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The probabilistic (or quantitative) modal μ-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS’s). Two semantics have been studied for this logic, both assigning to every process state a value in the interval [0, 1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic games. The two semantics have been proved to coincide on all finite PLTS’s. A first contribution of the thesis is to extend this coincidence result to arbitrary PLTS’s. A shortcoming of the probabilistic μ-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS’s such as Probabilistic Computation Tree Logic (PCTL). To address this limitation, we extend the logic with a new pair of operators: independent product and coproduct, and we show that the resulting logic can encode the qualitative fragment of PCTL. Moreover, a further extension of the logic, with the operation of truncated sum and its dual, is expressive enough to encode full PCTL. A major contribution of the thesis is the definition of appropriate game semantics for these extended probabilistic μ-calculi. This relies on the definition of a new class of games, called tree games, which generalize standard 2-player stochastic games. In tree games, a play can be split into concurrent subplays which continue their evolution independently. Surprisingly, this simple device supports the encoding of the whole class of imperfect-information games known as Blackwell games. Moreover, interesting open problems in game theory, such as qualitative determinacy for 2-player stochastic parity games, can be reformulated as determinacy problems for suitable classes of tree games. Our main technical result about tree games is a proof of determinacy for 2-player stochastic metaparity games, which is the class of tree games that we use to give game semantics to the extended probabilistic μ-calculi. In order to cope with measure-theoretic technicalities, the proof is carried out in ZFC set theory extended with Martin’s Axiom at the first uncountable cardinal (MAℵ1). The final result of the thesis shows that the game semantics of the extended logics coincides with the denotational semantics, for arbitrary PLTS’s. However, in contrast to the earlier coincidence result, which is proved in ZFC, the proof of coincidence for the extended calculi is once again carried out in ZFC +MAℵ1.

Melioration learning is an empirically well-grounded model of reinforcement learning. By means of computer simulations, this paper derives predictions for several repeatedly played two-person games from this model. The results indicate a likely convergence to a pure Nash equilibrium of the game. If no pure equilibrium exists, the relative frequencies of choice may approach the predictions of the mixed Nash equilibrium. Yet in some games, no stable state is reached.

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

Master of Arts / Journalism and Mass Communications / Soontae An / This study aims to investigate effects of external aid that can enhance motivation and performance of serious game playing to maximize learning effects. Based on self-determination theory, two types of rationales, intrinsic goal framing and extrinsic goal framing, were examined to know which message type is effective. Also, individual’s level of issue involvement was tested as a moderating variable. To investigate the effects of goal framing on motivation and moderating effects of involvement, this study designed a 2x2 (goal framing x involvement) between subjects with 120 middle school students in Korea. Three procedures involved for the experiment: 1) receiving message for manipulating involvement and goal framing 2) Playing actual serious game, and 3) answering survey questionnaire. Overall, providing intrinsic goal before a serious game was more effective to increase both motivation and performance than presenting extrinsic goal. However, there was no main effects and interaction effect in terms of issue involvement.

本文研究一個賣家和多個潛在買家對於一個不可分割商品的議價，且賣家之間存在與身份相關的外部性。我們考察一個有限期的展開形式的博弈，此博弈採用內生的匹配機制，在每一期的博弈中，賣家選擇和哪一個買家進行議價。 / 此博弈有一個唯一的平衡，且無論外部性爲正或負，在平衡中，協議總是立即達成。只有兩個買家時，若外部性爲負，商品必然售予效率買家；若外部性爲正，當買家的議價能力提高時，平衡結果可能會從無效率變爲有效率。若有超過兩個買家存在，無效率結果出現的可能性將會提高。 / This paper studies bargaining between one seller and multiple potential buyers on the sale of one indivisible good, in which indentity-dependent exernalities exist among buyers. We consider an extensive game with nite horizon and endogenous matching procedure, that is, the seller chooses the buyer whom to bargain with during each period of the bargaining game. / The bargaining game has a unique equilibrium with immediate agreement regardless of whether externalities are positive or negative. In a two-buyer game, the good is sold to the efficient buyer when externalities are negative. When externalities are positive, the outcome may change from inefficient to efficient by increasing the bargaining power of the buyers. Inefficient outcomes arise with higher probability in a game with more than two buyers. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhang, Xuechao. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 37-38). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Abstract --- p.i / Abstract in Chinese --- p.ii / Acknowledgements --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.5 / Chapter 3 --- The Model --- p.8 / Chapter 3.1 --- Bargaining Procedure --- p.8 / Chapter 3.2 --- Histories and Strategies --- p.9 / Chapter 3.3 --- Outcomes and Payos --- p.10 / Chapter 4 --- Equilibrium Analysis --- p.13 / Chapter 4.1 --- Equilibrium Dynamics --- p.13 / Chapter 4.2 --- Effi ciency Analysis --- p.20 / Chapter 5 --- Further Extensions --- p.32 / Chapter 5.1 --- Buyer-active Protocol --- p.32 / Chapter 5.2 --- Innite-horizon Framework --- p.34 / Chapter 6 --- Conclusion --- p.36 / References --- p.37

Nash's Theorem is a famous and widely used result in non-cooperative game theory which can be applied to games where each player's mixed strategy payoff function is defined as an expectation. Current proofs of this Theorem neither justify why this constraint is necessary or satisfactorily identifies its origins. In this Thesis we change this and prove Nash's Theorem for abstract games where, in particular, the payoff functions can be replaced by total orders. The result of this is a combinatoric proof of Nash's Theorem. We also construct a generalised simplicial complex model and demonstrate a more general form of Nash's Theorem holds in this setting. This leads to the realisation Nash's Theorem is not a consequence of a fixed-point theorem but rather a combinatoric phenomenon existing in a much more general mathematical model.