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Solving Endgames in Large Imperfect-Information Games such as Poker
Ha, Karel ; Hladík, Milan (advisor) ; Bošanský, Branislav (referee)
Title: Solving Endgames in Large Imperfect-Information Games such as Poker Author: Bc. Karel Ha Department: Department of Applied Mathematics Supervisor: doc. Mgr. Milan Hladík, Ph.D., Department of Applied Mathematics Abstract: Endgames have a distinctive role for players. At the late stage of games, many aspects are finally clearly defined, deeming exhaustive analysis tractable. Specialised endgame handling is rewarding for games with perfect information (e.g., Chess databases pre-computed for entire classes of endings, or dividing Go board into separate independent subgames). An appealing idea would be to extend this approach to imperfect-information games such as the famous Poker: play the early parts of the game, and once the subgame becomes feasible, calculate an ending solution. However, the problem is much more complex for imperfect information. Subgames need to be generalized to account for information sets. Unfortunately, such a generalization cannot be solved straightaway, as it does not generally preserve optimality. As a consequence, we may end up with a far more exploitable strategy. There are currently three techniques to deal with this challenge: (a) disregard the problem entirely; (b) use a decomposition technique, which sadly retains only the same quality; (c) or formalize improvements of...
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Spatial agent-based models of common pool resources
Vach, Dominik ; Gregor, Martin (advisor) ; Červinka, Michal (referee)
This thesis examines the application of the spatial aspect applied in the com- petitive models in the context of the natural resource economics. At first, the spatial models are thoroughly derived in one dimension. Then also their general properties such as the choice of the agents' location or their payoff function are examined. These properties are investigated for various distri- butions of the resource, and therefore they depend also on their parameters. The Nash equilibrium and local stability conditions are derived for the basic setups. In the second part, these competitive models are numerically tested also in a two-dimensional space. One of the results also suggests, that in the setup where the players have perfect information, the beginning player is not necessarily always better off than the second player. Throughout the entire thesis it is also comprehensively examined whether the existence of corners of the strategy space has an impact on the existence of the competition which was successfully demonstrated on several cases. JEL Classification Q20, Q22, C62, C68, C72 Keywords spatial models, natural resource exploitation, Nash equilibrium, fishery, computer simulations Author's e-mail vach.dominik@gmail.com Supervisor's e-mail martin.gregor@fsv.cuni.cz 1
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(Non)rationality of betting
Hamáček, Filip ; Hlaváček, Jiří (advisor) ; Cahlík, Tomáš (referee)
The subject of this bachelors thesis is nonrationality of betting. The first part of thesis is discussing different instruments of risk evaluation. The second part is about Petersburg paradox. This thesis tries to find alternative solution to Petersburg paradox using two methods, the first method is based on repeating high amount of petersburg games. The second method is maximizing the probability of economic survival, which is based on wealth of the player and on bound of economic survival. In the third part of this thesis, Sportka (Czech lottery) is compared with special tournament of poker. The goal of this part is to compare the expected return of investment on playing Sportka and on playing poker without even basic notion about rules and the game strategy. For the estimation of the expected value of poker tournament, different game scenarios are considered, probability of scenarios are based on players behavior according to the Nash equilibrium. Keywords: Betting, expected value, Petersburg paradox, lottery, poker, Nash equilibrium
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Evaluating public state space abstractions in extensive form games with an application in poker
Moravčík, Matej ; Hladík, Milan (advisor) ; Zimmermann, Karel (referee)
Efficient algorithms exist for finding optimal strategies in extensive-form games. However human scale problems, such as poker, are typically so large that computation of these strategies remain infeasible with current technology. State space abstraction techniques allow us to derive a smaller abstract game, in which an optimal strategy can be computed and then used in the real game. This thesis introduces state of the art abstraction techniques. Most of these techniques do not deal with public information. We present a new automatic public state space abstraction technique. We examine the quality of this technique in the domain of poker. Our experimental results show that the new technique brings significant performance improvement. Powered by TCPDF (www.tcpdf.org)
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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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