Aziza's Friendship Compendium, 1st Edition is my illustrated textbook that is part-compendium, part-manifesto. It provides a social, cultural and political analysis of friendship as an inherently anti-oppression, radical tool of resistance.  This study of friendship investigates the form and function of friendship, as well as how neoliberalism and the patriarchy cause friction in its functioning, and the ultimate fantasy of friendship that is possible through its untangling of systems of oppression. The Annotated and Expanded version of the Compendium shared here details the process and outcomes surrounding this publication and my master's degree project as a whole. This project is an argument for friendship, a celebration of friendship, a dissection of friendship and a resurrection and recontextualisation of friendship.  ˜”*°•♥ I hope you & your friends enjoy it ♥•°*”˜

The current president of the United States, Donald Trump, has been identified as the reason for a large shift in American foreign policy towards a doctrine closer to that of political realism. This claim has led us to examine if this transformation could be detected and described if we analyzed and compared Trump’s foreign policy doctrine with his predecessor, Barack Obama, through the lens Thomas Hobbes, whose ideas are at the core of the three modern schools of political realism. Accordingly, in this thesis, we deduce an analytical framework from the original corpus of Hobbes, where anarchy is divided into moral and empirical variables, identified as the primary factors for behavior in international settings. This is then applied inductively via a comparative qualitative content analysis to two primary documents, the National Security Strategies of 2010 containing the foreign policy doctrine of Obama, and the National Security Strategy of 2017 containing the doctrine of Trump. Our thesis shows a large shift in how the Presidents view the world in moral terms, or how they see it fit for the American executive to act on the international stage. And a relatively minor shift in empirical terms, or their perception of the foundational reality of the world system which they both consider to be of an anarchical nature closely connected to the theoretical model presented by our interpretation of Hobbes

Nous considérons des jeux de congestion sur des graphes. Dans les jeux non-atomiques, nous considérons un ensemble de joueurs infinitésimaux. Chaque joueur veut aller d'un sommet à un autre en choisissant une route de coût minimal. Le coût de chaque route dépend du nombre de joueur la choisissant. Dans les jeux atomiques divisibles, nous considérons un ensemble de joueurs ayant chacun une demande à transférer d'un sommet à un autre, en la subdivisant éventuellement sur plusieurs routes. Dans ces jeux, un équilibre de Nash est atteint lorsque chaque joueur a choisi une stratégie de coût minimal. L'existence d'un équilibre de Nash est assurée sous de faibles hypothèses. Les principaux sujets sont l'unicité, le calcul, l'efficacité et la sensibilité de l'équilibre de Nash. De nombreux résultats sont connus dans le cas où les joueurs sont tous impactés de la même façon par la congestion. Le but de cette thèse est de généraliser ces résultats au cas où les joueurs ont des fonctions de coût différentes. Nous obtenons des résultats sur l'unicité de l'équilibre dans les jeux non-atomiques. Nous donnons deux algorithmes capables de calculer un équilibre dans les jeux non-atomiques lorsque les fonctions de coût sont affines. Nous obtenons une borne sur le prix de l'anarchie pour certains jeux atomiques divisibles et prouvons qu'il n'est pas borné en général, même lorsque les fonctions sont affines. Enfin, nous prouvons des résultats sur la sensibilité de l'équilibre par rapport à la demande dans les jeux atomiques divisibles / We consider congestion games on graphs. In nonatomic games, we are given a set of infinitesimal players. Each player wants to go from one vertex to another by taking a route of minimal cost, the cost of a route depending on the number of players using it. In atomic splittable games, we are given a set of players with a non-negligible demand. Each player wants to ship his demand from one vertex to another by dividing it among different routes. In these games, we reach a Nash equilibrium when every player has chosen a minimal-cost strategy. The existence of a Nash equilibrium is ensured under mild conditions. The main issues are the uniqueness, the computation, the efficiency and the sensitivity of the Nash equilibrium. Many results are known in the specific case where all players are impacted in the same way by the congestion. The goal of this thesis is to generalize these results in the case where we allow player-specific cost functions. We obtain results on uniqueness of the equilibrium in nonatomic games. We give two algorithms able to compute a Nash equilibrium in nonatomic games when the cost functions are affine. We find a bound on the price of anarchy for some atomic splittable games, and prove that it is unbounded in general, even when the cost functions are affine. Finally we find results on the sensitivity of the equilibrium to the demand in atomic splittable games

