Combinatorial Theories of Games
- Tomáš Valla (guarantor)
- Jan Matyáš Křišťan
- Department of Theoretical Computer Science
Traditional game theory is a branch of mathematics, which has broad applications in economy, biology, politics and computer science. This theory studies the behaviour of agents (players) of a certain competitive process by designinng a mathematical model and investigating the strategies. The traditional task of classical game theory is to find the equilibria, which are the states of the game where no player wants to deviate from his strategy.
Historically, the second big development in game theory of two-player full-information combinatorial games, was by Conway, Berlekamp and Guy. They developed a theory, originally used for solving end-games in Go, into a full fledged field. The idea is to evaluate games such that otherwise incompatible games can be added, that is, played simultaneously. This led to the algrebraic approach to study combinatorial games.
The third most important step is the work of Beck, who established the theory of positional games (like tic-tac-toe and hex). In analysis of these game, one cannot escape the brute-force traversal of the game tree, which is no efficient. Beck introduced the „false probabilistic method“, which aims to tackhle this problem.
In this course we build the foundation of the theory of combinatorial and positional games.
We focus on theoretical analysis of games and building the theory, not on the programming aspects of game solving algorithms. The course requires independent work, ability to mathematically analyse, think and proof.
The course is also suitable for bachelors student in the third year, who attended introduction to graph theory, as well as for PhD students looking for research topics.
* do not fear the mathematics :)
* basics in graph theory, combinatorics and algebra
- Syllabus of lectures:
1) introduction to combinatorial games
2) formal definition of combinatorial games, game classes, strategy theorem
3) comparing games, game algebra (addition, negation, isomorphism)
4) no-number games, adding numbers
5) simplicity rule, dominating and reversible moves
6) infisitemal games, thermographs
7) strong and weak positinal games, strategy stealing, general tic-tac-toe
8) Hall theorem, pairing draw, resource counting
9) Erdös-Selfridge theorem, applications
10) ramsey games, other game versions
- Syllabus of tutorials:
Tutorials designed for deeper understanding the theory presented in the course and for analysis of simpler games.
- Study Objective:
* introduce the algebraic combinatorial game theory due to Conway
* introduce the positional game theory due to Beck
* learn how to apply the theory to analyse several simpler games
- Study materials:
Berlekamp, Conway, Guy: Winning Ways
J. Beck: Combinatorial Games, Tic-Tac-Toe Theory
Conway: On Numbers and Games
Albert, Nowakowski, Wolfe: Lessons in Play
Nisan, Roughgarden, Tardos, Vazirani: Algorithmic Game Theory
- Time-table for winter semester 2020/2021:
- Time-table is not available yet
- Time-table for summer semester 2020/2021:
Fri Thu Fri
- The course is a part of the following study plans:
- Computer Security, Presented in Czech, Version 2020 (elective course)
- Design and Programming of Embedded Systems, Presented in Czech, Version 2020 (elective course)
- Computer Systems and Networks, Presented in Czech, Version 2020 (elective course)
- Management Informatics, Presented in Czech, Version 2020 (elective course)
- Software Engineering, Presented in Czech, Version 2020 (elective course)
- System Programming, Presented in Czech, Version 2020 (elective course)
- Web Engineering, Presented in Czech, Version 2020 (elective course)
- Knowledge Engineering, Presented in Czech, Version 2020 (elective course)
- Specialization Computer Science, Presented in Czech, Version 2020 (elective course)
- Master's Program Informatics, Plan for students without Specialization, in Czech, 2020 (elective course)