A note on Bruss' stopping problem with random availability (Q2759596)

The authors consider a continuous-time generalization of the secretary problem. A decision maker finds an apartment during a fixed period \((0,T]\). Opportunities to inspect apartments occur at the epochs of a homogeneous Poisson process of an intensity \(\lambda\). The decision maker can rank a given apartment among all those inspected to date. The objective is to maximize the probability of selecting the best apartment from those available in \((0,T]\). \textit{R. Cowan} and \textit{J. Zabczyk} [Theory Probab. Appl. 23, 584-592 (1979) and Teor. Veroyatn. Primen. 23, 606-614 (1978; Zbl 0396.62063)] studied the problem when the intensity \(\lambda\) is known, and \textit{F. T. Bruss} [J. Appl. Probab. 24, No. 4, 918-928 (1987; Zbl 0596.60046)] investigated it when \(\lambda\) is unknown and the prior density of \(\lambda\) is exponential with a known parameter \(a\) \((\geq 0)\). The authors extend Bruss' problem to the problem in which each owner of apartment can accept the offer proposed by apartment's searcher with a fixed known probability \(p\) \((0<p\leq 1)\) and the decision maker is allowed to make at most \(m\) \((\geq 1)\) offers. They show that the optimal stopping rule for the problem is to make an offer to the first relatively best option after a time \(s^*_m=(T+a) \exp(-C^{(m)} (q))-a\), where \(q=1-p\). In particular, they give \(C^{(1)}(q)=1\), \(C^{(2)}(q)= 1+q/2\) and \(C^{(3)}(q)= 1+q/2+q^2/3 +q^3/8\). Finally they consider the case when the probability \(p\) depends on \(m\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00065].