L’anarchia désigne au sens propre la vacance du pouvoir dans la sphère militaire (la mort du chef sur le champ de bataille) et dans la sphère politique (la vacance de la magistrature, en particulier éponyme). La notion est vite devenue plus générale et exprime au figuré le manque ou le refus du commandement. Elle se rapproche des sens d’insubordination, insoumission, indiscipline, licence et désordre. L’expérience effective de la vacance du pouvoir dans les cités grecques, sa représentation et sa conceptualisation constituent les axes principaux de cette recherche. L’anarchia est envisagée à l’échelle de l’âme, de la famille, de la cité, voire de l’univers. Elle engage une réflexion sur l’articulation entre deux principes a priori antagonistes, l’aspiration à la liberté et la nécessité de l’ordre, et par conséquent sur les fondements de l’exercice légitime du pouvoir. L’étude repose sur l’analyse précise des emplois du terme anarchia dans les sources épigraphiques, historiques, littéraires et philosophiques. La première partie envisage les vacances effectives des magistratures dans le cadre du cours normal ou d’une rupture du fonctionnement des institutions ainsi que les palliatifs mis en place. Les occurrences du terme anarchia concernent les cités d’Athènes, Thasos, Téos, Syros et Bérénikè. S’ajoutent les emplois problématiques des termes akosmia à propos du régime crétois par Aristote et atagia dans les inscriptions thessaliennes. La deuxième partie envisage l’évolution sémantique du terme de l’absence de chef à l’anarchie chez les historiens et les poètes tragiques et la place de l’anarchia dans les théories du commandement élaborées par Xénophon, Platon et Aristote. / The term anarchia refers literally to the absence of power, in the military sphere (that caused by the loss of a commander), and the political sphere (the absence of archontes, specifically the eponymous archon). The concept quickly generalised, coming to designate in the figurative sense the lack and want of power or the rejection and negation of power. It approaches the meanings of insubordination, rebelliousness, unruliness, licentiousness and disorder. The actual experience of power vacuum in the cities of Ancient Greece and how the Greeks represented it and conceptualised it are the three main lines of this research. Anarchia is conceived not only in the city but also in the soul of the individual, in the family, or even in the universe. It demands reflection on the articulation between two seemingly antagonistic principles, the desire for freedom and the necessity of order, and consequently upon the foundations of legitimate authority. This work relies on a precise analysis of the term anarchia in the epigraphic, historical, literary and philosophical sources. The first part deals with actual periods of power vacuum in the ordinary course of political life or in the context of institutional disruption and the implemented remedies. The term anarchia is employed in the cities of Athens, Thasos, Teos, Syros and Berenike. One must add the problematical use of the terms acosmia by Aristotle regarding the Cretan regime and atagia in the Thessalian inscriptions. The second part deals with the semantic evolution of the term from the absence of ruler to anarchy in the work of historians and tragic poets and the role of anarchia in the theory of leadership developed by Xenophon, Plato and Aristotle.

Nosso objetivo principal, nesta tese, é fornecer um esboço de reconstrução programática do livro Os Limites da Liberdade, escrito por James Buchanan, aplicando os recursos da Metodologia da Teoria da Ciência e do Programa de Pesquisa, visando, especificamente, definir a nossa estrutura conceitual, que pretende reconstruir o problema e o modo de organizar a concepção de Estado ou da ordem constitucional, na perspectiva do autor James Buchanan. O núcleo da concepção deste autor é baseada na tese ontológica que considera o indivíduo uma entidade livre, auto-interessada e racional, dentro de uma perspectiva analógica, ligada com a noção econômica do mercado. O processo de reconstrução, deste trabalho, acompanha a idéia de emergência da cooperação, traduzida na ordem política pela noção do meio-termo entre a Anarquia e o Leviatã. Isto é feito buscando-se as regras metodológicas e as teses programáticas que controlam o equilíbrio ou balanço democrático da ordem política, em relação a esses dois extremos ou limites institucionais. A imagem deste equilíbrio, que é modelada pela analogia com a idéia econômica de mercado, está definida, apropriadamente, na expressão anarquia ordenada. A regra fundamental deste sistema institucional propõe otimizar a delegação de autoridade para o cidadão controlar poderes políticos, resgatando as premissas fundamentais da democracia, no sentido de evitar qualquer possibilidade da anarquia e do controle excessivo do Leviatã. A idéia básica da anarquia ordenada consiste em valorizar a racionalidade e a capacidade de negociação dos indivíduos, que são conhecedores de seus próprios interesses e limitados pelo interesse dos outros participantes. A operacionalização da anarquia ordenada é fundamentada no desenvolvimento