Both of the two major components of Game Theory, e.g., the non-cooperative game theory and the cooperative game theory, are becoming more and more closely related to the field of optimization, as the needs to study the analytical properties of games start to rise. The results presented in this thesis illustrate several connections between Optimization and Game Theory, and attempts are made to build a bridge between the cooperative game theory and the non-cooperative game theory, to characterize the co-existence of competition and cooperation in practice. We start by applying the properties of Polymatroid Optimization to the cooperative game theory, and show that both of the joint replenish game and the one warehouse multi retailer game are submodular games. In the next part, we show that the strategies promoting learning from history are convergent under certain conditions. This result can also be viewed as an efficient algorithm to compute the Nash Equilibrium of the game. Because the competitive routing game satisfies the condition, we know that if every user adapts with good enough memory, then asymptotically the system converges to Nash Equilibrium. Therefore, if the decision of cooperation is difficult to reverse, then it can be justified for the farsighted players to use the cost structure in the Nash Equilibrium point to decide if they should cooperate or not, instead of reacting to the immediate consequences as a basis to make decisions. With the optimization tools applied, we are able to show that in parallel network, the social cost and the cost of other players tend to decrease if two players cooperate. Also, the price of anarchy is higher when the flow demand of players are more evenly distributed. Using that structural result, we derive the exact upper bound of the price of anarchy for a given parallel network with fixed number of players. The exact upper bound of the price of anarchy for arbitrary parallel network with given number of players, which is independent to the network structure and parameters, can be derived consequently. / Simai He. / Adviser: Shuzhong Zhang. / Source: Dissertation Abstracts International, Volume: 72-11, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 97-103). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

In order to solve the NBM Nash formulates an optimization problem. The unique solution of this problem is the famous Nash Bargaining Solution (NBS). We revise this optimization problem and supplement some new selection criteria of profit allocation to develop some bargaining mechanisms appropriate for the two situations of our model respectively. Each bargaining mechanism offers the alliance one, two, or several reasonable profit distribution(s) which can be selected by these two parties. Subsequently for any situation we propose some novel dynamic programming algorithms with respect to several specific utility function structures involving job schedules respectively to implement those relevant mechanisms in pseudo-polynomial time. / In this dissertation we investigate a new cooperative game model, where two parties comprise an alliance to process a number of jobs offered by a customer and bargain about a reasonable processing profit distribution determined by a two-partition of these jobs. / In this model the non-negative integer-valued parameters of each job, which are the basic assumptions in traditional discrete scheduling models, are still adopted. We also assume each job is non-preemptive. Any party's utility function of the two-partition of these jobs does not possess any elegant continuous or concave property which is critical for the original Nash Bargaining Model (NBM), and furthermore we are only concerned with the integer-valued utility function. Consequently these assumptions result in a new discrete variation of the NBM. In this dissertation we highlight an important special case of our model, where after a two-partition of these jobs is given, each party's utility of processing the jobs assigned to him is related with an optimal schedule of these jobs which minimizes a cost (penalty) function. / This new model is motivated at least by the following real world phenomenon: after two manufacturers have jointly contracted with a customer for processing a number of jobs owing to their insufficient operation facilities, these two par ties need to negotiate a two-partition of these jobs to obtain a profit distribution acceptable for each one. In this dissertation we consider two situations of this model. In the first situation these two parties basically possess the same bargaining power. In the second situation one party possesses the greater bargaining power and can design some bargaining mechanisms more beneficial for himself attributed to his more operation techniques or facilities than the other's. / Chen, Quanle. / "November 2006." / Adviser: Xiaogiang Cai. / Source: Dissertation Abstracts International, Volume: 68-08, Section: B, page: 5489. