Game Theory Basics

Game theory is the science of interaction. This textbook, derived from courses taught by the author and developed over several years, is a comprehensive, straightforward introduction to the mathematics of non-cooperative games. It teaches what every game theorist should know: the important ideas and results on strategies, game trees, utility theory, imperfect information, and Nash equilibrium. The proofs of these results, in particular existence of an equilibrium via fixed points, and an elegant direct proof of the minimax theorem for zero-sum games, are presented in a self-contained, accessible way. This is complemented by chapters on combinatorial games like Go; and, it has introductions to algorithmic game theory, traffic games, and the geometry of two-player games. This detailed and lively text requires minimal mathematical background and includes many examples, exercises, and pictures. It is suitable for self-study or introductory courses in mathematics, computer science, or economics departments.

An Adversarial Search Method Based on an Iterative Optimal Strategy

A deeper game-tree search can yield a higher decision quality in a heuristic minimax algorithm. However, exceptions can occur as a result of pathological nodes, which are considered to exist in all game trees and can cause a deeper game-tree search, resulting in worse play. To reduce the impact of pathological nodes on the search quality, we propose an iterative optimal minimax (IOM) algorithm by optimizing the backup rule of the classic minimax algorithm. The main idea is that calculating the state values of the intermediate nodes involves not only the static evaluation function involved but also a search into the future, where the latter is given a higher weight. We experimentally demonstrated that the proposed IOM algorithm improved game-playing performance compared to the existing algorithms.

Simplicial Complexes are Game Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known parameters of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.

APPLICATION OF COMPLEX GAME-TREE STRUCTURES FOR THE HSU GRAPH IN THE ANALYSIS OF AUTOMATIC TRANSMISSION GEARBOXES

In the article was discussed the possibility of structures and information systems complex game trees for the analysis of automatic gearboxes. The purpose of modelling an automatic gearbox with graphs can be versatile, namely: determining the transmission ratio of individual gears, analysing the speed and acceleration of individual rotating elements. In a further step, logic tree-decision methods can be used to analyse functional schemes of selected transmission gears. Instead, for graphs that are models of transmission, parametrically acting tree structures can be used. This allows for the generalization and extension of the algorithmic approach, furthermore in the future it will allow further analyses and syntheses, such as checking the isomorphism of the proposed solutions, determining the validity of construction and / or operating parameters of the analysed gears. The game tree structure describes a space of possible solutions in order to find optimum objective functions. There is the connection with other graphical structures which can be graphs in another sense, or even decision trees with node and/or branch coding.

Object-Oriented Modeling of Matching to Systemic Classifiers

In this paper we present a version of object-oriented implementation of models of constructive regularized mental systems, mentals, and systemic classifiers introduced in [1] as well as algorithms for matching them to situations. We experiment the adequacy of the models and algorithms for the chess representing kernels of the class of combinatorial problems, where space of solutions can be represented by Reproducible Game Trees (RGT).