Eigen-distribution on random assignments for game trees

In this Letter, we investigate a special distribution, called eigen-distribution, on random assignments for a class of game trees T₂^k. There are two cases, where the assignments to leaves are independently distributed (ID) and correlated distributed (CD). In the ID case, we prove that the distributional probability ρ{variant} belongs to [frac(sqrt(7) - 1, 3), frac(sqrt(5) - 1, 2)], and ρ{variant} is a strictly increasing function on rounds k ∈ [1, ∞). In the CD case, we propose a reverse assigning technique (RAT) to form two particular sets of assignments, 1-set and 0-set, then show that the E¹-distribution (namely, a particular distribution on the assignments of 1-set such that all the deterministic algorithms have the same complexity) is the unique eigen-distribution for T₂^k in the global distribution.