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (p. 150-153). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

by Jan, Ting-Wai and Ng, To-Tung. / Thesis (M.B.A.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 66-67). / ABSTRACT --- p.i / TABLE OF CONTENTS --- p.ii / LIST OF FIGURES --- p.iv / LIST OF TABLES --- p.v / CHAPTER / Chapter I --- INTRODUCTION --- p.1 / Objective --- p.1 / The Industrial Gases Industry --- p.1 / The Products --- p.2 / Competition within an Industrial Park --- p.3 / A SCENARIO IN THE INDUSTRY --- p.3 / Chapter II --- DESIGN OF THE GAME --- p.4 / The Environment --- p.4 / The Time Line --- p.5 / The Game in Extensive Form --- p.7 / Chapter III --- EVALUATION OF THE EXPECTED PAYOFFS --- p.10 / Payoff criterion: Net Present Value Vs Internal Rate of Return --- p.10 / Calculation of Payoffs at End Nodes --- p.12 / Cash Flow Assumptions --- p.13 / The Product Selling Price --- p.13 / Utilities Consumption in Relation to Plant Utilization --- p.14 / Chapter IV --- ANALYSIS OF THE GAME --- p.16 / The First Iteration --- p.16 / The Right Side of the Tree --- p.17 / The Left Side of the Tree --- p.18 / Payoffs of Portion A (Investment Options) --- p.19 / If Firm F2 sells at Year -1 --- p.20 / Is It a Fair Deal for Firm F1 at Node 4? --- p.21 / Can Firm F2 Increase Its Payoff at Year -1 ? --- p.22 / Determine a New Product Selling Price Between Firms F1 and F2 --- p.24 / Revised Expected Payoffs for Nodes 6 to 9 at Year-1 --- p.25 / The second Iteration (Final Outcome) --- p.27 / Risk if Firms Make Irrational Decisions --- p.27 / Chapter V --- INDUSTRY PRACTICE IN LONG TERM CAPACITY PLANNING --- p.29 / The Decision Tree of Firm F1 --- p.29 / The Decision Tree of Firm F2 --- p.30 / The Implications of Making Decisions Based on Individual Decision Trees --- p.31 / By Firm F1 and F2 --- p.31 / Chapter VI --- CONCLUSION --- p.33 / APPENDIX A SUMMARY OF ASSUMPTIONS IN CASH FLOW --- p.34 / APPENDIX B SUMMARY OF CASH FLOW FOR VARIOUS PAYOFFS AT FIRST ITERATION --- p.35 / APPENDIX C SUMMARY OF CASH FLOW FOR VARIOUS PAYOFFS FOR FINAL OUTCOME --- p.55 / BIBLIOGRAPHY --- p.66

Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-08-28T20:37:21Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) TCC - Carlos.pdf: 1307050 bytes, checksum: e3104a4272da62c638840e38dced1527 (MD5) / Made available in DSpace on 2014-08-28T20:37:21Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) TCC - Carlos.pdf: 1307050 bytes, checksum: e3104a4272da62c638840e38dced1527 (MD5) Previous issue date: 2014-04-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we have exposure to some elements of game theory and certain resolution procedures games using matrices, probability and especially optimization, ie, we optimize the moves with mathematical background. For this we will use a Theorem, two Propositions and discuss several examples to game theory, applying what we are working and show how one can proceed in several games so that the reader can understand and use such a theory. The objective is to disseminate the ideas of game theory, which has applications in several areas, including economy and military art. / Neste trabalho teremos a exposição de alguns elementos da Teoria dos Jogos e certos procedimentos de resolução de jogos usando matrizes, probabilidade e principalmente otimização, ou seja, vamos otimizar as jogadas com embasamento matemático. Para tal usaremos um Teorema, duas Proposi ções e vários exemplos para discorrer sobre a Teoria dos Jogos, aplicando o que estamos trabalhando e mostrar como se pode proceder em vários jogos para que o leitor possa compreender e usar tal teoria. O objetivo deste trabalho é divulgar as ideias da Teoria dos Jogos, as quais tem aplicação em várias áreas, entre elas economia e arte militar.

