Equilibrium Points of an AND-OR Tree: under Constraints on Probability

We study a probability distribution d on the truth assignments to a uniform binary AND-OR tree. Liu and Tanaka [2007, Inform. Process. Lett.] showed the following: If d achieves the equilibrium among independent distributions (ID) then d is an independent identical distribution (IID). We show a stronger form of the above result. Given a real number r such that 0 < r < 1, we consider a constraint that the probability of the root node having the value 0 is r. Our main result is the following: When we restrict ourselves to IDs satisfying this constraint, the above result of Liu and Tanaka still holds. The proof employs clever tricks of induction. In particular, we show two fundamental relationships between expected cost and probability in an IID on an OR-AND tree: (1) The ratio of the cost to the probability (of the root having the value 0) is a decreasing function of the probability x of the leaf. (2) The ratio of derivative of the cost to the derivative of the probability is a decreasing function of x, too.Comment: 13 pages, 3 figure

多様なゲームについての論理的研究

要約のみTohoku University田中一之課

Lookahead Pathology in Monte-Carlo Tree Search

Monte-Carlo Tree Search (MCTS) is an adversarial search paradigm that first found prominence with its success in the domain of computer Go. Early theoretical work established the game-theoretic soundness and convergence bounds for Upper Confidence bounds applied to Trees (UCT), the most popular instantiation of MCTS; however, there remain notable gaps in our understanding of how UCT behaves in practice. In this work, we address one such gap by considering the question of whether UCT can exhibit lookahead pathology -- a paradoxical phenomenon first observed in Minimax search where greater search effort leads to worse decision-making. We introduce a novel family of synthetic games that offer rich modeling possibilities while remaining amenable to mathematical analysis. Our theoretical and experimental results suggest that UCT is indeed susceptible to pathological behavior in a range of games drawn from this family

Weakly balanced multi-branching AND-OR trees: Reconstruction of the omitted part of Saks-Wigderson (1986) (Theory and Applications of Proof and Computation)

We investigate variants of the Nash equilibrium for query complexity of Boolean functions. We reconstruct some omitted proofs and definitions in the paper of Saks and Wigderson (1986). In particular, by extending observation by Arimoto (2020), we introduce concepts of “weakly balanced multi-branching tree” as modified versions of “nearly balanced tree” of Saks and Wigderson, and we show recurrence formulas of randomized complexity for weakly balanced multi-branching trees

The phenomenon of Decision Oscillation: a new consequence of pathology in Game Trees

Random minimaxing studies the consequences of using a random number for scoring the leaf nodes of a full width game tree and then computing the best move using the standard minimax procedure. Experiments in Chess showed that the strength of play increases as the depth of the lookahead is increased. Previous research by the authors provided a partial explanation of why random minimaxing can strengthen play by showing that, when one move dominates another move, then the dominating move is more likely to be chosen by minimax. This paper examines a special case of determining the move probability when domination does not occur. Specifically, we show that, under a uniform branching game tree model, whether the probability that one move is chosen rather than another depends not only on the branching factors of the moves involved, but also on whether the number of ply searched is odd or even. This is a new type of game tree pathology, where the minimax procedure will change its mind as to which move is best, independently of the true value of the game, and oscillate between moves as the depth of lookahead alternates between odd and even