Dissertations / Theses on the topic 'Equilibria'

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 183-188).
The main contribution of this thesis is a new solution concept for symmetric games (of complete information in strategic form), the exchangeable equilibrium. This is an intermediate notion between symmetric Nash and symmetric correlated equilibrium. While a variety of weaker solution concepts than correlated equilibrium and a variety of refinements of Nash equilibrium are known, there is little previous work on "interpolating" between Nash and correlated equilibrium. Several game-theoretic interpretations suggest that exchangeable equilibria are natural objects to study. Moreover, these show that the notion of symmetric correlated equilibrium is too weak and exchangeable equilibrium is a more natural analog of correlated equilibrium for symmetric games. The geometric properties of exchangeable equilibria are a mix of those of Nash and correlated equilibria. The set of exchangeable equilibria is convex, compact, and semi-algebraic, but not necessarily a polytope. A variety of examples illustrate how it relates to the Nash and correlated equilibria. The same ideas which lead to the notion of exchangeable equilibria can be used to construct tighter convex relaxations of the symmetric Nash equilibria as well as convex relaxations of the set of all Nash equilibria in asymmetric games. These have similar mathematical properties to the exchangeable equilibria. An example game reveals an algebraic obstruction to computing exact exchangeable equilibria, but these can be approximated to any degree of accuracy in polynomial time. On the other hand, optimizing a linear function over the exchangeable equilibria is NP-hard. There are practical linear and semidefinite programming heuristics for both problems. A secondary contribution of this thesis is the computation of extreme points of the set of correlated equilibria in a simple family of games. These examples illustrate that in finite games there can be factorially many more extreme correlated equilibria than extreme Nash equilibria, so enumerating extreme correlated equilibria is not an effective method for enumerating extreme Nash equilibria. In the case of games with a continuum of strategies and polynomial utilities, the examples illustrate that while the set of Nash equilibria has a known finite-dimensional description in terms of moments, the set of correlated equilibria admits no such finite-dimensional characterization.
by Noah D. Stein.
Ph.D.

We modify the concept of consistent expectations equilibria introduced in Hommes and Sorger (1998) in two ways: (i) the consistency condition requires that the probability that the agents reject their perceived law of motion in any period does not exceed a given level and (ii) there may exist exogenous stochastic shocks. The concept is illustrated by two examples using a linear economic system. In one of the examples consistency implies rational expectations, in the other example it does not. (authors' abstract)
Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

The problem of finding equilibria in non-cooperative games and understanding their properties is a central problem in modern game theory. After John Nash proved that every finite game has at least one equilibrium (so-called Nash equilibrium), the natural question arose whether we can compute one efficiently. After several years of extensive research, we now know that the problem of finding a Nash equilibrium is PPAD-complete even for two-player normal-form games, making the task of finding approximate Nash equilibria one of the central questions in the area of equilibrium computation. In this thesis our main goal is a new study of the complexity of various variants of the approximate Nash equilibrium. Specifically, we study algorithms for additive approximate Nash equilibria in bimatrix and multi-player games. Then, we study algorithms for relative approximate Nash equilibria in multi-player games. Furthermore, we study algorithms for optimal approximate Nash equilibria in bimatrix games and finally we study the communication complexity of additive approximate Nash equilibria in bimatrix games.

This thesis studies various equilibrium concepts in the context of finite games of infinite duration and in the context of bi-matrix games. We considered the game settings where a special player - the leader - assigns the strategy profile to herself and to every other player in the game alike. The leader is given the leeway to benefit from deviation in a strategy profile whereas no other player is allowed to do so. These leader strategy profiles are asymmetric but stable as the stability of strategy profiles is considered w.r.t. all other players. The leader can further incentivise the strategy choices of other players by transferring a share of her own payoff to them that results in incentive strategy profiles. Among these class of strategy profiles, an 'optimal' leader resp. incentive strategy profile would give maximal reward to the leader and is a leader resp. incentive equilibrium. We note that computing leader and incentive equilibrium is no more expensive than computing Nash equilibrium. For multi-player non-terminating games, their complexity is NP complete in general and equals the complexity of computing two-player games when the number of players is kept fixed. We establish the use of memory and study the effect of increasing the memory size in leader strategy profiles in the context of discounted sum games. We discuss various follower behavioural models in bi-matrix games assuming both friendly follower and an adversarial follower. This leads to friendly incentive equilibrium and secure incentive equilibrium for the resp. follower behaviour. While the construction of friendly incentive equilibrium is tractable and straight forward the secure incentive equilibrium needs a constructive approach to establish their existence and tractability. Our overall observation is that the leader return in an incentive equilibrium is always higher (or equal to) her return in a leader equilibrium that in turn would provide higher or equal leader return than from a Nash equilibrium. Optimal strategy profiles assigned this way therefore prove beneficial for the leader.

It has been supposed since the middle of this century that it is the global magnetic field surrounding a quiescent prominence that provides the force to prevent its collapse due to the sun's gravitational field. Many theoretical models, assuming that the prominence plasma is supported in a dip in the magnetic field lines associated by the magnetic tension force, have since been put forward. The aim of this thesis is to propose further models of quiescent prominences to widen our understanding and knowledge of these remarkable features. A short overview over the magnetohydrodynamic equations used to describe solar prominences, or most of the solar phenomena for that matter, are discussed in chapter 2, and a short summary of prominence observations and attempts to model them is given in chapter 3. A brief description of the numerical code used in chapters 5 and 7 is given in chapter 4. Observations of Kim (1990) and Leroy (1985) have found that most large quiescent prominences are of inverse polarity type for which the magnetic field passes through the prominence in the opposite direction to that expected from the photospheric magnetic field. Many theoretical models have been proposed, but failed. Hence, in chapter 5 we investigate first – without the inclusion of a prominence sheet – when an inverse polarity magnetic field must have the correct topology for an inverse polarity configuration before the formation of the prominence itself. Only very recently, the first basic successful model of an I-type polarity prominence was proposed by Low (1993). In chapter 6 we examine this model and investigate current sheets more complicated and realistic than the one used by Low. These analytical models deal with the force-free solution, which is matched onto an external, unsheared, potential coronal magnetic field. These solutions are mathematically interesting and allow an investigation of different profiles of the current intensity of the magnetic field vector and of the mass density in the sheet. The prominence properties predicted by these models have been examined and have been found to match the observational values. The mathematics of current sheets in general is also briefly discussed. Chapter 7 deals with numerical solutions of inverse polarity prominences embedded in a force-free magnetic flux tube, matched onto an unsheared potential coronal field. Unfortunately the solutions gained are quite sensitive to the boundary conditions imposed on them through the numerical box, showing a loss of convergence and a tendency for the solution to blow up. Finally, a short summary as well as possible future work is given in chapter 8.