Chan, Chun Him = 論競爭環境中出現的協作現象 / 陳俊謙. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 139-141). / Abstracts in English and Chinese. / Chan, Chun Him = Lun jing zheng huan jing zhong chu xian de xie zuo xian xiang / Chen Junqian. / Title / Abstract --- p.i / Acknowledgements --- p.v / Table of Contents --- p.vii / Chapter 1 --- Prologue --- p.1 / Chapter 2 --- Prisoners' Dilemma and Snowdrift Game --- p.6 / Chapter 2.1 --- Introduction --- p.6 / Chapter 2.2 --- Formulation --- p.7 / Chapter 2.3 --- The Prisoners，Dilemma (PD) --- p.8 / Chapter 2.4 --- The Snowdrift Game (SG) --- p.9 / Chapter 2.5 --- Parameterizing Payoffs in PD --- p.10 / Chapter 2.5.1 --- The Temptation Reward Punishment Sucker System (TRPS system) --- p.10 / Chapter 2.5.2 --- "Our Expanded 2-parameter System (r, s system)" --- p.10 / Chapter 2.5.3 --- Our 1-parameter System (r system) --- p.11 / Chapter 2.5.4 --- The Benefit and Cost System (BC system) --- p.11 / Chapter 2.6 --- Parameterizing Payoffs in SG --- p.12 / Chapter 2.6.1 --- A Common 1-parameter System (rh system) --- p.13 / Chapter 2.6.2 --- Our 1-parameter System (r system) --- p.13 / Chapter 3 --- Iterated Prisoners' Dilemma (IPD) --- p.15 / Chapter 3.1 --- Tragedy of One-shot PD Game --- p.15 / Chapter 3.2 --- The First Cooperation --- p.16 / Chapter 3.2.1 --- Repeated PD Game --- p.16 / Chapter 3.2.2 --- Axelrod´ةs Tournament and Strategy “Tit-for-Tat´ح (TFT) --- p.17 / Chapter 3.3 --- The Second Cooperation --- p.18 / Chapter 3.3.1 --- Nowak and Sigmund Evolutionary Experiment --- p.18 / Chapter 3.3.2 --- Stochastic Reactive Strategies (SRSs) and Evolutionary Infinite IPD --- p.19 / Chapter 3.3.3 --- Experimental Setup --- p.20 / Chapter 3.3.4 --- Experimental Results --- p.21 / Chapter 3.3.5 --- Conclusion: TFT as a Pivot for Cooperation --- p.22 / Chapter 4 --- Evolutionary IPD with Strategy Lattices --- p.23 / Chapter 4.1 --- Sensitivity to Initial Conditions and Numerical Accuracy in IPD --- p.24 / Chapter 4.2 --- Modifications and Justifications --- p.24 / Chapter 4.3 --- Simulation Results --- p.26 / Chapter 4.3.1 --- Strategy Frequencies Xi(t) versus Time --- p.27 / Chapter 4.3.2 --- "Mean Final State in the Whole r, s Space" --- p.32 / Chapter 4.3.3 --- Time Evolution of (p) and (q) --- p.38 / Chapter 4.3.4 --- Mean Final State of a Strategy Lattice with Inclined Frequencies --- p.42 / Chapter 4.3.5 --- Conclusion --- p.50 / Chapter 5 --- Egoistic Exploiters Induced Global Generosity in Evolutionary IPD --- p.52 / Chapter 5.1 --- Introduction --- p.52 / Chapter 5.2 --- Modifications and Justifications --- p.52 / Chapter 5.3 --- "The Three Initial Conditions: Cases TO, T1 and T2" --- p.53 / Chapter 5.4 --- Simulation Results --- p.54 / Chapter 5.4.1 --- Distribution of the Final Dominant Strategies --- p.54 / Chapter 5.4.2 --- Case TO (100 Random Strategies) --- p.56 / Chapter 5.4.3 --- Case T1 (99 Random Strategies and a TFT-like Strategy) --- p.58 / Chapter 5.4.4 --- Case T2 (98 Random Strategies and both a TFT-like and an AllD-like Strategies) --- p.58 / Chapter 5.4.5 --- Comparing the Three Cases --- p.61 / Chapter 5.4.6 --- Discussion --- p.64 / Chapter 5.5 --- Analytic Manipulations for Small r Region --- p.65 / Chapter 5.5.1 --- Values of (pmps) and (qmps) for Case T2 --- p.65 / Chapter 5.5.2 --- Values of {pmps) and {qmps) for Case TO and T1 --- p.68 / Chapter 5.6 --- Conclusion --- p.71 / Chapter 6 --- The Basics of Networks --- p.72 / Chapter 6.1 --- Fully-connected Networks and Well-mixed Networks --- p.72 / Chapter 6.2 --- Simple Hypercubic Lattices --- p.73 / Chapter 6.3 --- Barabasi-Albert (BA) Scale-free Growing Networks --- p.74 / Chapter 7 --- Proposing the N-person Snowdrift Game (NSG) --- p.76 / Chapter 7.1 --- Introduction --- p.76 / Chapter 7.2 --- Limitations of 2-person Games on Networks --- p.76 / Chapter 7.3 --- The Existing N-person Games --- p.77 / Chapter 7.3.1 --- The Public Good Game (PGG) --- p.78 / Chapter 7.3.2 --- The N-person Battle of Sexes Games (NBOS) --- p.78 / Chapter 7.4 --- The NSG Scenario and Payoffs --- p.79 / Chapter 7.5 --- Everyday Examples of NSG --- p.80 / Chapter 7.6 --- Preview of Studies in Evolutionary NSG --- p.82 / Chapter 8 --- Evolutionary NSG in Well-mixed Populations --- p.84 / Chapter 