Two measures of efficiency for the secretary problem with multiple items at each rank

For $2\le k\in\mathbb{N}$, consider the following adaptation of the classical secretary problem. There are $k$ items at each of $n$ linearly ordered ranks. The $kn$ items are revealed, one item at a time, in a uniformly random order, to an observer whose objective is to select an item of highest rank. At each stage the observer only knows the relative ranks of the items that have arrived thus far, and must either select the current item, in which case the process terminates, or reject it and continue to the next item. For $M\in\{0,1,\cdots, kn-1\}$, let $\mathcal{S}(n,k;M)$ denote the strategy whereby one allows the first $M$ items to pass, and then selects the first later arriving item whose rank is \it either equal to or greater than\rm\ the highest rank of the first $M$ items (if such an item exists). Let $W_{\mathcal{S}(n,k;M)}$ denote the event that one selects an item of highest rank using strategy $\mathcal{S}(n,k;M)$ and let $P_{n,k}(W_{\mathcal{S}(n,k;M)})$ denote the corresponding probability. We obtain a formula for $P_{n,k}(W_{\mathcal{S}(n,k;M)})$, and for $\lim_{n\to\infty}P_{n,k}(W_{\mathcal{S}(n,k;M_n)})$, when $M_n\sim ckn$, with $c\in(0,1)$. In the classical secretary problem, the asymptotically optimal strategy yields a probability of success of $\frac1e\approx 0.368$. For $k=2$, the asymptotically optimal strategy yields yields a probability of success of about 0.701. For $k=3$, the optimal probability is above 0.85, for $k=7$, that probability exceeds 0.99, and for $k\ge12$, it is 1.000 to three decimal places. In the problem with multiple items at each rank, there is an additional measure of efficiency of a strategy besides the probability of selecting an item of highest rank; namely how quickly one selects an item of highest rank. We give a rather complete picture of this efficiency.

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