Previous research has identified linear boundaries within a normalized unit square for specific paired strategies within the iterated prisoner's dilemma schema. In this work, general methods of capturing linear boundaries are developed and demonstrated on a wider variety of paired strategies. The method is also tested using an alternate scoring method. An application of Burnside's Lemma simplifies the number of neighbourhood configurations to be considered. In addition, Shannon entropy is used as a means of evaluating diversity of agents evolved with different payoff matrices, by which one might locate a game that is as balanced as possible.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: There are many examples of cooperation in action in society and in nature. In some cases cooperation leads to the increase of the overall welfare of those involved, and in other cases cooperation may be to the detriment of the larger society. The presence of cooperation seems natural if there is a direct bene t to individuals who choose to cooperate. However, in examples of cooperation this bene t is not always immediately obvious. The so called prisoner's dilemma is often used as an analogy to study cooperation and tease out the factors that lead to cooperation. In classical game theory, each player is assumed to be rational and hence typically seeks to select his strategy in such a way as to maximise his own expected pay-o . In the case of the classical prisoner's dilemma, this causes both players to defect. In evolutionary game theory, on the other hand, it is assumed that players have limited knowledge of the game and only bounded rationality. Games in evolutionary game theory are repeated in rounds and players are a orded the opportunity to adapt and learn as this repetition occurs. Past studies have revealed that cooperation may be a viable strategy if the prisoner's dilemma is placed in an evolutionary context, where the evolutionary tness of a strategy is directly related to the pay-o achieved by the player adopting the strategy. One of the mechanisms that promote the persistence of cooperation in the evolutionary prisoner's dilemma is structured interaction between players. A mathematical framework for representing the evolutionary prisoner's dilemma (ESPD) is developed in this thesis. The mathematical framework is used to undertake an analytical approach (i.e. avoiding the use of simulation) towards investigating the dynamics of the ESPD with a path, cycle, plane grid or toroidal grid as underlying graph. The objective of this investigation is to determine the likelihood of the emergence of persistent cooperation between players. The ESPD on a path or a cycle admits two fundamentally di erent parameter regions; large values of the temptation-to-defect parameter are not capable of inducing persistent cooperation, while small values of this parameter allow for the possibility of persistent cooperation. It is found that the likelihood of cooperation increases towards certainty as the order of the underlying graph increases if the underlying graph is a path or cycle. The state space of the ESPD with a plane or toroidal grid graph as underlying graph grows very quickly as a function of the graph order. The automorphism classes of game states are enumerated to determine exactly how fast the size of the state space of the game grows as a function of the order of the underlying graph. Finally, the dynamics of the ESPD is investigated for a grid graph as underlying graph (in cases where the state space is small enough) by means of constructing the corresponding state graphs of the ESPD. / AFRIKAANSE OPSOMMING: Daar is baie voorbeelde van samewerking in the gemeenskap en in die natuur. In sommige gevalle lei samewerking tot 'n toename in die algehele welvaart van die betrokkenes, terwyl samewerking in ander gevalle tot nadeel van die bre er gemeenskap mag wees. Die voorkoms van samewerking blyk natuurlik te wees indien daar 'n direkte voordeel vir die individue is wat kies om saam te werk. In voorbeelde van samewerking is s o 'n voordeel egter nie altyd voor-diehand- liggend nie. Die sogenaamde prisoniersdilemma word dikwels as voorbeeld in die studie van samewerking gebruik om die faktore wat na samewerking lei, te ontbloot. In klassieke speleteorie word daar aangeneem dat elke speler rasioneel is en dus poog om sy spelstrategie op s o 'n manier te kies dat sy eie verwagte uitbetaling gemaksimeer word. In die geval van die klassieke prisoniersdilemma veroorsaak dit dat beide spelers mekaar verraai. In evolusion^ere speleteorie, daarenteen, word daar slegs aangeneem dat elke speler oor beperkte kennis van die spel en begrensde rasionaliteit beskik. Spele in evolusion^ere speleteorie word in rondtes herhaal en spelers word die geleentheid gebied om gedurende hierdie herhalingsproses aan te pas en te leer. Vorige studies het getoon dat samewerking 'n lewensvatbare strategie is indien die prisoniersdilemma in 'n evolusion^ere konteks gespeel word, waar die evolusion^ere ksheid van 'n strategie direk afhang van die uitbetaling van 'n speler wat die strategie volg. Een van die meganismes wat volhoubare samewerking in die evolusion^ere prisoniersdilemma voortbring, is gestruktureerde interaksie tussen spelers. 'n Wiskundige raamwerk word vir die voorstelling van die evolusion^ere prisoniersdilemma in hierdie tesis ontwikkel. Hierdie wiskundige raamwerk word gebruik om 'n analitiese studie (met ander woorde sonder die gebruik van simulasie) van die dinamika van die prisoniersdilemma op 'n pad, siklus, rooster in die vlak, of rooster op die torus as onderliggende gra ek van stapel te stuur. Die doel van hierdie studie is om die waarskynlikheid vir die ontstaan van volhoubare samewerking tussen spelers te bepaal. Die prisoniersdilemma op 'n pad of siklus as onderliggende gra ek het twee fundamenteel verskillende parametergebiede tot gevolg; groot waardes van die versoeking-om-te-verraai parameter lei nie tot volhoubare samewerking nie, terwyl volhoubare samewerking wel vir klein waardes van hierdie parameter moontlik is. Daar word gevind dat die kans vir volhoubare samewerking toeneem tot sekerheid namate die orde van die onderliggende gra ek groei. Die toestandsruimte van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek groei baie vinnig as 'n funksie van die orde van die gra ek. Die outomor smeklasse van die speltoestande word getel met die doel om te bepaal presies hoe vinnig die toestandsruimte van die spel as 'n funksie van die orde van die onderliggende gra ek groei. Die dinamika van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek word laastens deur middel van konstruksies van die ooreenstemmende toestandsgra eke ondersoek (in gevalle waar die toestandsruimte klein genoeg is).

