Katsunori Ano

Lower Bounds for Bruss’ Odds Problem with Multiple Stoppings

This paper addresses Bruss’ odds problem with multiple stopping chances. A decision maker sequentially observes a sequence of independent 0/1 (failure/success) random variables to correctly predict the last success with multiple stopping chances. First, we give a nontrivial lower bound of the probability of win (obtaining the last success) for the problem with m -stoppings. Next, we show that the asymptotic value for each classical secretary problem with multiple stoppings attains our lower bound. Finally, we prove a conjecture on the classical secretary problem, which gives a connection between the probability of win and the threshold values of the optimal stopping strategy.

Compare the ratio of symmetric polynomials of odds to one and stop

In this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last l successes, given a sequence of independent Bernoulli trials of length N , where k and l are predetermined integers satisfying 1≤ k ≤l N . This problem includes some odds problems as special cases, e.g. Bruss’ odds problem, Bruss and Paindaveine’s problem of selecting the last l successes, and Tamaki’s multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton’s inequalities and optimization technique, which gives a unified view to the previous works.

A secretary problem with restricted offering chances and random number of applications

Abstract This article considers a modification of the secretary problem with random number of applicants discussed by Presman and Sonin [1]. Our problem also allows both uncertainty of employment and restriction of offering chances, that is, an offer of acceptance is declined by the applicant with a fixed known probability 1 − p , (0≤ p ≤ 1) and the offering chances, before the decision maker gets one applicant, are at most M times. Section 2 gives the sufficient condition for the problem tto be monotone and the optimal strategy of the monotone problem. In Section 3 we give the examples for uniform distribution of random number of applicants.