping on the last success is considered. Both homogeneous Markov chains and nonhomogeneous Markov…

A NOTE ON BRUSS' STOPPING PROBLEM WITH RANDOM AVAILABILITY (Decision Theory in Mathematical Modelling)

- Mathematics
- 1999

Bruss (1987) has studied a continuous-time generalization of the so-called secretary problem, where options arise according to homogeneous Poisson processes with an unknown intensity of ?. In this…

A Prophet Inequality Related to the Secretary Problem

- Mathematics
- 1992

Let Z1, Zz , .. . , Zn be iudP.pcndent 0-1 -vahtt-d ranrlom vari ables. A gambler gels a. reward 1 if he st op8 a.t the time of the last success anrl otherwise gets no reward. A simple comparison…

A NATURAL VARIATION OF THE STANDARD SECRETARY PROBLEM

- Mathematics
- 2000

We consider a natural variation of the standard secretary problem: N groups of applicants are to be interviewed sequentially (in groups) by a manager and the manager wants to find a strategy which…

Invariant record processes and applications to best choice modelling

- Mathematics
- 1988

Let X1, X2,...be identically distributed random variables from an unknown continuous distribution. Further let Ir(1), Ir(2),...be a sequence of indicator functions defined on X1, X2,...by Ir(k) = 0…