In the theory of open quantum systems, a quantum Markovian master equation, the Lindblad equation, reveals the most general form for the generators of a quantum dynamical semigroup. In this thesis, we present the derivation of the Lindblad equation and several examples of Lindblad equations with their analytic and numerical solutions. The graphs of the numerical solutions illuminate the dynamics and the stabilization as time increases. The corresponding von Neumann entropies are also presented as graphs. Moreover, to illustrate the difference between the dynamics of open and isolated systems, we prove two theorems about the conditions for stabilization of the solutions of the von Neumann equation which describes the dynamics of the density matrix of open quantum systems. It shows that the von Neumann equation is not satisfied for modelling dynamics in the cognitive contextin general. Instead, we use the Lindblad equation to model the mental dynamics of the players in the game of the 2-player prisoner’s dilemma to explain the irrational behaviors of the players. The stabilizing solution will lead the mental dynamics to an equilibrium state, which is regarded as the termination of the comparison process for a decision maker. The resulting pure strategy is selected probabilistically by performing a quantum measurement. We also discuss two important concepts, quantum decoherence and quantum Darwinism. Finally, we mention a classical Neural Network Master Equation introduced by Cowan and plan our further works on an analogous version for the quantum neural network by using the Lindblad equation.

Cooperating in the Prisoner's Dilemma is irrational and some supporting mechanism is needed to stabilize cooperation. Indirect reciprocity based on reputation is one such mechanism. Assessing an individual's reputation requires first-order information, i.e. knowledge about its previous behavior, as it is utilized under image scoring. But there seems to be an agreement that in order to successfully stabilize cooperation, higher-order information is necessary, i.e. knowledge of others' previous reputations. We show here that such a conclusion might have been premature. Tolerant scoring, a first-order assessment rule with built-in tolerance against single defections, can lead a society to stable cooperation. (author's abstract)