de um projeto institucional que disponibiliza tecnologias, regras, sistemas e critérios de decisão que, supostamente, podem resolver problemas relacionados ao conflito dos interesses públicos e privados, através da interação democrática dos indivíduos. Nesta perspectiva, são elementos relevantes no contexto público do desenho institucional da anarquia ordenada, os sistemas político e judiciário, onde as tecnologias legais e as inovações legislativas podem desempenhar significativo papel na implementação da visão democrática da anarquia ordenada. / Our first aim at this thesis is to provide an outline of a rational reconstruction of the book The Limits of Liberty by James Buchanan with the methodological resources of the Methodology of Theory of Science and the Research Program with the purpose of defining our conceptual framework out of which we pretend to build up the problem and the specific way of organizing the conception of state or institutional order in Buchanan\'s view. The core of this conception is based upon the ontological thesis according to which the only entity existing is the individual, free, self interested and rational and also the guideline of pursuing the analogy with the notion of market in economics.This process of reconstruction follows, therefore, the idea of a emergency of cooperation that it is translated into the notion of political order as a middle way between anarchy and Leviathan. This is done pursuing the methodological rules and theses that controls its balance or skewness at the political order towards one of the boundaries.The image of this balance, that is moulded by this analogy with the idea of market, is properly defined as ordered anarchy. The main rule coordinating the other rules, a radical one, is that one should optimize the delegation of authority for citizen to control the political powers in order to work out the basic postulates of democracy in order to avoiding, on the one hand, any possibility of anarchy and in the other hand, Leviathan\'s excessive control. It follows the basic idea of valuing the rationality and capacity of negotiation of the individual who knows best its own interest but it is also limited by the interest of the others. The operationalization of this rule is based upon the development of a institutional design with resources of technologies, rules and systems and criteria of decision that are suposed to solve the problems related to the conflict of public and private interests coming from the interactions among the individuals in the ordered anarchy. In this perspective, a relevant element of this institutional design is that one of the electoral and judiciary system inside of which the technologies and its innovations will play a very meaningfull role for implementing this view of ordered anarchy.

Submitted by Maicon Juliano Schmidt (maicons) on 2015-06-03T14:54:25Z No. of bitstreams: 1 Leandro Rodrigues de Oliveira_.pdf: 1362296 bytes, checksum: 487da756b68c0545f7af4eff93ac8240 (MD5) / Made available in DSpace on 2015-06-03T14:54:25Z (GMT). No. of bitstreams: 1 Leandro Rodrigues de Oliveira_.pdf: 1362296 bytes, checksum: 487da756b68c0545f7af4eff93ac8240 (MD5) Previous issue date: 2015-04-17 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho se propõe a pensar a crise das referências como um pressuposto por onde se localiza uma problemática contemporânea de fundamentação ética. Tendo por base a reflexão sobre o niilismo, o que se pretende é entender como se deu o processo de enfraquecimento da tradição de pensamento moral na visão ocidental, diagnosticada pelo pensador Friedrich Nietzsche, como desvalorização dos valores supremos. Este trabalho contemplará diretamente o pensamento do filósofo alemão; contudo, manter-se-á numa leitura construída sob a ótica do filosofo jesuíta francês Paul Valadier, cujas obras nos convidam a compreender o problema do niilismo, tendo como perspectiva central uma avaliação que se desdobra em uma análise a respeito da crise valores e suas consequências para o mundo contemporâneo. Para o autor francês, a partir dos avanços da racionalidade científica com o advento da era moderna, as tradições foram postas em causa, interrogadas ou mesmo recusadas. Com isso, instala-se a impossibilidade de respostas às questões morais que se apresentam, uma vez que o ser humano passa a ser o responsável por definir seu agir e seu pensar no mundo. O resultado disso é o surgimento do relativismo moral que se instala como consequência do processo de anarquia dos valores, na condição de ausência do princípio de ordenação e hierarquização destes, originários da concepção de sentido e de mundo ordenado, antes provenientes das tradições. Por isso, faz-se perceber a importância de Nietzsche. Para Valadier, não há dúvida de que o filósofo alemão contribuiu mais do que muitos para este abalo moral. Daí que o diagnóstico nietzschiano seja tomado neste trabalho considerando-se sua relevância por vários aspectos: tanto no que concerne à crítica dos valores, à moral cristã, quando pela importância dada ao método genealógico pelo qual se torna possível compreender os fatores que levaram a tradição perder sua força de atuação, quando é revelado