Teaching the applications of optimisation in game theory's zero sum and non-zero sum games

We apply linear and non-linear programming to find the solutions for Nash equilibriums and Nash arbitration in game theory problems . Linear programming was shown as a viable method for solving mixed strategy zero-sum games. We review this methodology and suggest a class of zero-sum game theory problems that are well suited for linear programming. We applied this theory of linear programming to non-zero sum games. We suggest and apply a separate formulation for a maximising linear programming problem for each player. We move on the Nash arbitration method and remodel this problem as a non-linear optimisation problem. We take the game's payoff matrix and we form a convex polygon. Having found the status quo point (x*, y*), we maximise the product (x - x*)(y - y*) over the convex polygon using KTC non-linear optimisation techniques. The results give additional insights into game theory analysis

Teaching the applications of optimisation in game theory's zero sum and non-zero sum games

We apply linear and non-linear programming to find the solutions for Nash equilibriums and Nash arbitration in game theory problems. Linear programming was shown as a viable method for solving mixed strategy zero-sum games. We review this methodology and suggest a class of zero-sum game theory problems that are well suited for linear programming. We applied this theory of linear programming to non-zero sum games. We suggest and apply a separate formulation for a maximising linear programming problem for each player. We move on the Nash arbitration method and remodel this problem as a non-linear optimisation problem. We take the game's payoff matrix and we form a convex polygon. Having found the status quo point (x*, y*), we maximise the product (x-x*)(y-y*) over the convex polygon using KTC non-linear optimisation techniques. The results give additional insights into game theory analysis.nonlinear optimisation; game theory; linear programming; Nash equilibrium; Nash arbitration; non-zero sum games; zero sum games.

Sustainable development in a complex world

The thesis examines the development of theoretical models and practical tools for understanding and making decisions regarding sustainable development (SD) in a complex world. It seeks to answer three questions: • What is the meaning of SD in a complex world? • What are the implications for our policies and institutions? • What tools can we use to assist decision-making for SD in a complex world? The thesis examines these issues by bringing together thinking and research within complex systems theory, cultural theory and management theory. It also draws on the environmental and SD literatures, which are in turn associated with several disciplines including geography, economics and environmental science. The thesis is in four parts: The first part explores the diversity of SD definitions and the two main policy models for SD. It identifies the need for a theoretical explanation of the plurality of perspectives in the SD debate, and also for an integration of insights from complexity theory into SD definitions and policy models. The second part introduces complexity theory and cultural theory, and examines how they can support development of an improved understanding of sustainability. The third part applies complexity theory and cultural theory to analyse SD policies. It argues that these theories yield significant new insights about sustainability, including highlighting the importance of plurality and resilience in policy making for sustainability. The final part explores how the theories and tools developed by management science can be applied to sustainability, especially in increasing the plurality and resilience of SD decision making. It demonstrates that scenarios can promote the organisational learning and institutional resilience needed for sustainability, and that cultural theory provides an appropriate theoretical basis for scenario-building. The thesis concludes by developing a set of SD scenarios and analysing the results from their application

Experimetrics: A Survey

This monograph aims to survey a range of econometric techniques that are currently being used by experimental economists. It is likely to be of interest both to experimental economists who are keen to expand their skill sets, and also the wider econometrics community who may be interested to learn the sort of econometric techniques that are currently being used by Experimentalists. Techniques covered range from the simple to the fairly advanced. The monograph starts with an overview of treatment testing. A range of treatment tests will be illustrated using the example of a dictator-game giving experiment in which there is a communication treatment. Standard parametric and non-parametric treatment tests, tests comparing entire distributions, and bootstrap tests will all be covered. It will then be demonstrated that treatment tests can be performed in a regression framework, and the important concept of clustering will be explained. The multilevel modelling framework will also be covered, as a means of dealing with more than one level of clustering. Power analysis will be covered from both theoretical and practical perspectives, as a means of determining the sample size required to attain a given power, and also as a means of computing ex-post power for a reported test. We then progress to a discussion of different data types arising in Experimental Economics (binary, ordinal, interval, etc.), and how to deal with them. We then consider the estimation of fully structural models, with particular attention paid to the estimation of social preference parameters from dictator game data, and risky choice models with between-subject heterogeneity in risk aversion. The method maximum simulated likelihood (MSL) is promoted as the most suitable method for estimating models with continuous heterogeneity. We then consider finite mixture models as a way of capturing discrete heterogeneity; that is, when the population of subjects divides into a small number of distinct types. The application used as an example will be the level-k model, in which subject types are defined by their levels of reasoning. We then consider other models of behaviour in games, including the Quantal Response Equilibrium (QRE) Model. The final area covered is models of learning in games