Part I focuses on studying the extent of cooperation in networked entities, within the context of the Prisoner's Dilemma (PD) and the Snowdrift Game (SG). The iterated prisoner's dilemma (IPD) is studied in the full payoff space spanned by two parameters beta and gamma. A theoretical study on two-strategy IPD is presented. We then numerically study the IPD in the full payoff space, with four different initial configurations. It is found that including the Tit-for-tat-like (ETFT) and Always-defecting-like (EAllD) strategies as initial strategies can maximize the dominating area of generous strategies in the payoff space at equilibrium. The roles played by ETFT and EAllD are further studied on the diagonal and anti-diagonal lines of the payoff space. / Part II focuses mainly on studying the optical properties of grating within the Rigorous Coupled-Wave Analysis (RCWA) method. The surface plasmon (SP) dispersion relation in a system consisting of a thin metallic film sandwiched between a linear dielectric and nonlinear dielectric of arbitrary non- linearity is derived, based on a generalized "first integral" approach. The SP dispersion relation in a system consisting of a thin metallic film sandwiched in a symmetric nonlinear dielectric environment is then derived. The changes in SP dispersion relations on film thicknesses are discussed for both cases. / The optical properties of two samples of one-dimensional metallic reflection gratings are studied. The numerical results of the zeroth order reflectance are in good agreement with experimental data. The Wood's anomaly occurs when a particular diffraction order emerges or disappears, thus inducing a change in the efficiency of other diffraction orders. This phenomenon is studied by calculating and measuring the efficiencies of all allowed diffraction orders. Numerical results of the near field patterns show a coupling between the waveguide and SP modes. We also study the controllable enhanced trans- mission in a semiconductor grating. The dielectric constant of a semiconductor becomes a tensor in the presence of a static magnetic field parallel to the slit. Numerical results based on RCWA reveal that the zeroth order transmission peaks at normal incidence can be shifted to longer wavelengths and the peak values of transmission can largely be enhanced when a moderate magnetic field is applied. A single-mode theory incorporating anisotropy is developed. The analytic results are in quantitative agreement with RCWA, indicating that the tunability in the transmission stems from the waveguide mode. / The Snowdrift Game is regarded as an important alternative to PD in studying the emergence of cooperation in competing populations. The phase transitions in spatial snowdrift games are introduced. By studying the relative alignments of the payoffs of C and D nodes, the phase transitions are analytically explained. As an extension to the standard two-person SG, an N-person Snowdrift Game (NPSG) is proposed to include generic multi-person interactions. NPSG in a well-mixed population is studied analytically by using the replicator dynamics. A simulation algorithm is developed. We also study NPSG on lattices and find a suppressed cooperation frequency, when compared with the well-mixed case. For NPSG played on 1D chain, the problem can be solved analytically. We further extend our work to study co-evolving dynamics. We propose and study a model in which the connections are driven to evolve by the dynamics of SG. Compared with played on static network, the cooperation frequency is promoted. A semi-analytic theory is proposed, with the results qualitatively agree with simulation results. / The thesis consists of two independent parts. Part I focuses on evolutionary games in networked entities and Part II focuses on calculations on optical properties of metallic gratings. / Yin, Haiping = 競爭群體中合作的產生及具規則結構之金屬薄膜的光學性質 / 尹海平. / Adviser: Hui Pak Ming. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 185-200). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Yin, Haiping = Jing zheng qun ti zhong he zuo de chan sheng ji ju gui ze jie gou zhi jin shu bo mo de guang xue xing zhi / Yin Haiping.

Because the beer market in Taiwan had been opened up, the monopoly, Taiwan Beer, has been facing intense competition. However, state-owned Taiwan Beer uses flexible strategies to respond to these challenges and always is the leading company in the industry. This study tried to use Game Theory to analyze the competitive strategy of Taiwan Beer in different period of time and find out the relationships between Taiwan Beer and competitor. At the beginning, this research used Game theory to describe the competition in the beer industry. Then, it used Prisoner's Dilemma, the Beach Ice Cream Selling Model, entry-deny strategies, multi-brand strategy and Co-opetition to explain Taiwan Beer¡¦s interaction with others. Finally, this study did not see the competition as the only way that can help companies survive. When the company is thinking about how to defeat others, it also has to be thinking about how to cooperate with others to derive more advantages.