o seu sentido niilista escondido sob os valores considerados mais supremos. / This paper proposes to think the crisis of references as an assumption by where is a contemporary issue of ethical reasoning. Based on the reflection on nihilism, the aim is to understand how was the weakening of the moral tradition of thought in Western view, previously diagnosed from the thinker Friedrich Nietzsche, as devaluation of the highest values. This work directly contemplate the thought of German philosopher; however, will remain in a reading built from the perspective of the French Jesuit philosopher Paul Valadier, whose works invite us to understand the nihilism of the problem, with the central perspective an assessment that unfolds in an analysis about the crisis and its values consequences for the contemporary world. The French author, from the advances of scientific rationality with the advent of the modern era, the traditions have been challenged, questioned, or even rejected. With this, install the impossibility of answers to moral questions that arise, since the human being becomes responsible for defining their actions and their thinking in the world. The result is the emergence of moral relativism that develops as a consequence of the process of anarchy of values, provided that the absence of the ordering principle and ranking of these originating in design sense, orderly world, but from the traditions. In this sense, is made to realize the importance of Nietzsche. To Valadier, there is no doubt that the German philosopher contributed more than many for this moral shock. Hence Nietzsche's diagnosis is taken in this work considering their relevance for various aspects: both as regards the criticism of values, Christian morality, when the importance given to genealogical method by which it becomes possible to understand the factors that led to tradition lose its force of action, when it is revealed its nihilistic sense hidden under the values considered most supreme.

Depuis les années 1990, le Nouveau Management Public a transformé l’hôpital public, on parle d’un tournant gestionnaire. L’apparition des pôles d’activité médicale en 2005, consiste en une nouvelle gouvernance hospitalière. Nous avons cherché comprendre comment les pôles d’activité médicale font-ils évoluer les processus de décisions ? Nous avons choisi de mobiliser le concept de configuration structurelle (Mintzberg, 1982) et la théorie de l’anarchie organisée (Cohen, March et Olsen, 1972). La question est désormais de savoir si le changement de configuration structurelle va accroître ou réduire l’anarchie organisée ? Pour y répondre nous avons réalisé deux phases de récolte de données dans un CHU : l’analyse de la gestion quotidienne de trois pôles et de deux projets de pôle. Notre étude de cas a permis de mettre en évidence que l’hôpital n’est pas devenu une structure divisionnalisée mais qu’il est en transition, dans une configuration structurelle hybride. Les pôles ont eu tendance à augmenter l’anarchie organisée et le slack organisationnel pour l’heure. Notamment à cause de l’influence du contexte économique, l’absence de contrats de pôle, le problème de formation des agents et une opposition constante entre gestionnaires et médicaux. Les pôles se présentent comme de nouvelles unités qui ont rendu plus complexe l’organisation interne. Cependant, l’arrivée des pôles s’est accompagnée d’une redistribution des rôles, de nouveaux outils de gestion et de nouveaux processus de gestion. Ainsi, le changement de configuration structurelle a impacté le processus de décision de l’hôpital, en modifiant les pratiques et les cadres de ces processus de décision. / Since the 1990s, the New Public Management has transformed the public hospital, we talk about a turning management. The emergence of medical activity centers in 2005, consists of a new hospital governance. We sought to understand how the poles of medical activity are changing decision-making processes? We have chosen to mobilize the concept of structural configuration (Mintzberg, 1982) and the theory of organized anarchy (Cohen, March and Olsen, 1972). The question now is whether the change in structural configuration will increase or reduce organized anarchy? To answer this question, we conducted two data collection phases in a CHU: the analysis of the daily management of three clusters and two cluster projects. Our case study revealed that the hospital has not become a divisionalized structure but is in transition, in a hybrid structural configuration. The poles tended to increase organized anarchy and organizational slack for the time being. Notably because of the influence of the economic context, the absence of pole contracts, the problem of training agents and a constant opposition between managers and medical staff. The poles are new units that have made the internal organization more complex. However, the arrival of the clusters has been accompanied by a redistribution of roles, new management tools and new management processes. Thus, the change in structural configuration has impacted the hospital's decision-making process by changing the practices and frameworks of these decision-making processes.