Thesis (Ph.D.)--Physics, Georgia Institute of Technology, 2009.
Committee Chair: Cvitanovic, Predrag; Committee Member: Bracco, Annalisa; Committee Member: Dieci, Luca; Committee Member: Goldman, Daniel; Committee Member: Grigoriev, Roman

The mathematical modelling of a market, and the proof of existence of equilibria have been of central importance in mathematical economics. Since the existence proof is non-constructive in general, a natural question is if computation of equilibria can be done efficiently. Moreover, the emergence of Internet and e-commerce has given rise to new markets that have completely changed the traditional notions. Add to this the pervasiveness of computing resources, and an algorithmic theory of market equilibrium becomes highly desirable. The goal of this thesis is to provide polynomial time algorithms for various market models. Two basic market models are the Fisher model: one in which there is a demarcation between buyers and sellers, buyers are interested in the goods that the sellers possess, and sellers are only interested in the money that the buyers have; and the Arrow-Debreu model: everyone has an endowment of goods, and wants to exchange them for other goods. We give the first polynomial time algorithm for exactly computing an equilibrium in the Fisher model with linear utilities. We also show that the basic ideas in this algorithm can be extended to give a strongly polynomial time approximation scheme in the Arrow-Debreu model. We also give several existential, algorithmic and structural results for new market models: - the *spending constraint* utilities (defined by Vazirani) that captures the "diminishing returns" property while generalizing the algorithm for the linear case. - the capacity allocation market (defined by Kelly), motivated by the study of fairness and stability of the Transmission Control Protocol (TCP) for the Internet, and more generally the class of Eisenberg-Gale (EG) markets (defined by Jain and Vazirani). In addition, we consider the adwords market on search engines and show that some of these models are a natural fit in this setting. Finally, this line of research has given insights into the fundamental techniques in algorithm design. The primal-dual schema has been a great success in combinatorial optimization and approximation algorithms. Our algorithms use this paradigm in the enhanced setting of Karush-Kuhn-Tucker (KKT) conditions and convex programs.

Thesis (Ph. D.)--University of Oregon, 2007.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 100-125). Also available for download via the World Wide Web; free to University of Oregon users.

Reservoir fluids are usually saturated with water at reservoir conditions and may form gas hydrates in transfer lines, which potentially may plug the system. For long subsea pipelines, methanol injection is the practical means for preventing hydrate formation and for decomposing blockages. For efficient and economical pipeline design and operation, phase boundaries, phase fractions and distribution of water and methanol among the equilibrium phases of the system must be accurately known. The system comprising reservoir fluids, water and methanol demonstrates a complex multiphase behaviour and currently no quantitatively adequate description for it has been detailed in the open literature. The problem is addressed in this thesis by a consistent application of classical equilibrium thermodynamics. At ordinary operating conditions any combination of as many as six phases can be potentially present. For the description of the vapour and all liquid phases, we use one cubic equation of state with nonconventional mixing rules developed as part of this work. Classical thermodynamics together with the cell theory of van der Waals and Platteeuw were employed for the development of a general model for the calculation of heat capacities of gas hydrates. A consistent methodology has also been developed for obtaining the potential parameters of the cell model. Thereafter, application of the model demonstrates that for nearly spherical guest molecules the classical cell theory is a strictly valid description of gas hydrates. However, complex guest molecules distort the hydrate lattice, resulting in variation of the numerical values of certain parameters of the model. This work presents an efficient algorithm for the solution of the problem of the identity of the equilibrium phases in multiphase systems where gas hydrates are potentially present. The algorithm is based on the alternative use of two equivalent forms of the Gibbs tangent plane criterion and it is believed to be more appropriate for systems involving gas hydrate equilibria than previous methods. Application of the proposed algorithm in several regions of the phase diagram of both binary and multicomponent systems shows that it can be used reliably to solve any phase equilibria problem, including the location of phase boundaries. In summary this work presents a consistent, efficient and reliable scheme for multiphase equilibrium calculations of systems containing reservoir fluids, water and methanol. Favourable results have been obtained by comparison with diverse experimental data reported in the open literature and it is believed that the proposed correlation can be used reliably for pipeline design and operation.

