Optimal depth-first algorithms and equilibria of independent distributions on multi-branching trees

The main purpose of this paper is to answer two questions about the distributional complexity of multi-branching trees. We first show that for any independent distribution d on assignments for a multi-branching tree, a certain directional algorithm DIRd is optimal among all the depth-first algorithms (including non-directional ones) with respect to d. We next generalize Suzuki-Niida's result on binary trees to the case of multi-branching trees. By means of this result and our optimal algorithm, we show that for any balanced multi-branching AND-OR tree, the optimal distributional complexity among all the independent distributions (ID) is (under an assumption that the probability of the root having value 0 is neither 0 nor 1) actually achieved by an independent and identical distribution (IID).