<p>This essay is called ”<em>The hunt for anarchists- a study</em> <em>about the police in Stockholm's collection of bandit portrait during the turn of the century 1900</em>”<em> </em>and it is about the criminal category ”Anarchists” who the police in Stockholm used at the turn of the century 1900.  In the archive from the police in Stockholm during the essays time perspective, 1899-1909,  there are about one hundred photographs in the category ”Anarchists”, about half of these pictures have no information besides the names of the people, but the other half, 48 persons, have some information about age, work title and where the person come from. The information showed that the people in the pictures where not from Sweden, and after controlling them in all different kinds of archives I found only ten of them have left any traces in Stockholm. What I realized then was that the people on the pictures are anarchists from different countries in Europe, mostly from Italy, and that the police in Stockholm had these pictures because different police stations around Europe had sent them to the police in Stockholm. The police in Stockholm where on the lookout for fugitive anarchists.</p><p>The literature about the anarchist movement in Italy during this time describe the hard situation for Italian anarchists. The police had persecuted, arrested and executed manyof them so many anarchists had fled abroad. The same was for Russian anarchists after the unsuccessful revolution in 1905. Eight of the ten anarchists of the police photographs who had been in Stockholm where Russians. They were a group who was accused of trying to kill the Russian czar visiting Stockholm in 1909.</p><p>None of the anarchists on the pictures have ever in Stockholm committed a political crime so to find out what a anarchist crime is have not been possible. But the general picture of the anarchists in the photographs is of a man in his 30’s with a working class job, in short: an everyday man.</p>

This essay is called ”The hunt for anarchists- a study about the police in Stockholm's collection of bandit portrait during the turn of the century 1900” and it is about the criminal category ”Anarchists” who the police in Stockholm used at the turn of the century 1900.  In the archive from the police in Stockholm during the essays time perspective, 1899-1909,  there are about one hundred photographs in the category ”Anarchists”, about half of these pictures have no information besides the names of the people, but the other half, 48 persons, have some information about age, work title and where the person come from. The information showed that the people in the pictures where not from Sweden, and after controlling them in all different kinds of archives I found only ten of them have left any traces in Stockholm. What I realized then was that the people on the pictures are anarchists from different countries in Europe, mostly from Italy, and that the police in Stockholm had these pictures because different police stations around Europe had sent them to the police in Stockholm. The police in Stockholm where on the lookout for fugitive anarchists. The literature about the anarchist movement in Italy during this time describe the hard situation for Italian anarchists. The police had persecuted, arrested and executed manyof them so many anarchists had fled abroad. The same was for Russian anarchists after the unsuccessful revolution in 1905. Eight of the ten anarchists of the police photographs who had been in Stockholm where Russians. They were a group who was accused of trying to kill the Russian czar visiting Stockholm in 1909. None of the anarchists on the pictures have ever in Stockholm committed a political crime so to find out what a anarchist crime is have not been possible. But the general picture of the anarchists in the photographs is of a man in his 30’s with a working class job, in short: an everyday man.