Thesis (MScEng)--University of Stellenbosch, 2000.
ENGLISH ABSTRACT: Supercritical extraction is being investigated as a possible alternative to the processes currently used in the fractionation of paraffinic waxes. By removing the lighter carbon fractions from the wax, the wax hardness will be improved and its melting temperature range reduced, hence improving the performance of the wax product in certain applications. In order to evaluate and operate such an extraction process optimally, it is necessary to have a thermodynamic model that accurately represents the process system. There are, however, currently no predictive models available for these systems. In order to fit present models to the systems, accurate phase equilibrium data of the supercritical solvent - n-alkane systems are needed. Unfortunately, the amount of reliable published data on these systems in the required operating range is very limited. A view cell was designed and developed with which these high pressure equilibria could be studied. The binary phase equilibria of supercritical CO2 with n-CI2, n-CI6, n-C20, n-C24, n-C28 and n-C36 and of supercritical ethane with n-CI6, n-C24 and n-C28 were measured in the temperature range 313 - 367 K. It was found that the systems with these two solvents have very different types of phase behaviour. The n-alkane solubility is much higher in ethane, but supercritical CO2 will provide a much better degree of control over the selectivity achieved in an extraction process. Of the various equations of state investigated, it was found that the Patel Teja equation of state provided the best fit of the CO2 - n-alkane systems and that the Soave-Redlich- Kwong equation fitted the ethane - n-alkane systems the best. The interaction parameters of both these equations of state display a functional relationship with temperature and nalkane acentric factor, making it possible to determine parameter values for application at other operating temperatures and with other n-alkane systems. It was found that the current equations of state were not able to represent the phase equilibria accurately over the entire range of operating conditions. The poor performance of the equations of state can be attributed to inherent flaws in the existing equations of state.
AFRIKAANSE OPSOMMING: Superkritiese ekstraksie word tans ondersoek as 'n moontlike altematief vir die prosesse wat huidiglik gebruik word om paraffiese wasse te fraksioneer. Die Iigter koolstofwasse word verwyder om die washardheid te verhoog en die temperatuurgebied waaroor die was smelt te verklein. Dit verbeter dan die was se kwaliteit en werkverrigting. Modelle wat die superkritiese ekstraksie proses akkuraat kan voorstel word egter benodig om die ekstraksie proses te kan evalueer en optimaal te bedryf. Daar is tans geen modelle beskikbaar wat die proses direk kan voorstel nie. Akkurate fase-ewewigsdata word benodig om bestaande modelle aan te pas vir gebruik in hierdie sisteme. Daar is egter baie min betroubare faseewewigsdata vir die superkritiese oplosmiddel - n-alkaan sisteme beskikbaar in die literatuur. 'n Sig-sel, waarrnee hierdie hoe druk data gemeet kan word, is ontwerp en ontwikkel. Die volgende binere fase ewewigte is in die temperatuur gebied 313 - 367 K gemeet: superkritiese CO2 met n-CI2, n-CI6, n-C20, n-C24, n-C28 en n-C36, en superkritiese Etaan met n-CI6, n-C24 en n-C28. Daar is gevind dat hierdie twee superkritiese oplosmiddelsisteme verskillende tipes fase-ewewigsgedragte openbaar. Die n-alkane het 'n baie boer oplosbaarheid in Etaan, maar deur superkritiese C02 in 'n ekstraksie kolom te gebruik, sal tot beheer oor die selektiwiteit van die ekstraksieproses lei. Uit die verskillende toestandsvergelykings wat ondersoek is, is daar gevind dat die Patel- Teja vergelyking die CO2 sisteme die beste kon beskryf en dat die Soave-Redlich-Kwong vergelyking die beste vergelyking was om die Etaan sisteme mee te modelleer. Beide die toestandsvergelykings se interaksie parameters het 'n funksionele verband met temperatuur en die n-alkaan asentrise faktor getoon. Dit is dus moontlik om waardes vir die parameters vir sisteme by ander temperature en met ander n-alkaan tipes te bepaal. Daar was gevind dat die bestaande toestandsvergelykings nie die die fase-ewewigte oor die hele eksperimenele gebied akkuraat kon voorstel nie. Dit kan toegeskryf word aan foute wat inherent is aan die vergelykings.

Phase equilibria are at the heart of many properties of substances, such as their solubility, manufacturability, and stability. They are of significant industrial and commercial interest, perhaps most importantly to the pharmaceutical industry where drug stability and solubility are two of the largest challenges of drug development. The focus of this thesis then was to develop a molecular level understanding of phase equilibria, and produce tools and models to predict phase stability. An emphasis was given to exploring solid-solid and solid-liquid equilibria and stability. Specifically, the work presented here aimed to elucidate what drives the formation of multicomponent crystals, improve available models for exploring phase equilibria phenomena and explore solubility prediction from first principles as a potentially more powerful alternative to correlation based methods. These three fundamental areas were explored by employing molecular simulation in combination with the machinery of statistical mechanics, utilising advanced sampling methods and free energy calculations. This approach has led to the development of a foundation for understanding multicomponent crystal formation in terms of molecular affinities and packing, the characterisation of a set of soft coarse-grained potentials for use in phase equilibria studies, which overcome the main limitations of the most widely used potential, and finally, the development of a novel method for solubility prediction from first principles. Here, this novel method was successfully applied to an ionic (aqueous sodium chloride) and small molecular (urea in methanol and aqueous urea) system. In the future, these results are expected to lead to a set of guidelines for predicting (and perhaps prohibiting) multicomponent crystal formation, the development of a higher class of coarse-grained transferable force field with utility in studying phase equilibria, and powerful approach for predicting solubility of even large, flexible molecules (such as pharmaceuticals).

The main objectives of this work are the study of solid-liquid equilibrium of salts in pure and mixed solvents, and of biomolecules, such as amino acids and peptides, in water. The correlation and prediction of properties for mixtures containing charged electric species, the electrolytes, is of great relevance for the chemical industry. A brief discussion about the whole interest of this work and the need of concentrating efforts to develop accurate models for electrolyte systems is initially focused. The fundamental concepts of electrolyte thermodynamics and industrial examples where electrolytes play an important role are given. The available different models to correlate and/or predict properties and phase equilibria for this kind of mixtures are reviewed and compared. An isothermal analytical method, which has been implemented to measure salt solubilities, is described in detail. The experimental solubilities obtained for NaCl, KCl, NaBr and KBr, in the pure solvents water, methanol, ethanol and in the mixed solvents water/methanol, water/ethanol and methanol/ethanol in the temperature range between 25 oC and 80 oC are given. The new experimental data is used together with additional information published by other authors, concerning solid-liquid equilibrium of salts in pure and mixed solvents and osmotic coefficients in pure solvents, in order to establish an extensive and reliable database. This is adopted for the development of consistent thermodynamic models. Two UNIQUAC based models are suggested: the UNIQUAC + Pitzer-Debye-Hückel model, and the UNIQUAC model with linear temperature dependent solvent/salt parameters. A new developed approach for correlating salt solubilities, based on the symmetric convention of normalization of the activity coefficients and on the mole fraction concentration scale on ionized basis is presented. In this way, it is possible the direct access to the salt solubility product in terms of its calorimetric properties such as the melting temperature, enthalpy of fusion and heat capacity change. The capabilities of these models for the correlation and prediction of solid-liquid equilibrium and other thermodynamic properties are discussed. The results indicate that this procedure and these models are satisfactory for solid-liquid equilibrium calculations. The work on weak electrolytes consists of the development of a new group-contribution method. This includes two terms: the UNIFAC model to account for the short-range interaction forces, and a Debye-Hückel expression for the long-range forces. New UNIFAC groups have been assigned to describe the amino acids and peptides studied, and the chemical equilibrium is taken into account simultaneously with the physical equilibrium. Using this approach, the temperature and pH effects on the solubilities of amino acids in aqueous solutions are taken into consideration. This model predicts very successfully the pH influence on the solubilities of amino acids and therefore may be used for engineering purposes.

