Abstract
We consider a generalization of the best choice problem to upward directed graphs. We describe a strategy for choosing a maximal element (i.e., an element with no outgoing edges) when a selector knows in advance only the number n of vertices of the graph. We show that, as long as the number of elements dominated directly by the maximal ones is not greater than c1n for some positive constant c1 and the indegree of remaining vertices is bounded by a constant D, the probability pn of the right choice according to our strategy satisfies lim infn→∞pnn≥δ>0, where δ is a constant depending on c1 and D.
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