1 Title: Statistical Physics of Frustrated Evolutionary Games Author: Miroslav Pištěk Department: Institute of Theoretical Physics Supervisor: RNDr. František Slanina, CSc. Supervisor's e-mail address: slanina@fzu.cz Abstract: In last two decades, the effort devoted to interdisciplinary research of bounded sources allocation is growing, examining complex phenomena as stock markets or traffic jams. The Minority Game is a multiple-agent model of inevitable frus- tration arising in such situations. It is analytically tractable using the replica method originated in statistical physics of spin glasses. We generalised the Mi- nority Game introducing heterogenous agents. This heterogeneity causes a con- siderable decrease of an average agent's frustration. For many configurations, we achieve even a positive-sum game, which is not possible in the original game variant. This result is in accordance with real stock market data. Keywords: frustrated evolutionary games, Minority Game, Replica method

O objetivo desta dissertação foi a implementação de uma plataforma de otimização por Programação Genética (PG) com o intuito de estudar e caracterizar as propriedades estatísticas de uma grande classe de problemas. Esta implementação foi feita através de programas escritos em LISP, executados num {\\it cluster} de computadores com o sistema operacional Linux. A plataforma foi usada para estudar uma versão do {\\it Jogo da Minoria} (JM) onde seus jogadores utilizam redes neurais para a realização de suas escolhas. Os jogadores foram divididos em dois grupos distintos. O primeiro formado por jogadores que apresentam estratégias estáticas e portanto não adquirem aprendizado. O segundo grupo é formado por jogadores que utilizam um algoritmo de aprendizado para alterar suas estratégias de identificação da minoria. Mostramos que, em determinadas condições, estes jogadores adaptativos conseguem identificar padrões nas escolhas dos jogadores não-adaptativos e assim optam pela decisão da minoria. Porém a eficiência nesta identificação depende do algoritmo de aprendizado utilizado. O algoritmo de aprendizado gerado pela PG se apresentou mais eficiente que outros algoritmos analisados, como, por exemplo, o algoritmo hebbiano. Esta eficiência é caracterizada por uma emergência espontânea de coordenação entre estes jogadores e que lhes proporcionam um melhor desempenho médio por jogador. / The aim of this work is to describe an optimization platform through Genetic Programming in order to study and characterize the statistical property of a wide class of problems. This implementation was written in LISP and executed in a cluster of computers running the Linux operational system. The platform was used to study a version of the Minority Game where the players used neural networks to make their choices. The players were divided into two distinct groups. The first group was made up of players that had quenched strategies and therefore could not learn. The second group had players that used learning algorithms to change their strategies for minority identification. We showed that, under some conditions, the adaptatives players are able to identify patterns in the choices of the non-adaptatives players and can thus benefit by choosing the minority decision. The efficiency in this identification depends on the learning algorithm. The algorithm generated by Genetic Programming is more efficiency than the others algorithms analysed such as hebbian perceptron learning. This efficiency is characterized by a spontaneous emergence of coordination between this players, which permits earning higher scores than average.

O objetivo desta dissertação foi a implementação de uma plataforma de otimização por Programação Genética (PG) com o intuito de estudar e caracterizar as propriedades estatísticas de uma grande classe de problemas. Esta implementação foi feita através de programas escritos em LISP, executados num {\\it cluster} de computadores com o sistema operacional Linux. A plataforma foi usada para estudar uma versão do {\\it Jogo da Minoria} (JM) onde seus jogadores utilizam redes neurais para a realização de suas escolhas. Os jogadores foram divididos em dois grupos distintos. O primeiro formado por jogadores que apresentam estratégias estáticas e portanto não adquirem aprendizado. O segundo grupo é formado por jogadores que utilizam um algoritmo de aprendizado para alterar suas estratégias de identificação da minoria. Mostramos que, em determinadas condições, estes jogadores adaptativos conseguem identificar padrões nas escolhas dos jogadores não-adaptativos e assim optam pela decisão da minoria. Porém a eficiência nesta identificação depende do algoritmo de aprendizado utilizado. O algoritmo de aprendizado gerado pela PG se apresentou mais eficiente que outros algoritmos analisados, como, por exemplo, o algoritmo hebbiano. Esta eficiência é caracterizada por uma emergência espontânea de coordenação entre estes jogadores e que lhes proporcionam um melhor desempenho médio por jogador. / The aim of this work is to describe an optimization platform through Genetic Programming in order to study and characterize the statistical property of a wide class of problems. This implementation was written in LISP and executed in a cluster of computers running the Linux operational system. The platform was used to study a version of the Minority Game where the players used neural networks to make their choices. The players were divided into two distinct groups. The first group was made up of players that had quenched strategies and therefore could not learn. The second group had players that used learning algorithms to change their strategies for minority identification. We showed that, under some conditions, the adaptatives players are able to identify patterns in the choices of the non-adaptatives players and can thus benefit by choosing the minority decision. The efficiency in this identification depends on the learning algorithm. The algorithm generated by Genetic Programming is more efficiency than the others algorithms analysed such as hebbian perceptron learning. This efficiency is characterized by a spontaneous emergence of coordination between this players, which permits earning higher scores than average.

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This paper analyzes two main frameworks that challenge the “mainstream” finance theory and the random walk hypothesis. The first framework is based on investor irrationality and is called Behavioural Finance. The second framework views the financial market as a chaotic system and is called Fractal Theory of a financial market. Behavioural Finance attacks the assumption of investor rationality, thus challenging the conventional finance theories on the micro level. Fractal Theory challenges the EMH and the “macroeconomics” of finance. This paper presents a step towards unifying the frameworks of Behavioural Finance and Fractal Theory. After a review of the relevant literature, a model of the financial market is suggested that rests on the predictions of both Behavioural Finance and Fractal Theory. As a next step, a mathematical algorithm is described that allows to test the financial market for consistency with the presented model. The mathematical algorithm is applied to 10 years of daily S&P500 price quotes, and consistent statistical evidence shows that the predicted fractal pattern reveals itself in the S&P500 prices. The new model outperforms the random walk in out-of-sample forecasting.