Bibliography: pages vi-viii.
Interest rates are fundamental in the explanation of equilibrium prices over time, because they provide the link between the present and the future. Capturing this dynamic feature, the overlapping generations model is particularly suitable to address the interest rate problem, as has been shown by Paul Samuelson, David Gale and Costas Azariadis. This thesis reviews their contribution to the theory of interest: with his consumption-loan model, Samuelson sets the analytical framework for subsequent research. Furthermore, he demonstrates that the optimal interest rate is unstable, implying that a competitive economy may fail to approach the social optimum. The Samuelson and classical sets of assumptions are consolidated in the intertemporal exchange model of Gale. Its equilibrium nature, however, ignores the sequential adjustment of disequilibrium interest rates to their equilibrium values. Consequently it is difficult to comment on the direction of causality involved in the interest rate determination, unless a clearing house is introduced which simultaneously resolves the starting-up, continuity and causality problems. Departing from the full certainty scenario, Azariadis analyses the existence and likelihood of self-fulfilling prophecies. It is shown that the implications of the economy's assumed Markovian structure are twofold: while facilitating the parametric treatment of the transition probabilities, it negates the question concerning the likelihood of sunspot equilibria. Within the specified framework it is impossible to explain how the economy arrives at such equilibria; it is only possible to identify the conditions that maintain (once they exist) these self-fulfilling prophecies.

In this work, we investigate numerically the evolution of perturbed Vlasov equilibria. according to the full nonlinear system with particular emphasis on analyzing the asymptotic states towards which the system evolves. The simulations are carried out with the numerical code that we have implemented on the Cray X-MP of the Pittsburgh Supercomputing Center and which is based on the splitting scheme algorithm. Maxwellian symmetric and one-sided bump-on-tail and two-stream type of equilibrium distributions are considered: the only distribution which seems to evolve towards a BGK equilibrium is the two-stream while the asymptotic states for the other distributions are better described by superpositions of possible BGK modes. Perturbations with wave-like dependence in space and both symmetric and non-symmetric dependence on velocity are considered. For weakly unstable modes, the problem of the discrepancy between different theoretical models about the scaling of the saturation amplitude with the growth rate is addressed for the first time with the splitting scheme algorithm. The results are in agreement with the ones obtained in the past with less accurate algorithms and do not exhibit spurious numerical effects present in those. Finally, collisions are included in the splitting scheme in the form of the Krook model and some simulations are performed whose results are in agreement with existing theoretical models.
Ph. D.

Dissertation research involves development of Mobile Order Theory thermodynamic models to mathematically describe and predict the solubility, spectral properties, protonation equilibrium constants and two-phase partitioning behavior of solutes dissolved in binary solvent mixtures of analytical importance. Information gained provide a better understanding of solute-solvent and solvent-solvent interactions at the molecular level, which will facilitate the development of better chemical separation methods based upon both gas-liquid and high-performance liquid chromatography, and better analysis methods based upon complexiometric and spectroscopic methods. Dissertation research emphasizes chemical equilibria in systems containing alcohol cosolvents with the understanding that knowledge gained will be transferable to more environmentally friendly aqueous-organic solvent mixtures.

Thesis (Ph. D.)--University of Oregon, 2000.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 374-384). Also available for download via the World Wide Web; free to University of Oregon users.

This thesis studies the application of geometric concepts and methods in the analysis of strategic-form games, in particular bimatrix games. Our focus is on three geometric concepts: the index, geometric algorithms for the computation of Nash equilibria, and polytopes. The contribution of this thesis consists of three parts. First, we present an algorithm for the computation of the index in degenerate bimatrix games. For this, we define a new concept, the “lex-index” of an extreme equilibrium, which is an extension of the standard index. The index of an equilibrium component is easily computable as the sum of the lex-indices of all extreme equilibria of that component. Second, we give several new results on the linear tracing procedure, and its bimatrix game implementation, the van den Elzen-Talman (ET) algorithm. We compare the ET algorithm to two other algorithms: On the one hand, we show that the Lemke-Howson algorithm, the classic method for equilibrium computation in bimatrix games, and the ET algorithm differ substantially. On the other hand, we prove that the ET algorithm, or more generally, the linear tracing procedure, is a special case of the global Newton method, a geometric algorithm for the computation of equilibria in strategic-form games. As the main result of this part of the thesis, we show that there is a generic class of bimatrix games in which an equilibrium of positive index is not traceable by the ET algorithm. This result answers an open question regarding sustainability. The last part of this thesis studies the index in symmetric games. We use a construction of polytopes to prove a new result on the symmetric index: A symmetric equilibrium has symmetric index +1 if and only if it is “potentially unique”, in the sense that there is an extended symmetric game, with additional strategies for the players, where the given symmetric equilibrium is unique.

