博弈理論己貫穿於許多不同的研究領域中，成為了一種非常有用的研究工具。在演化博弈中，個體可以通過比較彼此在競爭中表現的好壞，從一個策略改變至另一個策略。當依賴於收益參數的競爭個體系統處於穩定態時，可以是一個所有個體都採用同一策略的均勻系統，或是使用不同策略的個體共同存在的非均勻系統。近年來，均勻態和混合態或是均勻相和混合相之間的轉變成為物理學家們的研究熱點。 / 在論文的第一部份(第二及第三章)中，我們研究了有空間關聯的群體在規則晶格上演化雪堆博弄(ESG) 。在文獻中的分析方法大多數是基於對近似法(pair approximation) 。但是對近似法的結果，就算是在定性上，都未能捕捉到電算模擬結果中顯現的重要特性。例如，對近似法未能給出全c 態和混合態之間以及混合態與全D 態之間的轉變特徵，其中c和D 分別表示演化雪堆博奔中的兩種策略。而這些轉變作為收益參數的函數，卻可於ESG 在不同的規則品格中的電算模擬中被觀察到。對近似法的這些缺陷引發了本論文的研究。當不同局域競爭組態的分佈形式假設為二項式分佈，並將之引入到c 個體數目隨時間演變的動態公式中，就能得出全D 態。當進一步深入考慮ESG 的演化後，我們得知在收益參數較小時，D 個體被孤立的局部組態的比例在二項式分佈假設中被高估。通過修正D 個體被孤立的局部組態比例，就能得出全C 態。通過適當調整收益參數較小時D 個體被孤立的局部組態的比例，和收益參數較大時適當調整C個體被孤立的局部組態的比例，我們就能得出與模擬結果吻合較好的結果。然後，我們還將這個方法推廣至線性鏈和六角品格結構中。該種方法比對近似法得到的結果更好。 / 在論文的第二部份(第四章)中，我們研究了有本懲罰者的存在對演化雪堆博弈在正方晶格上演化的影響。該博弈中相應地有三種策略。懲罰者願意支付額外成本以對不合作者造成顯外的損失。同時懲罰者與其他合作者之間是合作的。我們展示了在收益參數組成的空間下得出的詳盡相圖。我們從動態演化後期出現的最終生存形態(last surviving patterns) 出發，討論了相圖中的相邊界成因。 / Game theoretical methods have become a useful tool in research across many fields. In evolutionary games, agents could switch from one strategy to another based on how well they perform as compared with others. Depending on a payoff parameter, a system of competing agents may form a homogeneous system with all agents taking on the same strategy or an inhomogeneous system with the coexistence of different strategies in the steady state. The transitions between the homogeneous and mixed states or phases are of much interest to physicists in recent years. / In Part I (Chapter 2 and 3) of the thesis, the evolutionary snowdrift game (ESG) is studied in spatially structured populations on regular lattices. Analytic approaches in the literature are largely based on the pair approximation. However, results of the pair approximation cannot capture the key features in simulation results even qualitatively. For example, the pair approximation fails to give the transition between an All-C and the mixed phase and the transition between the mixed phase and an All-D phase as a function of a payoff parameter observed in ESG on different regular lattices, where C and D are the two strategies in ESG. The failure motivated the present work. By incorporating different local competing configurations within an assumption of binomial distribution into the dynamical equation for the time evolution of the number of C-agents, the All-D phase readily emerges. Further consideration of the dynamics of ESG informs us that the proportion of isolated-D local configurations is over-estimated in the binomial distribution when the payoffparameter is small. By modifying the proportion of isolated-D congurations, the All-C phase results. By suitably adjusting the weighings of isolated-D congurations for small payoff parameters and isolated-C congurations for large payoff parameters, good agreement with simulation results is obtained. The approach is then generalized to linear chain and hexagonal lattice. The present approach performs much better than the pair approximation. / In Part II (Chapter 4) of the thesis, the effects of the presence of costly punishers in ESG are studied in a square lattice. There are, thus, three strategies in the game. The punishers are willing to pay an extra cost so as to inflict an extra damage to a defector, and they cooperate with other cooperators. Detailed phase diagrams are presented in the space formed by the payoff parameters. The phase boundaries are discussed within the idea of last surviving patterns at the late stage of the dynamics. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chan, Wan Hang = 晶格上演化雪堆博弈及相關問題的研究 / 陳運亨. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 81-86). / Abstracts also in Chinese. / Chan, Wan Hang = Jing ge shang yan hua xue dui bo yi ji xiang guan wen ti de yan jiu / Chen Yunheng. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Game Theory --- p.2 / Chapter 1.1.1 --- Prisoner Dilemma --- p.3 / Chapter 1.1.2 --- Snowdrift Game --- p.3 / Chapter 1.1.3 --- Costly Punishment --- p.4 / Chapter 1.2 --- Evolutionary Game Theory --- p.6 / Chapter 1.2.1 --- Updating Rules --- p.6 / Chapter 1.2.2 --- Updating Schemes --- p.6 / Chapter 1.2.3 --- Spatial Structures --- p.8 / Chapter 1.3 --- Analytic Approaches --- p.9 / Chapter 1.3.1 --- Mean-field Approach --- p.9 / Chapter 1.3.2 --- Last Surviving Patterns --- p.10 / Chapter 2 --- Evolutionary Snowdrift Game on Square Lattice --- p.11 / Chapter 2.1 --- Introduction --- p.11 / Chapter 2.2 --- Simulation Results --- p.12 / Chapter 2.3 --- Last Surviving Patterns --- p.15 / Chapter 2.4 --- Pair Approximation --- p.19 / Chapter 2.5 --- The Site Dynamical Equations --- p.22 / Chapter 2.5.1 --- Pure Binomial Approximation --- p.23 / Chapter 2.5.2 --- Modified Binomial Approximation --- p.24 / Chapter 2.5.3 --- Isolated Factor Correction --- p.27 / Chapter 2.5.4 --- The Zonal Correction --- p.31 / Chapter 2.6 --- Summary --- p.32 / Chapter 3 --- Evolutionary Snowdrift Game on Other Networks --- p.34 / Chapter 3.1 --- Snowdrift Game on Other Lattices --- p.34 / Chapter 3.1.1 --- Linear Chain --- p.34 / Chapter 3.1.2 --- Hexagonal Lattice --- p.40 / Chapter 3.2 --- Snowdrift Game on Small-world Network --- p.43 / Chapter 3.3 --- Summary --- p.50 / Chapter 4 --- Costly punishment in ESG on Lattices --- p.52 / Chapter 4.1 --- Introduction --- p.52 / Chapter 4.2 --- Model --- p.54 / Chapter 4.3 --- Simulation Results [1] --- p.56 / Chapter 4.3.1 --- Results with fixed α and β [1] --- p.56 / Chapter 4.3.2 --- Results with a fixed α [1] --- p.60 / Chapter 4.3.3 --- Consideration of special local structures --- p.65 / Chapter 4.3.4 --- Results of different α --- p.67 / Chapter 4.4 --- Pair Approximation Extended to Three Strategies [1] --- p.69 / Chapter 4.5 --- Summary --- p.76 / Chapter 5 --- Summary --- p.78 / Bibliography --- p.81 / Chapter A --- Uncertainty in Simulation Results --- p.87
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_328712 |
| Date | January 2012 |
| Contributors | Chan, Wan Hang., Chinese University of Hong Kong Graduate School. Division of Physics. |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography |
| Format | electronic resource, electronic resource, remote, 1 online resource (xiv, 88 leaves) : ill. (some col.) |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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