Hiring a secretary from a poset.

The secretary problem lies at the core of mechanism design for online auctions. In this work we study the generalization of the classical secretary problem in a setting where there is only a partial order be- tween the elements and the goal of the algorithm is to return one of the maximal elements of the poset. This is equivalent to the setting where the seller has a multidimensional objective function with only a partial order among the outcomes. We obtain an algorithm that succeeds with probability at least?1 + l k^{−k/(k−1)} ((1+log^{-1/(k-1)} k)^k -1) where k is the number of maximal elements in the poset and is the only information about the poset that is known to the algorithm. On the other hand, we prove an almost matching upper bound of k^{−1/(k−1)} on the success probability of any algorithm for this problem; this upper bound holds even if the algorithm knows the complete structure of the poset.

• Algorithms and Theory