Participants in deregulated electric power markets compete for financial transmission rights (FTRs) to hedge against losses due to transmission congestion by submitting bids to the independent system operator (ISO). The ISO obtains an FTR allocation, that maximizes sales revenue while satisfying simultaneous feasibility. This FTR allocation remains in place for a length of time during which the participants compete in the energy market to maximize their total payoff from both FTR and energy markets. Energy markets (bi-lateral, day ahead, real time) continue until the the end of the current FTR period, at which time the participants can choose to modify their FTR holdings for the next FTR period. As in any noncooperative game, finding Nash equilibrium bidding strategies is of critical importance to the participants in both FTR and energy markets. In this research, a two-tier matrix game theoretic modeling approach is developed that can be used to obtain equilibrium bidding behavior of the participants in both FTR and energy markets considering the total payoff from FTR and energy. The matrix game model presents a significant deviation from the bilevel optimization approach commonly used to model FTR and energy allocation problems. A reinforcement learning (RL) algorithm is also developed which uses a simulation model and a value maximization approach to obtain the equilibrium bidding strategies in each market. The model and the RL based solution approach allow consideration of multi-dimensional bids (for both FTR and energy markets), network contingencies, varying demands, and many participants. The value iteration based RL algorithm obtains pure strategy Nash equilibrium for FTR and energy allocation. A sample network with three buses and four participants is considered for demonstrating the viability of the game theoretic model for FTR market. A PJM network example with five buses, five generators and three loads is also considered to analyze equilibrium bidding behavior in joint FTR and energy markets. Several numerical experiments on the sample networks are conducted using the approach of statistical design of experiments (DOE) to assess impacts of variations of bid and network parameters on the market outcomes like participant payoffs and equilibrium strategies.

In this study cost-benefits problems concerning the knapsack problem of limited resources is studied and how this relates to an attacker perspective when choosing defense strategies. This is accomplished by adopting a cost-benefit method and merging it with game theory. The cost-benefit method chosen for this study is the Analytical Hierarchy Process and from the field of game theory the Bayesian Nash Equilibrium is used. The Analytical Hierarchy Process allows the user to determine internally comparable weights between elements, and to bring in a security dimension to the Analytical Hierarchy Process a sub category consisting of confidentiality, integrity and availability is used. To determine the attacker strategy and, in effect, determine the best defense strategy the Bayesian Nash Equilibrium is used.

We propose a new classification for multi-agent learning algorithms, with each league of players characterized by both their possible strategies and possible beliefs. Using this classification, we review the optimality of existing algorithms and discuss some insights that can be gained. We propose an incremental improvement to the existing algorithms that seems to achieve average payoffs that are at least the Nash equilibrium payoffs in the long-run against fair opponents. / Singapore-MIT Alliance (SMA)

This thesis considers resource management issues in wireless sensor networks (WSNs), wireless local area networks (WLANs), and cognitive radio (CR) networks. Since energy is a critical resource in WSNs, we consider energy minimization techniques based on explicit node cooperation and distributed source coding (DSC). The explicit node cooperation based on space time block codes (STBC) improves energy efficiency of WSNs, by reducing the energy consumption per bit of each sensor node. The DSC on the other hand exploits the spatial correlation in WSNs, and thus reduces the data generated in a WSN. For the purpose of our analysis, we model the spatial correlation according to a linear Gauss-Markov model. Through our numerical results, we observe that the node cooperation combined with DSC can improve energy efficiency for many cases of interest. A unique aspect of our work is we obtain important structural results using the concepts from monotone comparative statics. These structural results provide insights into the general design of WSNs. Through our numerical results, we also demonstrate that, the cooperation based transmission can achieve better mutual information (MI)-energy tradeoff than the non-cooperation based transmission scheme. From the perspective of WLANs, we propose a price based approach to regulate the channel occupancy of low rate users, which is known to be the primary cause for low overall throughput in WLANs. Owing to the decentralized nature of WLANs we use non-cooperative game theory as a tool for analysis. Specifically, we use supermodular game theory. Through our analysis, we show that an increase in price leads to an increase in rate of WLAN users. We also prove that the best response dynamics indeed converge to the Nash equilibrium of the underlying non-cooperative game. Through our numerical results, we demonstrate that by proper tuning of the price, the proposed price based approach can lead to an improvement in overall throughput of a WLAN. Finally from the perspective of CR networks, we consider the impact of number of channels captured by a secondary user on its transmission control protocol (TCP) throughput. From our simulation results it was found that, there exists a definite optimal number of channels a secondary user needs to capture, to maximize its TCP throughput.