Complexity underlying life is largely governed by the dynamics of interaction within and between living and nonliving entities. Evolutionary strategy games are extensively used in modelling and understanding complex behaviors in a wide range of fields including theoretical biology, social interactions, economics, politics, defense and security. Strategy games are said to distill the key elements of interactions be- tween real-world entities and organizations - one of the challenges lies in determining the mapping of complex real life situation dynamics to that of a certain game. That leads us to the two major research questions outlined below. In this thesis, we are taking evolutionary games a step further to investigate the interplay between local and global dynamics, where local dynamics are repre- sented by locally pairwise interactions among the population's players governed by the Iterated Prisoner's Dilemma game. To represent the global dynamics, two main modelling ideas are proposed, in the first model; a mixed evolutionary game is in- troduced where players are competing globally on the population level in a minority game. The interplay between local and global dynamics in this model represents the interplay between different scopes of competition between the same players. Sec- ondly, we introduce a model for studying the effect of sharing global information concerning a population of players, shedding light on how global information can alter the emerging dynamics of local interactions. Furthermore, the thesis addresses the question of whether games - with different dynamics - have unique signatures (footprints) that can be used in recognizing and differentiating among them, and whether these footprints are consistent along the evolutionary path of these games. We show here that by building winning networks between players, and determining network motifs of these winning networks, we can obtain motifs' counts signals that are sufficient to categorize and recognize the game's utility matrix used by the players. We also demonstrate that these footprints - motifs' counts - are consistent along the evolutionary path of the games, due to a hyper-cyclic behavior that exists between strategies. Finally, we show that this approach is capable of identifying whether a certain population is driven by local dynamics or both local and global dynamics using the proposed mixed game.

The common theme throughout the thesis is to explore the possibility of using a single framework, namely the systems theory framework, in modeling a few stylized features of a financial market. A systems theoretic model is developed, in this thesis in Chapter 3, for confidence bias of an individual. The effect of this bias on his investment decision is brought out explicitly. The phenomenon of excessive trading, arising due to overconfidence and optimism, has been explained. The concept of virtual capital, incorporating the ideas from prospect theory, is introduced. We have proposed a dynamical system framework to model limits to arbitrage and the herding behavior in financial markets in Chapter 4. The market evolves due to the participation of traders. It is instructive to look at the market as a system evolving from a set of initial conditions during every time interval. In the proposed model, herding is defined as a specific relation between the system responses. The proposed herding measure quantifies how far the individual is from clustering with others. It is also shown how this interpretation helps us to understand the effects of herding. There exists a risk when the market price variation, due to herding, is thought of as entirely due to the portfolio fundamentals. The generic dynamical system model that captures some aspects of the limits of arbitrage is also proposed wherein fundamental risk, noise trader risk, implementation risk, and model risk can be incorporated. The proposed model offers a single framework to study the Marginally Efficient Market and Synchronization Risk models. In Chapter 5, we have proposed a switching dynamical system with minority game rules incorporated within the framework. We have explored the possibility of developing a market model, in Chapter 6, in the same framework that has been used to develop models for arbitrage and herding. We have explored, in this thesis, the possibility of using a single framework to model stylized features of stock market. It will be a long way before a single model can capture all complex characteristic features of a stock market. We have attempted, in this thesis, to capture a few stylized features in a single framework, if not in a single model. Different models proposed for individual confidence bias, limits to arbitrage, herding, and switching model for incorporating minority games are all set up in system dynamics framework. This leads to a stage where one can explore incorporating other features, not addressed in this thesis, in system dynamics framework. If each feature is captured using a different framework like confidence bias as stochastic system, herding as pattern cluster, limits to arbitrage as rule-based agents, etc., it would be difficult to integrate them into a single framework. But, in the present work, we have captured the chosen stylized features using system dynamics framework though individual models differ from each other substantially. The challenges are many in creating a single framework. The vision of such framework may involve different components such as modeling decision making, considering risk profiles, devising investment strategies, etc. Stylized features would come as emergent properties of complex interactions among the components of the system. Emergence refers to the way in which multiplicity of simple interactions lead to complex behavior. Emergence of such features may include different time scales of causal relationships among components. System may have thresholds, determined by diversity of traders and nature of interactions, which is vital for features to become emergent. This can be seen in practice. Stock market regulates the relative prices of companies across the world. There is no single central agency to control the workings of the market. Traders have knowledge of only few companies within their portfolio, and to follow transaction rules. Trends and patterns are still emerging which are studied by technical analysts. Emergent properties are mostly signature of self-organizing complex system. Self-organization in complex system relies on four properties which are fundamental in system dynamics framework: positive feedback, negative feedback, multiple interactions, and balance among strategies. A complex adaptive stock market system which is self-organizing and exhibit stylized features as emergent property is a distant goal of system theorists around the world. The challenge does not end there. We have attempted to model and study the stylized features of a stock market in systems theory framework. The focus of our approach is to use the dynamical system modeling to study the features. We have not considered the investment aspects in a financial market. The investment models are very important in real life for individuals and policy-makers. Future extension of the ideas explored in this thesis could be along the lines of creating investment models for individuals and policy-makers. Creating such models using complex adaptive stock market system goes a long way in understanding a phenomenon that had started by Dutch East India Company issuing shares on Amsterdam Stock Exchange way back in 1602.