8.1 --- The Method of Replicator Dynamics --- p.84 / Chapter 8.2 --- The Simulation Algorithm --- p.90 / Chapter 8.3 --- The Simulated Equilibrium Contributor Fraction x*(r) --- p.91 / Chapter 8.4 --- Analytic Manipulations on the Algorithm --- p.92 / Chapter 8.4.1 --- Conversion Probabilities and Equilibrium Conditions --- p.92 / Chapter 8.4.2 --- Discussions --- p.94 / Chapter 8.5 --- Analytic Treatments on the Time Evolution of the Contribution Level x(t) --- p.96 / Chapter 8.5.1 --- Discrete Time Equation --- p.97 / Chapter 8.5.2 --- Continuous Approximations --- p.97 / Chapter 8.5.3 --- Analytical Solutions of N=2 and N=S --- p.98 / Chapter 8.6 --- Conclusion --- p.100 / Chapter 9 --- Evolutionary NSG in Simple Hypercubic Lattices --- p.101 / Chapter 9.1 --- Comparison between Well-mixed and Lattice Populations --- p.101 / Chapter 9.2 --- Simulation Parameters --- p.103 / Chapter 9.3 --- Simulation Results for ID Lattices --- p.103 / Chapter 9.4 --- Analytic Theory for ID Lattices by Analyzing on the Local Configurations --- p.105 / Chapter 9.4.1 --- The Absence of Connected Contributors --- p.105 / Chapter 9.4.2 --- Theory for ID k=2 Chain Lattice --- p.106 / Chapter 9.4.3 --- Theory for ID k=4 Chain Lattice --- p.109 / Chapter 9.5 --- Simulation Results for 2D Lattices --- p.112 / Chapter 9.6 --- Analytic Theory for 2D k=4 Square Lattice --- p.115 / Chapter 9.7 --- Simulation Results for 3D Cubic Lattice --- p.120 / Chapter 9.8 --- Conjectures for Approximate Solutions in High-Dimensional Hy- percubic Lattices --- p.120 / Chapter 9.9 --- Extracting the Relation Neff=2k+1 --- p.122 / Chapter 9.10 --- Conclusion --- p.123 / Chapter A --- Evolutionary NSG in Barabasi-Albert Networks --- p.125 / Chapter A.l --- The New Elements in BA Networks --- p.126 / Chapter A.2 --- The Two Implementations - Model A and Model B --- p.126 / Chapter A.3 --- Results in Model A (Varying-N) --- p.127 / Chapter A.4 --- Results in Model B (Fixed-N) --- p.133 / Chapter A.5 --- Conclusion --- p.135 / Chapter B --- Supplementary Equations for Chapter 9 --- p.136 / Chapter B.l --- Equations for ID k=4 Lattice --- p.136 / Chapter B.2 --- Equations for 2D k=4 Lattice --- p.137 / Bibliography --- p.139

This thesis consists of three essays in economic theory, two on search models with information acquisition and one on repeated games when precise information about discount factors is unavailable. In the first essay, I develop a model in which optimal costly information acquisition by individual firms causes adverse selection in the market as a whole. Each firm’s information acquisition policy determines which customers it serves, which in turn affects the distribution of remaining customers and hence other firms’ incentives. I show that when information acquisition is ‘smooth’, the adverse selection externality due to each firm is dampened, and in equilibrium all firms make positive profits. By contrast, with lumpy information acquisition, only a limited number of firms are profitable. I establish that my results apply to a broad class of continuous-time information acquisition processes. The second essay explores information acquisition in labor markets. Noting that African-Americans face shorter employment durations than similar whites, we hypothesize that employers discriminate in acquiring ability-relevant information. We construct a model with a binary information generating process, ‘monitoring’, at the disposal of firms. Monitoring black but not white workers is self-sustaining. This ‘bad’ equilibrium is not merely a matter of coordination; rather, it is determined by history and not easily reversed. The model’s additional predictions, lower lifetime incomes and longer unemployment durations for blacks, are both strongly empirically supported. In the third essay, we investigate the possibility of repeated games equilibria that are robust to the discount factors. We prove a negative result which shows that a sizable part of the set of feasible individually rational payoffs can never be produced by such equilibria. We find the cutoff defining this region and interpret it as a limit on the ability to punish deviations when future rewards for randomization cannot be finely calibrated. Furthermore, we present a robust folk theorem to support payoffs in the complementary region with strategies that remain Subgame Perfect Nash Equilibria at all greater discount factors.

