We study the multi-selection version of 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 we are given m (≥ 1) selection chances and find that the optimal rule has the form of 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 give closed-form formulas 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 infinity. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent to the corresponding limit for the classical secretary problem with two selection chances.
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.
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