Στα πλαίσια της διατριβής αναπτύξαμε δύο από τους πρώτους αλγορίθμους υπολογισμού μιας ε-προσεγγιστικής ισορροπίας Nash για την περίπτωση όπου το ε είναι κάποια σταθερά. Οι προσεγγίσεις που επιτυγχάνουν οι αλγόριθμοί μας είναι ε=3/4 και ε=(2+λ)/4 αντίστοιχα, όπου λ είναι το ελάχιστο, μεταξύ όλων των ισορροπιών Nash, κέρδος για έναν παίκτη. Επιπλέον, μελετήσαμε μια ευρεία κλάση τυχαίων παιγνίων δύο παικτών, για την οποία υπολογίσαμε μια πολύ καλή ε-προσεγγιστική ισορροπία Nash, με το ε να τείνει στο 0 καθώς το πλήθος των διαθέσιμων στρατηγικών των παικτών τείνει στο άπειρο. Οι αρχές της θεωρίας παιγνίων είναι χρήσιμες στην ανάλυση της επίδρασης που έχει στην καθολική απόδοση ενός συστήματος διαμοιραζόμενων πόρων η εγωιστική και ανταγωνιστική συμπεριφορά των χρηστών του. Προς την κατεύθυνση αυτή, εστιάσαμε στο πρόβλημα της εξισορρόπησης φορτίου. Μελετήσαμε διάφορα μοντέλα πληροφόρησης (π.χ. όταν όλα τα φορτία είναι άγνωστα ή όταν κάθε παίκτης γνωρίζει το μέγεθος του δικού του φορτίου) και αναλύσαμε για το καθένα το σύνολο και τις ιδιότητες των ισορροπιών Nash. Yπολογίσαμε επίσης φράγματα στο λόγο απόκλισης, ο οποίος εκφράζει την επίδραση που έχει στην απόδοση του συστήματος η εγωιστική συμπεριφορά των χρηστών του. Εκτός από τα υπολογιστικά θέματα που σχετίζονται με τη θεωρία παιγνίων, έχει ενδιαφέρον να μελετηθεί κατά πόσο μπορεί η θεωρία παιγνίων να βοηθήσει στην ανάπτυξη και ανάλυση αλγορίθμων για υπολογιστικά δύσκολα προβλήματα συνδυαστικής βελτιστοποίησης. Προς αυτήν την κατεύθυνση, μελετήσαμε από παιγνιοθεωρητική σκοπιά το πρόβλημα χρωματισμού των κορυφών ενός γραφήματος. Ορίσαμε κατάλληλα το παίγνιο χρωματισμού γραφήματος και αποδείξαμε ότι κάθε παίγνιο χρωματισμού γραφήματος έχει πάντα μια αγνή ισορροπία Nash, και ότι κάθε αγνή ισορροπία Nash αντιστοιχεί σε ορθό χρωματισμό του γραφήματος. Δείξαμε επίσης ότι υπάρχει πάντα μια αγνή ισορροπία Nash που χρησιμοποιεί βέλτιστο αριθμό χρωμάτων, δηλαδή ίσο με το χρωματικό αριθμό του γραφήματος. Επιπλέον, περιγράψαμε και αναλύσαμε έναν πολυωνυμικό αλγόριθμο που υπολογίζει μια αγνή ισορροπία Nash για ένα οποιοδήποτε παίγνιο χρωματισμού γραφήματος και χρησιμοποιεί συνολικά ένα πλήθος χρωμάτων που ικανοποιεί ταυτόχρονα τα περισσότερα κλασικά γνωστά φράγματα στο χρωματικό αριθμό. / We developed two algorithms for computing an e-approximate Nash equilibrium for the case where e is an absolute constant. The approximations achieved by our algorithms are e=3/4 and e=(2+l)/4 respectively, where $\lambda$ is the minimum, among all Nash equilibria, payoff of either player. Furthermore, we studied a wide class of random two player games, for which we showed how to compute an e-approximate Nash equilibrium, where e tends to zero as the number of strategies of the players tends to infinity. Game theoretic concepts are useful in determining the impact that selfish behavior plays on the global performance of a system involving selfish entities. Towards this direction, we focused on the problem of load balancing. We studied the case where the agents are not necessarily fully informed about the exact values of their loads. We focused on several models of information (e.g. when all agents know nothing about the loads, or when each agents knows her own load) and, for each model, we characterized the set of Nash equilibria and analyzed their properties. Moreover, we bounded the coordination ratio, a measure which captures the impact that selfish behavior has to the global performance of the system, in contrast to the performance achieved by an optimum centralized algorithm. Besides the computational issues related to game theory, it is interesting to investigate whether game theory can help us in developing and analyzing algorithms for computationally difficult combinatorial optimization problems. Towards this direction, we studied from a game theoretic point of view the problem of vertex coloring. In particular, we properly defined the graph coloring game and we proved that every graph coloring game has a pure Nash equilibrium, and each pure Nash equilibrium corresponds to a proper coloring of the graph. We also showed that there exists a pure Nash equilibrium that uses an optimum number of colors, i.e. equal to the chromatic number. Furthermore, we developed and analyzed a polynomial time algorithm that computes a pure Nash equilibrium for any graph coloring game, using a number of colors satisfying most of the known classical bounds on the chromatic number.