Evaluating reinforcement learning for game theory application learning to price airline seats under competition

Applied Game Theory has been criticised for not being able to model real decision making situations. A game's sensitive nature and the difficultly in determining the utility payoff functions make it hard for a decision maker to rely upon any game theoretic results. Therefore the models tend to be simple due to the complexity of solving them (i.e. finding the equilibrium).In recent years, due to the increases of computing power, different computer modelling techniques have been applied in Game Theory. A major example is Artificial Intelligence methods e.g. Genetic Algorithms, Neural Networks and Reinforcement Learning (RL). These techniques allow the modeller to incorporate Game Theory within their models (or simulation) without necessarily knowing the optimal solution. After a warm up period of repeated episodes is run, the model learns to play the game well (though not necessarily optimally). This is a form of simulation-optimization.The objective of the research is to investigate the practical usage of RL within a simple sequential stochastic airline seat pricing game. Different forms of RL are considered and compared to the optimal policy, which is found using standard dynamic programming techniques. The airline game and RL methods displays various interesting phenomena, which are also discussed. For completeness, convergence proofs for the RL algorithms were constructed

Immersive analytics for oncology patient cohorts

This thesis proposes a novel interactive immersive analytics tool and methods to interrogate the cancer patient cohort in an immersive virtual environment, namely Virtual Reality to Observe Oncology data Models (VROOM). The overall objective is to develop an immersive analytics platform, which includes a data analytics pipeline from raw gene expression data to immersive visualisation on virtual and augmented reality platforms utilising a game engine. Unity3D has been used to implement the visualisation. Work in this thesis could provide oncologists and clinicians with an interactive visualisation and visual analytics platform that helps them to drive their analysis in treatment efficacy and achieve the goal of evidence-based personalised medicine. The thesis integrates the latest discovery and development in cancer patients’ prognoses, immersive technologies, machine learning, decision support system and interactive visualisation to form an immersive analytics platform of complex genomic data. For this thesis, the experimental paradigm that will be followed is in understanding transcriptomics in cancer samples. This thesis specifically investigates gene expression data to determine the biological similarity revealed by the patient's tumour samples' transcriptomic profiles revealing the active genes in different patients. In summary, the thesis contributes to i) a novel immersive analytics platform for patient cohort data interrogation in similarity space where the similarity space is based on the patient's biological and genomic similarity; ii) an effective immersive environment optimisation design based on the usability study of exocentric and egocentric visualisation, audio and sound design optimisation; iii) an integration of trusted and familiar 2D biomedical visual analytics methods into the immersive environment; iv) novel use of the game theory as the decision-making system engine to help the analytics process, and application of the optimal transport theory in missing data imputation to ensure the preservation of data distribution; and v) case studies to showcase the real-world application of the visualisation and its effectiveness

Using features for automated problem solving

We motivate and present an architecture for problem solving where an abstraction layer of "features" plays the key role in determining methods to apply. The system is presented in the context of theorem proving with Isabelle, and we demonstrate how this approach to encoding control knowledge is expressively different to other common techniques. We look closely at two areas where the feature layer may offer benefits to theorem proving — semi-automation and learning — and find strong evidence that in these particular domains, the approach shows compelling promise. The system includes a graphical theorem-proving user interface for Eclipse ProofGeneral and is available from the project web page, http://feasch.heneveld.org

Reinforcing connectionism: learning the statistical way

Connectionism's main contribution to cognitive science will prove to be the renewed impetus it has imparted to learning. Learning can be integrated into the existing theoretical foundations of the subject, and the combination, statistical computational theories, provide a framework within which many connectionist mathematical mechanisms naturally fit. Examples from supervised and reinforcement learning demonstrate this. Statistical computational theories already exist for certainn associative matrix memories. This work is extended, allowing real valued synapses and arbitrarily biased inputs. It shows that a covariance learning rule optimises the signal/noise ratio, a measure of the potential quality of the memory, and quantifies the performance penalty incurred by other rules. In particular two that have been suggested as occuring naturally are shown to be asymptotically optimal in the limit of sparse coding. The mathematical model is justified in comparison with other treatments whose results differ. Reinforcement comparison is a way of hastening the learning of reinforcement learning systems in statistical environments. Previous theoretical analysis has not distinguished between different comparison terms, even though empirically, a covariance rule has been shown to be better than just a constant one. The workings of reinforcement comparison are investigated by a second order analysis of the expected statistical performance of learning, and an alternative rule is proposed and empirically justified. The existing proof that temporal difference prediction learning converges in the mean is extended from a special case involving adjacent time steps to the general case involving arbitary ones. The interaction between the statistical mechanism of temporal difference and the linear representation is particularly stark. The performance of the method given a linearly dependent representation is also analysed