The objective of this research is to develop the framework of empirical-evidence equilibria (EEEs) in stochastic games. This framework was developed while attempting to design decentralized controllers using learning in stochastic games. The overarching goal is to enable a set of agents to control a dynamical system in a decentralized fashion. To do so, the agents play a stochastic game crafted such that its equilibria are decentralized controllers for the dynamical system. Unfortunately, there exists no algorithm to compute equilibria in stochastic games. One explanation for this lack of results is the full-rationality requirement of game theory. In the case of stochastic games, full rationality imposes that two requirements be met at equilibrium. First, each agent has a perfect model of the game and of its opponents strategies. Second, each agent plays an optimal strategy for the POMDP induced by its opponents strategies. Both requirements are unrealistic. An agent cannot know the strategies of its opponents; it can only observe the combined effect of its own strategy interacting with its opponents. Furthermore, POMDPs are intractable; an agent cannot compute an optimal strategy in a reasonable time. In addition to these two requirements, engineered agents cannot carry perfect analytical reasoning and have limited memory; they naturally exhibit bounded rationality. In this research, bounded rationality is not seen as a limitation and is instead used to relax the two requirements. In the EEE framework, agents formulate low-order empirical models of observed quantities called mockups. Mockups have unmodeled states and dynamic effects, but they are statistically consistent; the empirical evidence observed by an agent does not contradict its mockup. Each agent uses its mockup to derive an optimal strategy. 1Since agents are interconnected through the system, these mockups are sensitive to the specific strategies employed by other agents. In an EEE, the two requirements are weakened. First, each agent has a consistent mockup of the game and the strategies of its opponents. Second, each agent plays an optimal strategy for the MDP induced by its mockup. The main contribution of this dissertation is the use of modeling to study stochastic games. This approach, while common in engineering, had not been applied to stochastic games.

This thesis concerns the computational problem of finding one Nash equilibrium of a bimatrix game, a two-player game in strategic form. Bimatrix games are among the most basic models in non-cooperative game theory, and finding a Nash equilibrium is important for their analysis. The Lemke—Howson algorithm is the classical method for finding one Nash equilib-rium of a bimatrix game. In this thesis, we present a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labelled facets in d-space. The construc-tion is extended to two classes of non-square games where, in addition to exponentially long Lemke—Howson computations, finding an equilibrium by support enumeration takes exponential time on average. The Lemke—Howson algorithm, which is a complementary pivoting algorithm, finds at least one solution to the linear complementarity problem (LCP) derived from a bimatrix game. A closely related complementary pivoting algorithm by Lemke solves more general LCPs. A unified view of these two algorithms is presented, for the first time, as far as we know. Furthermore, we present an extension of the standard version of Lemke's algorithm that allows one more freedom than before when starting the algorithm.

We consider the concept of coarse correlated equilibrium (CCE) in various contexts; games with quadratic payoff functions (which include Cournot duopoly, public good provision and emission abatement) and a linear duopoly game. For the games with quadratic payoffs we compute the largest feasible total utility in any CCE and show that it is achieved by a CCE involving only two strategy profiles. The improvement over and above the Nash equilibrium payoff is substantial in the various economic examples considered for this class of games. In case of the linear duopoly game, we prove that Nash-centric devices, involving a sunspot structure, are simple symmetric CCE, and any unilateral perturbation from such a structure fails to be an equilibrium.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Economics, 1985.
MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE.
Vita.
Bibliography: leaves 187-189.
by Jonathan Lewis Burke.
Ph.D.

Bibliography: pages 167-169.
The computation of chemical and phase equilibria is an essential aspect of chemical engineering design and development. Important applications range from flash calculations to distillation and pyrometallurgy. Despite the firm theoretical foundations on which the theory of chemical equilibrium is based there are two major difficulties that prevent the equilibrium state from being accurately determined. The first of these hindrances is the inaccuracy or total absence of pertinent thermodynamic data. The second is the complexity of the required calculation. It is the latter consideration which is the sole concern of this dissertation.

Doutoramento em Engenharia Química
Em consequência de uma série de problemas ambientais, económicos e políticos relacionados com o uso de combustíveis convencionais, vários países estão agora a focar as suas atenções em combustíveis alternativos. O biodiesel está na linha da frente das alternativas ao petróleo no sector dos transportes, sendo considerado uma opção a curto prazo visto que o seu preço é competitivo e não são necessárias mudanças nos motores para implementar o seu uso. De entres os possíveis processos de produzir biodiesel, a reacção de transesterificação com catálise básica é o método preferido. Depois da reacção são sempre necessários processos de purificação de modo ao biodiesel produzido cumprir os standards definidos para os combustíveis alternativos, reduzindo problemas de motor e consequentemente aumentando a sua aceitação por parte dos consumidores. De entre as especificações encontram-se o conteúdo em água, em álcool e em glicerol. Ser-se capaz de descrever correctamente o equilíbrio de fases de sistemas que são de interesse para os processos de purificação de biodieseis numa gama alargada de condições termodinâmicas é uma condição necessária para uma correcta simulação do processo industrial, de modo a se atingir uma elevada produtividade a baixos custos de operação. O uso de moléculas oxigenadas como combustíveis representa uma alteração significativa em termos da termodinâmica de soluções. Para combustíveis baseados em petróleo as equações de estado cúbicas e os modelos clássicos de coeficientes de actividade mostraram ser apropriados, no entanto para combustíveis novos como o biodiesel, sendo mais complexos do ponto de vista das interacções intermoleculares com formação de dipolos e pontes de hidrogénio, são necessários modelos termodinâmicos mais complexos para descrever essas interacções. Neste trabalho a CPA EoS (Cubic-Plus-Association Equation of State) será desenvolvida de modo a permitir uma descrição adequada dos equilíbrios líquido-vapor e líquido-líquido para uma serie de sistemas binários e multicomponentes contendo água, ácidos gordos, ésteres de ácidos gordos, glicerol e álcoois.
As a consequence of a range of environmental, economical and political problems related to the use of conventional petroleum based fuels, several countries are now focusing their attention on alternative fuels. Biodiesel is at the forefront of the alternatives to petroleum based fuels in the transportation sector, being considered an important short-time option since its price can be competitive with conventional diesel and no motor changes are required. Among the various approaches to produce biodiesel, basic catalyzed transesterification is the preferable method. After the transesterification reaction purification steps are always necessary in order to provide the fuel with the quality levels required by the standards for alternative fuels, reducing engine problems and consequently increasing consumers’ acceptance. Among the specifying minimums are the water, the alcohol and the glycerol contents. Being able to correctly describe the phase equilibria of systems of interest for the biodiesel purification processes in a broad range of thermodynamic conditions is a necessary condition for a correct simulation of the industrial process, in order to achieve high productivity and low operating costs. The use of oxygenated molecules as fuels represents a significant change in terms of solution thermodynamics. While for petroleum-based fuels, cubic equations of state and classic activity coefficient models have proved to be appropriate, new fuels, such as biodiesel, are more complex from the point of view of intermolecular interactions with dipoles and hydrogen bonding being important on these systems. To be able to represent such interactions more complex engineering thermodynamics models are required. In this work the CPA EoS (Cubic – Plus – Association Equation of State) will be developed to provide an adequate description of the vapor-liquid and liquidliquid phase equilibria of several binary and multicomponent systems containing water, fatty acids, fatty acid esters, glycerol and alcohols.

