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

Search and planning under incomplete information : a study using Bridge card play

This thesis investigates problem-solving in domains featuring incomplete information and multiple agents with opposing goals. In particular, we describe Finesse --- a system that forms plans for the problem of declarer play in the game of Bridge. We begin by examining the problem of search. We formalise a best defence model of incomplete information games in which equilibrium point strategies can be identified, and identify two specific problems that can affect algorithms in such domains. In Bridge, we show that the best defence model corresponds to the typical model analysed in expert texts, and examine search algorithms which overcome the problems we have identified. Next, we look at how planning algorithms can be made to cope with the difficulties of such domains. This calls for the development of new techniques for representing uncertainty and actions with disjunctive effects, for coping with an opposition, and for reasoning about compound actions. We tackle these problems with a..

Learning The Game: Implementations Of Convolutional Networks In Automated Strategy Identification

Games can be used to represent a wide variety of real world problems, giving rise to many applications of game theory. Various computational methods have been proposed for identifying game strategies, including optimized tree search algorithms, game-specific heuristics, and artificial intelligence. In the last decade, systems like AlphaGo and AlphaZero have significantly exceeded the performance of the best human players in Chess, Go, and other games. The most effective game engines to date employ convolutional neural networks (CNNs) to evaluate game boards, extract features, and predict the optimal next move. These engines are trained on billions of simulated games, wherein the strategies become increasingly refined as more games are played. To explore the trade-offs inherent in developing CNNs, we will train them to play the game Connect-4, which is relatively small and has known optimal strategies. In this setting, we experiment with a variety of neural structures and other related factors with only a few hundred thousand simulated games. The results will allow us to compare how different aspects of the neural network impact performance. We propose a framework for this training process which generalizes to any two-player board games meeting some necessary criteria

The key node method: a highly-parallel alpha-beta algorithm

Journal ArticleA new parallel formulation of the alpha-beta algorithm for minimax game tree searching is presented. Its chief characteristic is incremental information sharing among subsearch processes in the form of "provisional" node value communication. Such "eager" communication can offer the double benefit of faster search focusing and enhanced parallelism. This effect is particularly advantageous in the prevalent case when static value correlation exists among adjacent nodes. A message-passing formulation of this idea, termed the "Key Node Method", is outlined. Preliminary experimental results for this method are reported, supporting its validity and potential for increased speedup

Studies in Machine Learning Using Game Playing�

Computer Scienc

The Expected-Outcome Model of Two-Player Games

This paper introduces a new, crisp definition of two-player evaluation functions. These functions calculate a node's expected-outcome value. or the probability that a randomly chosen leaf beneath it will represent a win. The utility of these values to game programs will be assessed by a series of experiments that compare the performance of expected-outcome functions with that of some popular, previously studied evaluators. To help demonstrate the domain-independence of these new functions, the experiments will be run on variants of several games, including tic-tac-toe, Othello, and chess. In addition, the paper outlines a. new probabilistic model of game-trees which involves rethinking many long-accepted assumptions in light of the newly defined expected-outcome functions

Intelligent strategy for two-person non-random perfect information zero-sum game.

Tong Kwong-Bun.Thesis submitted in: December 2002.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 77-[80]).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- An Overview --- p.1Chapter 1.2 --- Tree Search --- p.2Chapter 1.2.1 --- Minimax Algorithm --- p.2Chapter 1.2.2 --- The Alpha-Beta Algorithm --- p.4Chapter 1.2.3 --- Alpha-Beta Enhancements --- p.5Chapter 1.2.4 --- Selective Search --- p.9Chapter 1.3 --- Construction of Evaluation Function --- p.16Chapter 1.4 --- Contribution of the Thesis --- p.17Chapter 1.5 --- Structure of the Thesis --- p.19Chapter 2 --- The Probabilistic Forward Pruning Framework --- p.20Chapter 2.1 --- Introduction --- p.20Chapter 2.2 --- The Generalized Probabilistic Forward Cuts Heuristic --- p.21Chapter 2.3 --- The GPC Framework --- p.24Chapter 2.3.1 --- The Alpha-Beta Algorithm --- p.24Chapter 2.3.2 --- The NegaScout Algorithm --- p.25Chapter 2.3.3 --- The Memory-enhanced Test Algorithm --- p.27Chapter 2.4 --- Summary --- p.27Chapter 3 --- The Fast Probabilistic Forward Pruning Framework --- p.30Chapter 3.1 --- Introduction --- p.30Chapter 3.2 --- The Fast GPC Heuristic --- p.30Chapter 3.2.1 --- The Alpha-Beta algorithm --- p.32Chapter 3.2.2 --- The NegaScout algorithm --- p.32Chapter 3.2.3 --- The Memory-enhanced Test algorithm --- p.35Chapter 3.3 --- Performance Evaluation --- p.35Chapter 3.3.1 --- Determination of the Parameters --- p.35Chapter 3.3.2 --- Result of Experiments --- p.38Chapter 3.4 --- Summary --- p.42Chapter 4 --- The Node-Cutting Heuristic --- p.43Chapter 4.1 --- Introduction --- p.43Chapter 4.2 --- Move Ordering --- p.43Chapter 4.2.1 --- Quality of Move Ordering --- p.44Chapter 4.3 --- Node-Cutting Heuristic --- p.46Chapter 4.4 --- Performance Evaluation --- p.48Chapter 4.4.1 --- Determination of the Parameters --- p.48Chapter 4.4.2 --- Result of Experiments --- p.50Chapter 4.5 --- Summary --- p.55Chapter 5 --- The Integrated Strategy --- p.56Chapter 5.1 --- Introduction --- p.56Chapter 5.2 --- "Combination of GPC, FGPC and Node-Cutting Heuristic" --- p.56Chapter 5.3 --- Performance Evaluation --- p.58Chapter 5.4 --- Summary --- p.63Chapter 6 --- Conclusions and Future Works --- p.64Chapter 6.1 --- Conclusions --- p.64Chapter 6.2 --- Future Works --- p.65Chapter A --- Examples --- p.67Chapter B --- The Rules of Chinese Checkers --- p.73Chapter C --- Application to Chinese Checkers --- p.75Bibliography --- p.7