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Maximum principles for elliptic systems of partial differential equations
Bílý, Michael ; Bulíček, Miroslav (advisor) ; Kaplický, Petr (referee)
We consider nonlinear elliptic Bellman systems which arise in the theory of stochastic differential games. The right hand sides of the equations (which are called Hamiltonians) may have quadratic growth with respect to the gradient of the unknowns. Under certain assumptions on Lagrangians (from which the Hamiltonians are derived), that are satisfied for many types of stochastic games, we establish the existence and uniqueness of a Nash point and develop structural conditions on the Hamiltonians. From these conditions we establish a certain version of maximum and minimum principle. This result is then used to establish the existence of a bound solution. 1
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Balancing rules of prototype board game Souboj živlů
Holubyev, Dmytro ; Gemrot, Jakub (advisor) ; Surynek, Pavel (referee)
The main goal of this thesis is to analyze the mechanisms of the prototype board game Clash of the elements. Using knowledge from the field of game theory and simulations using the shortcomings of the rules, which are unbalanced between players. The simulator has to offer players and developers a comprehensive and flexible tool that thanks to sophisticated artificial gamers can efficiently test the changes without the risk of disturbing the gaming policies and strategies.
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Algorithmic complexity of solution concepts in selected classes of non-cooperative games
Wichera, Adam ; Majer, Ondřej (advisor) ; Kroupa, Tomáš (referee)
Title: Algorithmic complexity of solution concepts in selected classes of non-cooperative games Author: Adam Wichera Department: Department of Logic Supervisor: RNDr. Ondřej Majer, CSc. Supervisor's e-mail address: majer@ u.cas.cz Abstract In the presented work we study natural algorthmic problems rising from the concept of Nash Equilibrium. The problem of it's existence is trivial, because it follows from Nash The- orem of completeness of Nash Equilibria. Even related search problem doesn't seem to belong to NP-complete class, the reason being the very fact, that existence of Nash Equilibria is certain. Interesting observation is that every natural extension of this problem seems to be NP-complete. Many of such problems have been proven to be NP-complete through reduction of SAT problem, Klike problem or problem of searching subcover of certain size. The question, wheather the pro- blem of existence of assymmetric Nash equilibria of symmeric game ts with the others, in being NP-complete, has been an open problem. Here we show how to alternate the proof from [? ] and apply the construction to problem of existence of assymetric equilibria and therefore prove its NP-completness. Keywords: Nash equilibrium, Algorithmic complexity, Non-cooperative games, Game Theory, Assymetric equilibria, 1
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Evolutionary games and their applications to economic conflicts
Kuzmiak, Maroš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
At the beginning of my Master's thesis we define basic terms such as payoff, strategy, best reply and Nash equilibrium. Furthermore, we introduce the population perspective, in which during a random meeting of a pair of players, these players interact according to their strategies and they receive payoffs. We define the criterion of evolutionary stability, which shows a link between payoffs in the game and strategy spreading among population. The most common description of this evolution is based on the replicator equations. We analyze their basic properties and examine the relationship between the stationary points of this system and the concepts of Nash equilibrium and evolutionary stability. In the following practical part, we apply the introduced theory to model the Cournot duopoly. Its aim is to analyze the model characteristics in terms of evolutionary stability and to determine the duopolist's behavior in the long run.
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Mandatory disclosure
Hrycej, Martin ; Gregor, Martin (advisor) ; Džmuráňová, Hana (referee)
The bachelor thesis firstly introduces the problems of mandatory disclosure and describes different attitudes to it. Following is the essential part of the work containing the definition and specification of the studied models of monopoly and duopoly. These models are further analysed and solved using the game theory tools and microeconomics methods for profit optimization. Subsequently the numerical results are evaluated and presented. The obtained results are suggesting whether voluntary or mandatory disclosure regime is more profitable for whom under different conditions - the models parameters in other words. Finally the discussion of the results, the results' relevance and interpretations in the context of related literature are presented. The particular numerical results are of a reasonable nature and rather fit to the literature context than not.