Poker is a complex game to analyze. In this project we will use the mathematics of game theory to solve some simplified variations of the game. Probability is the building block behind game theory. We must understand a few concepts from probability such as distributions, expected value, variance, and enumeration methods to aid us in studying game theory. We will solve and analyze games through game theory by using different decision methods, decision trees, and the process of domination and simplification. Poker models, with and without cards, will be provided to illustrate optimal strategies. Extensions to those models will be presented, and we will show that optimal strategies still exist. Finally, we will close this paper with an original work to an extension that can be used as a medium to creating more extensions and, or, different games to explore.

This thesis is composed of three chapters. Chapter 1 considers the existence of equilibria in games with complete information, where players may have non-ordered and discontinuous preferences. Chapter 2 studies the issues on the existence of pure and behavioral strategy equilibria in games with incomplete information and discontinuous payoffs. We consider the standard setting with Bayesian preferences as well as the case in which players may face ambiguity. Chapter 3 extends the classical results on the Walras-core existence and equivalence to an ambiguous asymmetric information economy, where agents maximize maximin expected utilities (MEU). These results are based on the papers He and Yannelis (2014, 2015a,b,c, 2016a,b). In the first chapter, we propose the condition of "continuous inclusion property" to handle the difficulty of discontinuous payoffs in various general equilibrium and game theory models. Such discontinuities arise naturally in economic situations, including auction, price competition of firms and also patent races. Based on the continuous inclusion property, we establish the equilibrium existence result in a very general framework with discontinuous payoffs. On one hand, this condition is sufficiently general from the methodological point of view, as it unifies almost all special conditions proposed in the literature. On the other hand, our condition is also potentially useful from the realistic point of view, as it could be applied to deal with many economic models which cannot be studied before because of the presence of the discontinuity. In the second chapter, I study the existence problem of pure and behavioral strategy equilibria in discontinuous games with incomplete information. The framework of games with incomplete information is standard as in the literature, except for that we allow players' payoffs to be discontinuous. We illustrate by examples that the Bayesian equilibria may not exist in such games and the previous results are not applicable to handle this problem. We propose some general conditions to retain the existence of both pure strategy and behavioral strategy Bayesian equilibrium, and show that our condition is tight. In addition, we study the equilibrium existence problem in discontinuous games under incomplete information and ambiguity, and show that the maximin framework solves the equilibrium existence issue without introducing any additional condition. In the last chapter, I study a general equilibrium model with incomplete information by adopting the maximin expected utilities. The model is powerful enough to describe the behaviors of risk averse agents that cannot be explained by the standard assumption of subjective expected utilities. I use this new formulation to extend many classical results in general equilibrium theory by incorporating ambiguity into the model. In addition, the desirable incentive compatibility property is shown in our model with maximin expected utilities, while this property will typically fail in the traditional setup. Specifically, the existence results are shown for various equilibrium notions in a general equilibrium model, and the incentives can be guaranteed when all agents use the maximin expected utilities.