Mestrado em Medição de Equilíbrios Gás Líquido a Altas Pressões
Este trabalho tem como objectivo a medição da solubilidade de gases em líquidos a altas pressões. Para a realização experimental foi usada uma célula de altas pressões de volume variável e um método de observação directa para a detecção das transições de fase. As condições usadas neste trabalho experimental foram temperaturas entre 276.00 e 373.15 K e pressões ate 100 MPa. Numa primeira fase foram efectuadas medições de solubilidade de metano em anilina pura e para uma solução aquosa de composição mássica igual a 5% em anilina. A anilina é fundamentalmente usada em processos como a produção de borracha, de poliuretanos, pigmentos e tintas, fármacos, herbicidas e fungicidas. É usualmente produzida através de processos de redução do nitrobenzeno por reacção com o HCl. De facto, esta síntese é um processo químico complexo onde um grande número de processos compete entre si. Alterações nas condições do processo com a possível formação de produtos intermediários podem afectar a eficiência do processo. Para a sua melhoria foi sugerido que o hidrogénio usado fosse simultaneamente produzido e gasto no reactor principal. Neste caso e para a produção do mesmo, era necessário adicionar metano e água ao reactor. Tendo por base a ideia de que as reacções onde o hidrogénio é reagente e produto ocorrem em simultâneo, torna-se clara a importância do estudo da solubilidade do metano em anilina pura e em soluções aquosas desta. Numa segunda fase foi estudada a solubilidade do dióxido de carbono em soluções aquosas de tri-iso-butil(metil)fosfónio tosilato, com composições molares são de 4, 8 e 12% em líquido iónico. Este pertencente à família dos fosfónios. Possui uma viscosidade e densidade elevadas, é térmica e quimicamente estável e ainda possui uma elevada polaridade. Apresenta uma miscibilidade completa em água e nos solventes mais usuais, como o diclorometano e tolueno, não sendo no entanto míscivel em hexano. O tri-isobutil( metil)fosfónium tosilato é usado como solvente nos processos de hidroformilação de olefinas e ainda em processos de captura e conversão de dióxido de carbono. Neste trabalho experimental, a temperatura e a pressão foram inicialmente aumentadas até o sistema atingir o equilíbrio. A pressão é diminuída lentamente até se verificar o aparecimento/desaparecimento da última bolha de gás. A pressão à qual a última bolha de gás desaparece representa a pressão de equilíbrio para aquela temperatura. Este procedimento foi efectuado para vários sistemas e várias temperaturas.
The aim of this work is the measurement of the solubilities of different gases in different liquids systems at high pressures. A variable volume high pressure cell was used to perform the measurements based on the visual synthetic method for the detection of phase transitions. The conditions used in this work were temperatures ranging from 276.00 to 373.15 K and pressures up to 100 MPa. The first step of this work was the measurement of methane’s solubility in pure aniline and for one aqueous solution of 5% (w/w) in aniline. The main applications of aniline are the manufacture of polyurethane, rubber, dyes and pigments, drugs and agrochemicals (herbicides and fungicides). This compound is normally produced by reduction of nitrobenzene by reaction with HCl. In fact, that synthesis is a complex chemical process where a number of competing processes contribute to the final product. Changes in process conditions with the possible formation of intermediates can act as catalyst poisons that change the process’s efficiency. For the improvement of the production process it was suggested that the hydrogen, used for the reduction of the nitrobenzene, could be simultaneously generated and used in the main reactor. In this case for the in situ production of hydrogen it would be necessary the addition of methane and water in the reactor. Having in mind that these reactions occur simultaneously, it becomes clear the importance of studying the solubility of methane in aniline and in its aqueous solutions. In the second step the solubility of carbon dioxide in aqueous solutions of triiso- butyl(methyl)phosphonium tosylate with molar compositions of 4, 8 and 12% molar of ionic liquid, was studied. This compound belongs to the phosphonium - based ionic liquids family. It is characterized by a high viscosity and density. It is thermal and chemically stable and has a high polarity. This compound is totally miscible with water as well as with common organic solvents such as dicloromethane and toluene but is not miscible with hexane. The main applications of Tri-iso-butyl(methyl)phosphonium tosylate are like solvents in the hydroformylation of olefins and in processes of capture and conversion of carbon dioxide. In this experimental work, temperature and pressure were slowly increased until the system becomes homogeneous. After that, the pressure is slowly decreased until the appearance/disappearance of the last bubble of gas. The pressure at witch the last bubble disappears represents the equilibrium pressure for that temperature. This procedure was repeated for several systems and several temperatures.