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Social contract as a rational choice
Froněk, Martin ; Wintr, Jan (advisor) ; Kühn, Zdeněk (referee)
Social contract as a rational choice The thesis is intended to be an introduction into contemporary social contract theories which are elaborated within the methodological framework of the rational choice theory. John Rawls' book A Theory of Justice is a seminal work in this field. The thesis, however, does focus not only on the approach adopted by Rawls, but on the theories of other authors as well - that of J. Buchanan, D. Gauthier, T. Scanlon and R. Nozick. These remain quite unknown in the Czech context. The structure of the thesis should allow for a comparison between the respective theories as the author starts with the object of the social contract, proceeds to the parties to the contract and, finally, to its content. Special attention is devoted to the game theory which models some of the typical interpersonal interactions and, using its theoretical apparatus, can illustrate problems of social cooperation - the key theme of virtually all social contract theories. Following Schelling's theory of focal points an expressive theory of law is discussed. In the next part of the thesis the author illustrates mechanisms of maintaining of social order with the help of latest empirical studies. Eventually, the author tries to outline the way analytic philosophy views the social contract as a system...
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Game Theory for Gifted Secondary School Students
Skálová, Alena ; Hykšová, Magdalena (advisor) ; Staněk, Jakub (referee)
The thesis contains a textbook for gifted secondary school students. Its aim is to give to these students (or to their teachers) a Czech-written text covering fundamental principles in the field of game theory. In the first part we introduce the combinatorial games and some elementary methods of their solution. The second part is devoted to the game of Nim, to the Sprague-Grundy function and to the sums of the combinatorial games. It also contains a necessary introduction to the binary numeral system. In the third part we present the concept of matrix and bimatrix games. There is a lot of exercises and examples in the textbook. At the end we bring solutions to the most of them, providing the active reader with the opportunity of checking their own solutions.
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Application of game theory on oligopoly structures
Kuzmiak, Maroš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
This bachelor thesis is devoted to the games theory and its application to economic models of oligopolies. We focus on determining optimal strategies in non-cooperative conflict and on the issue of coalitions forming and rewards redistribution in cooperative conflict. Lessons learned and characteristics obtained are afterwards applied to different models of oligopolies. Individual chapters are supplemented by short numerical examples which are used to demonstrate the explained theory. The application of this work is to be found in areas where conflict situations are modeled using the knowledge on the choice of optimal strategies.
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Rules in homeless communities: an institutional analysis
Ehrlich, Matěj ; Schwarz, Jiří (advisor) ; Janotík, Tomáš (referee)
In this thesis, we are interested in rules of coexistence among homeless persons in Prague. At first, we will generally describe the emergence of order in society then we will focus on informal institutions and various ways how to enforce them. Subsequently, we will turn our attention to the topic of homelessness. We will provide a definition of homelessness suitable for our purposes and describe functioning of informal social networks that emerged among homeless people in different parts of the world. This will lead us to the empirical part of the thesis, which is based on unstructured interviews and results in description of ob- served regularities and enforcement mechanisms connected to them. For some of those regularities, we will use game theoretical approach to explain how did they emerge and how are they sustained. Keywords: homelessness, informal institutions, institutional analysis, game theory, private enforcement Author's e-mail: matej.ehrlich@gmail.com Supervisor's e-mail: jiri.schwarz@gmail.com
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Does payoff dominance matter?: an experiment
Polena, Michal ; Cingl, Lubomír (advisor) ; Levely, Ian Vandemark (referee)
Risk dominance and Payoff dominance are considered to be the most important selection criteria in Stag-hunt games. In contrast, the main finding of Schmidt et al. (2003) is that players do not respond to changes in Payoff dominance parameter in these games. There might be, however, other explanations for results of Schmidt et al. (2003). Moreover, Dubois et al. (2011) and Battalio et al. (2001)'s experimental results suggest that sufficiently large changes in Payoff dominance parameter may play a role. We, therefore, proposed three Stag-hunt games in order to examine whether players respond to large changes in Payoff dominance parameter. Furthermore, we tested the predictive power of Relative riskiness. Our main finding is that even large changes in Payoff dominance parameter do not induce players to change their choices. An in- significant trend in players' choices, caused by Relative riskiness, was detected in our second finding. Possible explanations are discussed.
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