Traditionally, most consumers of electricity pay for their consumption according to a fixed-rate. The few existing implementations of real time pricing have been restricted to large industrial consumers, where the benefits could justify the high implementation cost. With the advancement of Smart Grid technologies, large scale implementation of variable-rate metering will be more practical. Consumers will be able to control their electricity consumption in an automated fashion, where one possible scheme is to have each individual maximize their own utility as a noncooperative game. In this thesis, noncooperative games are formulated among the consumers of Smart Grid with two real-time pricing schemes, where the Nash equilibrium operation points are investigated for their uniqueness and load balancing properties. The first pricing scheme charges a price according to the average cost of electricity borne by the retailer and the second charges according to a time-variant increasing-block price. The zero revenue model and the constant revenue rate model, are the two revenue models being considered. The relationship between these games and certain congestion games, known as atomic flow games from the computer networking community, is demonstrated. It is shown that the proposed noncooperative game formulation falls under the class of atomic splittable flow games. It is shown that the Nash equilibrium exists for four different cases, with different pricing schemes and revenue models, and is shown to be unique for three of the cases, under certain conditions. It is shown that both pricing schemes lead to similar electricity loading patterns when consumers are interested only in the minimization of electricity costs. Finally, the conditions under which the increasing-block pricing scheme is preferred over the average cost based pricing scheme are discussed.

This thesis considers resource management issues in wireless sensor networks (WSNs), wireless local area networks (WLANs), and cognitive radio (CR) networks. Since energy is a critical resource in WSNs, we consider energy minimization techniques based on explicit node cooperation and distributed source coding (DSC). The explicit node cooperation based on space time block codes (STBC) improves energy efficiency of WSNs, by reducing the energy consumption per bit of each sensor node. The DSC on the other hand exploits the spatial correlation in WSNs, and thus reduces the data generated in a WSN. For the purpose of our analysis, we model the spatial correlation according to a linear Gauss-Markov model. Through our numerical results, we observe that the node cooperation combined with DSC can improve energy efficiency for many cases of interest. A unique aspect of our work is we obtain important structural results using the concepts from monotone comparative statics. These structural results provide insights into the general design of WSNs. Through our numerical results, we also demonstrate that, the cooperation based transmission can achieve better mutual information (MI)-energy tradeoff than the non-cooperation based transmission scheme. From the perspective of WLANs, we propose a price based approach to regulate the channel occupancy of low rate users, which is known to be the primary cause for low overall throughput in WLANs. Owing to the decentralized nature of WLANs we use non-cooperative game theory as a tool for analysis. Specifically, we use supermodular game theory. Through our analysis, we show that an increase in price leads to an increase in rate of WLAN users. We also prove that the best response dynamics indeed converge to the Nash equilibrium of the underlying non-cooperative game. Through our numerical results, we demonstrate that by proper tuning of the price, the proposed price based approach can lead to an improvement in overall throughput of a WLAN. Finally from the perspective of CR networks, we consider the impact of number of channels captured by a secondary user on its transmission control protocol (TCP) throughput. From our simulation results it was found that, there exists a definite optimal number of channels a secondary user needs to capture, to maximize its TCP throughput.