The Lyapunov function method is used in proving stability, asymptotic or globally asymptotic stability of discrete dynamic systems. We show that the slightly relaxed versions of the well known sufficient conditions are also necessary. The stability of the equilibria of time-invariant nonlinear dynamical systems with discrete time scale is investigated. We present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle, then the equilibrium maybe unstable, or stable, or even asymptotically stable, which show that the necessary condition cannot be further restricted in general. In addition, the necessary condition is given in terms of spectral radius and matrix norms. The asymptotic stability of equilibria in a number of discrete dynamic oligopolies is analyzed. First the equivalence of the equilibrium problem of a large class of nonlinear games and the equilibrium problem of a class of discrete dynamic systems is verified. Stability conditions are then derived for a certain class of dynamic models, and these results are finally applied to single-product oligopolies, multiproduct oligopolies, and labor-managed oligopolies. The economic interpretation of the stability conditions are also presented. The stability properties of a special class of homogeneous dynamic economic systems are examined. The nonlinearity of the models and the presence of eigenvalues with zero real parts in a normally hyperbolic invariant set make the application of the classical theory impossible. Some principles of the modern theory of dynamical systems and invariant manifolds are applied. The local and global strong attractivity of the set of equilibria is verified under mild conditions. As an application, special labor-managed oligopolies are investigated.

Constructive mathematics is mathematics with intuitionistic logic (together with some appropriate, predicative, foundation)-it is often crudely characterised as mathematics without the law of excluded middle. The intuitionistic interpretation of the connectives and quantifiers ensure that constructive proofs contain an inherent algorithm which realises the computational content of the result it proves, and, in contrast to results from computable mathematics, these inherent algorithms come with fixed rates of convergence. The value of a constructive proof lies in the vast array of models for constructive mathematics. Realizability models and the interpretation of constructive ZF set theory into Martin Löf type theory allows one to view constructive mathematics as a high level programing language, and programs have been extracted and implemented from constructive proofs. Other models, including topological forcing models, of constructive set theory can be used to prove metamathematical results, for example, guaranteeing the (local) continuity of functions or algorithms. In this thesis we have highlighted any use of choice principles, and those results which do not require any choice, in particular, are valid in any topos. This thesis looks at what can and cannot be done in the study of the fundamental fixed point theorems from analysis, and gives some applications to mathematical economics where value is given to computability.

The main focus of this thesis is the dynamics of a two-degree-of-freedom Hamiltonian system near an elliptic equilibrium point in 1 : ±2 resonance, as described by its integrable resonant normal form approximation. After applying singular reduction, the system is studied on the reduced phase space that Kummer called the "orbit manifold". We give a complete description of the critical values of the energy–momentum mapping which than enables us to study the topology of the regular, and more importantly, the singular fibres. We then derive equations for the period of the reduced system, for the rotation number and the non-trivial action. Although some of these results are not new, our approach does not rely on compact fibration and is based on a similar approach introduced earlier by S. Vũ Ngoc for focus–focus points. The non-trivial action of the system enables us to establish fractional monodromy very elegantly by deriving the transition matrix for the actions directly. Both the isoenergetic non-degeneracy condition and the Kolmogorov non-degeneracy condition of KAM theory are derived and analysed for the resonant case. It tuns out that the twist vanishes in a neighbourhood of the equilibrium point for the sign-indefinite case. The Kolmogorov condition, however, is always satisfied near 1 : ±2 resonant equilibria.

This thesis is dedicated to the exploration of fluid phases confined in nanoporous materials using Nuclear Magnetic Resonance (NMR) techniques, with an aim to benefit catalysis research. Included in this report are studies of pure fluids and their mixtures, confined in titania and silica catalyst supports. These investigations are conducted at industrially-relevant, high-temperature (≥ 180 °C) and high-pressure conditions (up to 13 bar), made possible by a pilot-scale chemical reactor unit, designed to operate inside the strong magnetic fields of an NMR spectrometer. NMR spectroscopy, relaxation and pulsed field gradient (PFG) diffusion experiments were performed on each of the systems discussed in this report. Cyclohexane was initially studied inside a porous titania catalyst support at 188 °C and various pressures up to 13 bar. The adsorption and desorption processes of the cyclohexane were observed, revealing a number of previously unobserved phenomena. In addition to an overall, averaged diffusion coefficient, a slow diffusion coefficient was observed within the PFG NMR data attributable to surface diffusive processes occurring within the material. Additionally, T1 relaxation studies were found to provide experimental evidence for the differing configurations of adsorbed layers on the adsorption and desorption branch of the isotherm. Cyclohexane was subsequently studied alongside fluorobenzene in a series of silica catalyst supports of 6 nm, 10 nm and 20 nm pore size. In doing this, it was hoped that the multiple phenomena observed in the titania experiments might be deconvoluted, allowing a greater level of insight. The diffusivities of the fluids were found to differ significantly between the materials, and greater evidence was found of the slow-diffusing surface phase in each of the materials. Additionally, concentrations of cyclohexane and fluorobenzene in the gas and adsorbed layers inside the pore space were calculated via the results of the PFG NMR experiments, providing a map of confined phase behaviour. Competitive adsorption effects were found to become more significant, the smaller the pore size of the material. The results of the cyclohexane and fluorobenzene in silica studies were modelled, using approaches available in the literature, which were found to give varying levels of prediction. The data set acquired in this thesis was found to provide a useful standard, against which current and future models of confined phase behaviour might be verified.