Combinatorial games are intriguing and have a tendency to engross students and lead them into a serious study of mathematics. The engaging nature of games is the basis for this thesis. Two combinatorial games along with some educational tools were developed in the pursuit of the solution of these games. The game of Nim is at least centuries old, possibly originating in China, but noted in the 16th century in European countries. It consists of several stacks of tokens, and two players alternate taking one or more tokens from one of the stacks, and the player who cannot make a move loses. The formal and intense study of Nim culminated in the celebrated Sprague-Grundy Theorem, which is now one of the centerpieces in the theory of impartial combinatorial games. We study a variation on Nim, played on a graph. Graph Nim, for which the theory of Sprague-Grundy does not provide a clear strategy, was originally developed at the University of Colorado Denver. Graph Nim was first played on graphs of three vertices. The winning strategy, and losing position, of three vertex Graph Nim has been discovered, but we will expand the game to four vertices and develop the winning strategies for four vertex Graph Nim. Graph Theory is a markedly visual field of mathematics. It is extremely useful for graph theorists and students to visualize the graphs they are studying. There exists software to visualize and analyze graphs, such as SAGE, but it is often extremely difficult to learn how use such programs. The tools in GeoGebra make pretty graphs, but there is no automated way to make a graph or analyze a graph that has been built. Fortunately GeoGebra allows the use of JavaScript in the creation of buttons which allow us to build useful Graph Theory tools in GeoGebra. We will discuss two applets we have created that can be used to help students learn some of the basics of Graph Theory. The game of thrones is a two-player impartial combinatorial game played on an oriented complete graph (or tournament) named after the popular fantasy book and TV series. The game of thrones relies on a special type of vertex called a king. A king is a vertex, k, in a tournament, T, which for all x in T either k beats x or there exists a vertex y such that k beats y and y beats x. Players take turns removing vertices from a given tournament until there is only one king left in the resulting tournament. The winning player is the one which makes the final move. We develop a winning position and classify those tournaments that are optimal for the first or second-moving player.

The most common method of diagnosing Obsessive-Compulsive Disorder is the Yale-Brown Obsessive Compulsive Scale, which measures the severity of symptoms without regard to compulsions. However, this scale is limited to only considering the quantifiable time and energy lost to compulsions. Conversely, current systems of brain imaging arrest mobility and thus make it virtually impossible to observe compulsions at all, focusing instead on neurological responses to external stimuli. There is little research which merges both approaches, to consider the neuro-physiological effects of obsessions as well as the physical response through compulsions. As such, this research is focused on developing a model of compulsivity based upon neurological chemical pathways. The objective is to develop a model which would predict, given a set of environmental parameters, the probability of an individual with OCD performing compulsive behavior and the prevalence of such behavior. By applying this concept to a neural system known as the worry circuit, a computer program was composed and simulations run by this program suggest that the likelihood of compulsive behavior can be predicted using a function of the number of compulsions performed previously. In this model, each neurological agent in the worry circuit, represented by an automaton, has a certain probability of reacting to a stimulus and moving into one of two distinct excited states. Based on the final state of the automaton, the agent will send excitatory or inhibitory signals to surrounding agents, which also have a certain probability of changing states. If the final agent within the cycle shifts into an excited state, the subject will perform a compulsion. These results may be considered preliminary, given the sample size of the case study and the primitive nature of the model.