This thesis consists of five chapters, based on four different articles. All of them are devoted to different aspects of spatial heterogeneity and its impact on economic equilibrium in space. The concept of heterogeneous continuous space is discussed in the introductory chapter.The first model "Equilibrium in Continuous Space under Decentralized Production" addresses the issue of the impact of differences across locations in exogeneous productivity on the structure of equilibrium prices, production and trade. The goal is to describe the general equilibrium in a spatially decentralized economy, when production, consumption and markets are distributed in continuous space and transportation costs are essentially linear. It is shown that an autarky equilibrium can exist only if transport costs are high enough. In the general case, the general equilibrium in this model includes some endogeneously determined trade areas, with flows of goods across space, and autarky areas where production and consumption activities take place only at the same point. An analytical solution in explicit functions is obtained; it contains equilibrium prices, labor supply and flows of goods as functions of the spatial variable. The model can be applied to a set of practical questions in regional economics. In particular, it is able to describe persistent price differentials across regions and non-local consequences of road construction and transportation cost shocks for the economy. The differences across locations in population density may have either historical or economic reasons.The second model "Hotelling's Revival" extends a well-known research of H.Hotelling (1929) to the two-dimensional case with spatially heterogeneous demand density, preserving the rest of his classical assumptions. It is shown that the problem of demand discontinuity in the one-dimensional model, which was discovered by d'Aspremont, Gabszewich and Thisse (1979), disappears in this case. This also holds for any bounded distribution of consumers on any compact set on a plane, which can describe real geographical situations. Demand continuity still holds for any transport costs, strictly increasing in distance and not necessarily linear. Although this is sufficient for the existence of Nash equilibrium in mixed strategies, in pure strategies it exists only for some subset of cases. Examples of both existence and non-existence are constructed, and for some family of densities the separation point between the two cases is found.The third model addresses locational choice of heterogeneous consumers, when land is also heterogeneous in quality. It is based on two articles. The first, "Dacha Pricing", is presented in chapter 4 and studies the problem of locational rent in a city-neighbourhood when utility includes both the impact of transport costs and time for transportation. For the case of identical agents the problem is solved explicitly and comparative statics with respect to exogeneous changes in transport cost and speed is studied. For the case of agents who are heterogeneous with respect to their income, a solution is also obtained. The model explains some evidence about dacha pricing in Russia and its dynamics during the transition period. The second article related to this model is "Location and Land Size Choice by Heterogeneous Agents". It generalizes the first one and form a separate chapter 5. A new approach about the general equilibrium allocation of heterogeneous divisible good (like land) among a continuum of heterogeneous consumers is proposed. The model is based on continuity of primitives which allow not only to finding a general equilibrium solution in a class of continuous functions, but also to treat the solution to a continuous problem as the limit of the corresponding sequence of discrete problems. This solves one of Berliant's paradoxes, related to spatial economics. The multiplicity of equilibria is shown to take place.

Neste trabalho demonstram-se os teoremas de ponto fixo de Brouwer e Kakutani com o objetivo de provar a existência do equilíbrio de Nash em jogos finitos em forma normal. No primeiro capítulo apresentam-se as definições de teoria dos jogos, começando com jogos finitos em forma normal e terminando com o conceito de equilíbrio de Nash. Na primeira seção do capítulo dois desenvolve-se a teoria de simplexes, em Rn, e se demonstra o teorema de Brouwer. Na seção seguinte, são relacionadas as propriedades de semi-continuidade superior e gráfico fechado em set functions, para então provar os teoremas de Celina e von Neumann que, em conjunto com o teorema de Brouwer, resultam no teorema de Kakutani no fim da seção. Como último resultado é demonstrado o teorema de existência do equilíbrio de Nash em jogos finitos em forma normal através do teorema de Kakutani, mostrando que o equilíbrio de Nash é um ponto fixo de uma set function. / In this work, the fixed-point theorems of Kakutani and Brouwer are proved with the intention of showing the existence of Nash equilibrium in finite normal-form games. In the first chapter the needed definitions of game theory are shown, starting with finite normal-form games and ending with the concept of Nash equilibrium. In the first section of chapter two, simplex theory in Rn is developed and then the Brouwer fixer point theorem is proved. In the next section, some relations of upper hemi-continuity and closed graph in set functions are shown, then proving the theorems of Celina and von Neumann that, along with Brouwer theorem, result in Kakutani fixed-point theorem in the end of the section. As the last result, the existence of Nash equilibrium in finite normal-form games is proved through Kakutani’s theorem, relating the Nash equilibrium to the fixed-point of a set function.

