Partially ordered secretaries

The elements of a finite nonempty partially ordered set are exposed at independent uniform times in [0,1] to a selector who, at any given time, can see the structure of the induced partial order on the exposed elements. The selector’s task is to choose online a maximal element. This generalizes the classical linear order secretary problem, for which it is known that the selector can succeed with probability 1/e and that this is best possible. We describe a strategy for the general problem that achieves success probability at least 1/e for an arbitrary partial order.