published_or_final_version / Psychology / Master / Master of Philosophy

Separate essays on entrepreneurship, joint venturing, and trade comprise this thesis. The emergence of entrepreneurship is common in the real world but relatively less so in classical economic models. If industry incumbents are attributed with full rationality and perfect foresight, then there are few, if any, profitable opportunities left for new entrants (entrepreneurs) to exploit. This essay explains how entrepreneurs can emerge in a dynamic world when firms must choose between a technology strategy that is either statically or dynamically efficient. A model is developed which shows how such opportunities for new entry can occur when incumbents are caught in a Prisoners’ Dilemma game involving technology strategy. A relevance measure and policy implications are then explored. Joint ventures, especially of the R&D type, are becoming increasingly important as a way to gain needed technological and market competencies. Unfortunately, many joint ventures have the characteristics of a Prisoners’ Dilemma. Firms may cooperate or defect in the venture. If contracts, side-payments, and third-party verification of the venture outcome are unavailable, then the dominant solution to the Prisoners’ Dilemma (mutual defection) results. This paper proposes the use of an ex-ante auction to obtain a Pareto-improvement for these ventures. A Pareto-improvement is assured when non-transferable costs and benefits of firms are not conditional on joint venture strategies. When this condition is not met restrictions are required to obtain the Pareto-improvement. The problem of trade between countries that share an international open access resource is becoming significant as the world reaches the limits of critical shared resource stocks. It is modelled as a world with one primary factor, two intermediate goods, one final good (harvested from the open access resource), and two nations where it is assumed that either the trading takes place over one stage (nations are price-takers), or two stages (nations have market power). Imperfect competition and open access generated externalities affect the trading efficiency. To maximize world welfare this essay recommends subsidizing R&D where comparative advantage exists, and creating international agreements to ensure the one-stage game structure is used when trading.

The Learning Classifier System (LCS) and its descendant, XCS, are promising paradigms for machine learning design and implementation. Whereas LCS allows classifier payoff predictions to guide system performance, XCS focuses on payoff-prediction accuracy instead, allowing it to evolve "optimal" classifier sets in particular applications requiring rational thought. This research examines LCS and XCS performance in artificial situations with broad social/commercial parallels, created using the non-Markov Iterated Prisoner's Dilemma (IPD) game-playing scenario, where the setting is sometimes asymmetric and where irrationality sometimes pays. This research systematically perturbs a "conventional" IPD-playing LCS-based agent until it results in a full-fledged XCS-based agent, contrasting the simulated behavior of each LCS variant in terms of a number of performance measures. The intent is to examine the XCS paradigm to understand how it better copes with a given situation (if it does) than the LCS perturbations studied.Experiment results indicate that the majority of the architectural differences do have a significant effect on the agents' performance with respect to the performance measures used in this research. The results of these competitions indicate that while each architectural difference significantly affected its agent's performance, no single architectural difference could be credited as causing XCS's demonstrated superiority in evolving optimal populations. Instead, the data suggests that XCS's ability to evolve optimal populations in the multiplexer and IPD problem domains result from the combined and synergistic effects of multiple architectural differences.In addition, it is demonstrated that XCS is able to reliably evolve the Optimal Population [O] against the TFT opponent. This result supports Kovacs' Optimality Hypothesis in the IPD environment and is significant because it is the first demonstrated occurrence of this ability in an environment other than the multiplexer and Woods problem domains.It is therefore apparent that while XCS performs better than its LCS-based counterparts, its demonstrated superiority may not be attributed to a single architectural characteristic. Instead, XCS's ability to evolve optimal classifier populations in the multiplexer problem domain and in the IPD problem domain studied in this research results from the combined and synergistic effects of multiple architectural differences.

Thesis (M.A.) - Carleton University, 2005. / Includes bibliographical references (p. 63-68). Also available in electronic format on the Internet.