Ce travail présente une optimisation bi-niveau pour la conception de réseaux durables de ressources dans les parcs éco-industriels (EIP). Tout d'abord, les méthodes d'optimisation multiobjectif sont explorées afin de gérer la nature multicritère des problèmes de conception de réseaux dans les EIP. Ensuite, différents cas d’étude sont explorés et analysés afin de maintenir un équilibre concernant les coûts opératoires des usines, tout en minimisant la consommation des ressources naturelles. Ainsi, le problème est modélisé selon une structure bi-niveau reprenant les concepts de la théorie des jeux, où les usines des entreprises jouent un jeu de Nash entre elles, tout en étant dans une structure de jeu de Stackelberg avec l'autorité environnementale. Cette structure définit un modèle qui doit être transformé en un problème MOPEC (Multiple Optimization Problems with Equilibrium Constraints). Différents cas d’étude sont explorés : le premier cas est le réseau d'eau mono-polluant d’un EIP dans lequel l’influence des paramètres opératoires des usines est étudiée afin de déterminer ceux qui favorisent la symbiose entre les usines. Le réseau d'eau est composé d'un nombre fixe de procédés et d’unités de régénération où les concentrations maximales d’entrée et de sortie des polluants sont définies a priori. L'objectif est alors de déterminer quelles sont les allocations entre procédés et unités de régénération. Les résultats obtenus mettent en évidence les avantages de la structure du modèle proposée par rapport aux approches multiobjectif traditionnelles, en obtenant des gains économiques équilibrés d’usines différentes (gains entre 12-25%) tout en maintenant une faible consommation globale des ressources. Ensuite, d'autres études de cas sont abordées à l'aide de la structure bi-niveau : il s’agit d'inclure simultanément les réseaux d'énergie et d’eau dans une formulation multileader multi-follower où les deux «autorités » environnementales sont supposées jouer un jeu non-coopératif de Nash. Dans un premier cas, le gain économique est plus important en incluant des réseaux d'énergie dans la structure de l’EIP. La deuxième étude de cas industriel explore un modèle de réseau d’utilités offre-demande où l'autorité environnementale vise à minimiser les émissions totales de CO2 dans le parc. La conclusion des différents cas explorés montre des résultats extrêmement favorables en termes de coût et d’impact environnemental ce qui vise à encourager les entreprises à participer à l'EIP. / This work presents a bilevel programming framework for the design of sustainable resource networks in eco-industrial parks (EIP). First, multiobjective optimization methods are explored in order to manage the multi-criteria nature of EIP network design problems. Then, different case studies are modeled in order to minimize and maintain in equilibrium participating plants operating costs while minimizing resource consumption. Thus, the structure of the model is constituted by a bilevel programming framework where the enterprises’ plants play a Nash game between them while being in a Stackelberg game structure with the authority. This structure defines a model which, in order to be solved, has to be transformed into a MOPEC (Multiple Optimization Problems with Equilibrium Constraints) structure. Regarding the case studies, monocontaminant water networks in EIP are studied first, where the influence of plants operating parameters are studied in order to determine the most important ones to favor the symbiosis between plants. The water network is composed of a fixed number of process and water regeneration units where the maximal inlet and outlet contaminant concentrations are defined a priori. The aim is to determine which processes are interconnected and the water regeneration allocation. Obtained results highlight the benefits of the proposed model structure in comparison with traditional multiobjective approaches, by obtaining equilibrate different plants operating costs (i.e. gains between 12-25%) while maintaining an overall low resource consumption. Then, other case studies are approached by using the bilevel structure to include simultaneously energy networks in a multi-leader-multi-follower formulation where both environmental authorities are assumed to play a noncooperative Nash game. In the first case study, economic gain is proven to be more significant by including energy networks in the EIP structure. The second industrial case study explores a supply-demand utility network model where the environmental authority aims to minimize the total equivalent CO2 emissions in the EIP. In all cases, the enterprises’ plants are encouraged to participate in the EIP by the extremely favorable obtained results.

Nesse trabalho, aplicando métodos da Topologia Algébrica, nós obtivemos novas versões do teorema de equilíbrio de Nash. Nós definimos um conceito de equilíbrio local para jogos não cooperativos, o chamado equilíbrio local fraco, e demonstramos sua existência quando os espaços de estratégia são variedades diferenciáveis e as funções payoff são continuamente diferenciáveis. Nós demonstramos a ineficiência do equilíbrio local fraco no sentido de Pareto / In this work, applying methods of Algebraic Topology, we obtain new versions of the Nash equilibrium theorem. We define a concept of local equilibrium for non-cooperative games, the socalled weak local equilibrium, and we prove its existence when the spaces of strategies are differentiable manifolds and the payoff functions are continuously differentiable. We prove the ineffciency of weak local equilibrium in the Pareto sense