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Game theory and poker
Schmid, Martin ; Hladík, Milan (advisor) ; Zimmermann, Karel (referee)
This thesis introduces the basic concepts of the game theory. Necessary models and solution concepts are described. Follows the summary of the computational complexity of these concepts and corresponding algorithms. Poker is formalized as one of the game theory game models. State of the art algorithms for the ex- tensive form games are explained with the application to the Poker. The thesis also introduces the Annual Computer Poker Competition and participating pro- grams. Finally, new result about the extensive form games with many actions is presented. Keywords: Game theory, Poker, Nash equilibrium, Extensive form games
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Probabilistic semantics for Independence-friendly Logics
Seidl, Julian ; Majer, Ondrej (advisor) ; Švarný, Petr (referee)
(in English): Character of the work is purely theoretical and it pursues game theory in the perspective of mathematical logic and probability. The work is divided into two parts. Introductory part compiles basic concepts and definitions, summarizing the game theory and basics of syntax and semantics of mathematical logic and its extensions suitable for work in the field of game theory. Introductory part also explains following terms: extensive and strategic form of games, Nash equilibrium, pure and mixed strategies, winning strategies or independence-friendly logic. The problems solved in the second part of the work such as question of existence of Nash equilibrium in the games with infinite models or issue which arises when trying to uniformly distribute the probability of strategies in the same class of games are sketched out. The second part continues with analysis of strategic games with imperfect information aiming to the solution of nontrivial problems earlier proposed. Second part also introduces basic concepts and definitions of the probability theory, which helps comprehending the problems mentioned above. The last part of the work before the very presentation of some results induced by the area of infinite games is conversion between strategic and extensive games form. In the end of the...
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Game of Markets
Dóczy, Aneta ; Novotná, Veronika (referee) ; Chvátalová, Zuzana (advisor)
This diploma thesis deals with conict economic situations based on game theory. In the beginning, basic models of conict situations and current popular software tools are dened not only for the general support of student education or for science, but also for solving economic problems in game theory. Based on this analysis, the conicting situation of two competing rms is being solved. Gradually, work goes deeper into areas of delay dierential equations that better show the behavior of two players on the market. Subsequently, these delayed dierential equations are projected into the Cournot model, for which a critical value is identied that switches the stability of two rms on the market due to the delayed realization of their outputs.
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Habitat selection game
Slavík, Jakub ; Pražák, Dalibor (advisor) ; John, Oldřich (referee)
In the presented work we study an application of evolutionary game theory in behavioral ecology, specifically the habitat selection game, which describes the distribution of population into a finite number of patches. We also show the existence, uniqueness and evolutionary stability of the ideal free distribution (IFD) observed in natural environments. To describe the process of the distri- bution we specify the dynamics of the habitat selection game using dispersion dynamics, and we show the stability of the IFD for different types of dispersion dynamics using the classical theory of ordinary differential equations and the theory of ordinary differential equations with discontinuous righthand sides. 1
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Limit behavior of the Nash equlibrium
Kovařík, Vojtěch ; Spurný, Jiří (advisor) ; Bárta, Tomáš (referee)
The subject of study of game theory - games - serves as mathematical models for real-life problems. In every game there are two or more players who aim to maximize their own profit by choosing their actions. A situation where no player can benefit from changing his own action alone has got particular importance in the study of games - it is called Nash equilibrium. Games with a finite number of players have certain advantages over those with an infinite number of players. For one, problems whose model is a game with a finite number of players are quite common. Moreover, one of the classical results of game theory is that (with certain additional assumptions) in every game with a finite number of players there exists a Nash equilibrium. On the other hand, when trying to describe the properties of a game with an infinite number of players we might be able to use calculus instead of going trough all possibilities (as is common for games with a finite number of players), which tends to be computationally demanding. However, if we want to use these advantages of games with an infinite number of players, it is important first to know whether there is any relationship between games with a finite and infinite number of players at all. The goal of this thesis is to define terms and to introduce tools which would allow...
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Artificial intelligence for Texas Holdem poker game
Moravčík, Matej ; Petříčková, Zuzana (advisor) ; Sýkora, Ondřej (referee)
Recently there has been a great expansion of poker. This includes live games, as well as games on the internet. For beginners, it may be difficult to find opponents skilled enough and thus improve their gaming performance without deposit of their own funds. Using of artificial intelligence seems as good solution for the problem, but there are only few suitable programs available. This thesis describes the overall design and development of such an application, specially designed for tournament variant of Texas Hold'em poker. Most attention is devoted to the artificial intelligence. There are two main approaches discussed - approximate Nash equilibrium and the use of expert system. Emphasis is placed on the first option. The main contribution of this thesis is detailed description and comparison of three algorithms for calculating the approximation of Nash equilibrium. Two of them are original heuristics algorithms, that take advantage of specific structure of poker game. Algorithms have been implemented and their properties have been empirically evaluated. The final result is a full-featured application designed for end users. It simulates poker game and provides a powerful artificial intelligence with attractive graphical user interface.
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Systems of Polynomial Equations in Economics
Šramková, Kristína ; Tomáš, Jiří (referee) ; Kureš, Miroslav (advisor)
Bachelor thesis is based on application of mathematical apparatus for the analysis of economic models, in particular models that lead to a system of polynomial equations. One of the parts is a summary of basic knowledge of algebra focused on Gröbner basis. Hereinafter are discussed economic models in which solution Gröbner basis are applied using the program Wolfram Mathematica. Own software package is implemented into this program as a concept of solution to simplify the calculation and work with models.
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Game Theory in Waste Management
Osička, Ondřej ; Vašík, Petr (referee) ; Hrdina, Jaroslav (advisor)
V této práci je vytvořen model rozhodovací situace v odpadovém hospodářství využívající metody teorie her. Model tvoří nekooperativní hra pro reprezentaci konfliktu zpracovatelů odpadu a kooperativní hra pro reprezentaci konfliktu producentů odpadu. Pro konflikt zpracovatelů odpadu je k nalezení strategií při volbě cen na bráně využit koncept Nashovy rovnováhy, takto nalezené stabilní strategie mohou sloužit jako předpověď budoucí situace. Pro zpřesnění množin strategií jsou určeny dolní a horní meze. Pro konflikt producentů odpadu se uvažuje spolupráce všech producentů a určuje se pro ni přerozdělení nákladů pomocí Shapleyho hodnoty a nucleolu. Pro konflikt více producentů jsou vyvinuty aproximační algoritmy pro Shapleyho hodnotu i nucleolus. Tyto algoritmy jsou založeny na předpokladu, že se vzdálení hráči vzájemně neovlivňují. Model je aplikován na situaci v České republice. Pro konflikt zpracovatelů odpadu je nalezen jeden bod Nashovy rovnováhy. Pro konflikt producentů odpadu jsou určeni někteří producenti s vysokým kooperativním potenciálem.
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Modelling of Dynamic Games with Imperfect Information
Straka, Richard ; Smrčka, Aleš (referee) ; Hrubý, Martin (advisor)
This work concentrates on the study of dynamic games; in particular games with imperfect information, implementation of a model and experimentation with the model. The game DarkElf was chosen for this project. It is a strategic, turn based game with economic and military features, where the decisions made by players are simultaneously implemented at a predetermined time.
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