The secretary problem on an unknown poset

Abstract

We consider generalizations of the classical secretary problem, also known as the problem of optimal choice, to posets where the only information we have is the size of the poset and the number of maximal elements. We show that, given this information, there is an algorithm that is successful with probability at least $\frac{1}{e}$. We conjecture that if there are $k$ maximal elements and $k \geq 2$ then this can be improved to $\sqrt[k-1]{\frac{1}{k}}$, and prove this conjecture for posets of width $k$. We also show that no better bound is possible.