Esta tesis consiste en cuatro trabajos independientes. En amplios términos los dos temas generales que trata se pueden clasificar como fusiones horizontales y comparaciones de los parámetros del mercado del equilibrio entre competencia en precios y cantidades.
En el articulo, "Fusiones con funciones de oferta", analizo los efectos de una fusión en una industria que produce un bien homogéneo cuando las empresas compiten eligien funciones de oferta. Las empresas eligen las funciones que relacionan su oferta con el precio de mercado. La competencia a la Cournot donde las empresas eligen a las cantidades, y la competencia a la Bertrand donde las empresas eligen los precios son casos especiales de la competencia en funciones de oferta. Considero una industria con un nivel fijo de capital. El capital una empresa define su tecnología de producción. Una fusión crea no solamente una empresa más grande, pero también una con una estructura más eficiente del coste que cualesquiera de sus componentes. Se muestra que cualquier fusión hace que todas las empresas que bajen su producción. La disminución de la producción por las empresas no participantes hace cualquier fusión rentable. Cualquier fusión en una industria con el capital social igualmente distribuido conduce a una pérdida del bienestar. Finalmente, una fusión es más probable que aumente el bienestar si aumenta la "simetría" en la industria.
En mi trabajo "Fusiones bajo la incertidumbre", considero las decisiones de fusionar de empresas que producen bienes diferenciados en un mercado con choques idiosincrásicos de la demanda. Las empresas toman sus decisiones de fusionar antes de que choques afecten a la industria. Todas las empresas comparten la misma información sobre las condiciones de mercado en cualquier momento. En particular, después de un choque, todas las empresas saben todos los parámetros relevantes del mercado. Así, una fusión no produce ninguna ventaja informativa. Exploro el modelo bajo dos modos de competencia. Comparando los incentivos para fusionar con los incentivos en el caso de un mercado determinista, encuentro que la característica incierta en el modelo aumentan la rentabilidad de una fusión bajo competencia de precio, mientras que con competencia en cantidades, su efecto sobre la rentabilidad de la fusión es ambiguo. Sin embargo, si hay incentivos a fusionar en el caso determinista, después de la incertidumbre, aumenta la rentabilidad de la fusión.
En mi trabajo "Comparando los precios de Bertrand y de Cournot: Un caso de substitutos y de complementos " confirma la sabiduría común según la cual la competencia del precio es más competitiva que la competencia de la cantidad. Resultados anteriores que apoyan esta visión en mercados de productos substitutos o en mercados de productos complementarios. Considero el caso cuando los complementos y los substitutos coexisten en un mercado de productos diferenciado.
El artículo "Innovación en un modelo asimétrico: Comparando los equilibrios de Cournot y de Bertrand ". Este artículo compara los resultados y la eficiencia dinámica de los equilibrios de Cournot y de Bertrand en un duopolio diferenciado, donde solamente una empresa puede reducir el coste marginal antes de la competencia en el mercado de producto. Demostramos que, con alta sustitución y costes bajos de innovación: - los niveles de I&D pueden ser más altos bajo competencia de precio si la difusión es baja; - la producción, el exceso del consumidor y el bienestar total pueden ser más grandes bajo Cournot si la difusión es alta. Un nuevo resultado de este artículo es que con la innovación de proceso, los consumidores y los productores pueden encontrarse en mejor situación bajo competencia de cantidad. Proporcionamos ejemplos numéricos bien definidos que demuestran que estos resultados no dependen de la asimetría del modelo.
This dissertation consists of four pieces of independent work. In broad terms two general themes that it addresses can be classified as horizontal mergers and comparison of equilibrium market parameters in Cournot and Bertrand competition.
In my paper, "Mergers with supply functions", I analyze the equilibrium effects of a merger in a homogenous good industry when firms compete by choosing supply schedules. Firms choose functions that relate their supply to the market price. Cournot competition where firms commit to quantities, and Bertrand competition where firms commit to prices are special cases of supply function competition - they exogenously impose horizontal and vertical supply schedules on the firms. I consider an industry with a fixed capital stock. A merger creates not only a larger firm, but also one with more efficient cost structure than any of its constituents. I find that any merger results in all firms lowering supply. The decrease in supply by non-participating firms makes any merger profitable. This differs from the effects of mergers under Cournot competition where the non-participating firms expand their supply, reducing the profitability of a merger. Any merger in an industry with equally distributed capital stock leads to a welfare loss. Finally, a merger is more likely to be welfare enhancing if it increases the "symmetry" in the industry.
In my paper, titled "Mergers under uncertainty", I consider the merger decisions of firms producing differentiated products in a market with idiosyncratic demand shocks. Firms make their merger decisions taking into account that future shocks will hit the industry. All firms share the same information about the market conditions at any time. In particular, after the arrival of a shock all firms know all the relevant market parameters. Thus, a merger does not produce any informational advantage. I explore the model under two competition modes. By comparing the incentives for merger with the incentives in a benchmark case of a deterministic market, I find that the uncertain feature in the model increases the attractiveness of a merger under price competition, while with Cournot competition its effect on merger profitability is ambiguous. However, if there are incentives to merge in the deterministic case, then uncertainty increases the profitability of the merger. .
In my short paper, "Comparing Bertrand and Cournot prices: A case of substitutes and complements", I reasses the common wisdom according to which price competition is more competitive than quantity competition. Previous results supporting this view deal with either substitute products markets or symmetric complement products markets. I consider the case when complements and substitutes co-exist in a differentiated products market. I introduce a set of symmetry criteria for this market, and show that when they are fulfilled, there exists a symmetric Bertrand equilibrium with a lower price than the price resulting from any symmetric Cournot equilibrium.
The last piece of work in the thesis, "Innovation in an Asymmetric Setting: Comparing Cournot and Bertrand Equilibria", is a joint work with Ioana Chioveanu. This paper compares the outcomes and the dynamic efficiency of Cournot and Bertrand equilibria in a differentiated duopoly with substitute goods, where only one firm can reduce marginal cost before product market competition. We show that, with high substitutability and low innovation costs:
-R&D levels can be higher under price competition if spillovers are low;
-output, consumer surplus and total welfare can be larger under Cournot if spillovers are high.
A new result of this article is that with process innovation, both consumers and producers can be better off under quantity competition. We provide well-defined numeric examples showing that these findings do not depend on the asymmetry of the model. The fact that innovation levels may be higher in Bertrand competition is consistent with the findings of Bester and Petrakis (1993). We identify a parameter equivalence that extends all their results from process to product innovation: In particular, incentives to improve quality may be larger under price competition, if substitutability is high.