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ADVERTIMENT. L'accés Als Continguts D'aquesta Tesi Doctoral I La Seva Utilització Ha De Respectar Els Drets De La Persona Autora

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ADVERTIMENT. L'accés Als Continguts D'aquesta Tesi Doctoral I La Seva Utilització Ha De Respectar Els Drets De La Persona Autora ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 del Text Refós de la Llei de Propietat Intel·lectual (RDL 1/1996). Per altres utilitzacions es requereix l'autorització prèvia i expressa de la persona autora. En qualsevol cas, en la utilització dels seus continguts caldrà indicar de forma clara el nom i cognoms de la persona autora i el títol de la tesi doctoral. No s'autoritza la seva reproducció o altres formes d'explotació efectuades amb finalitats de lucre ni la seva comunicació pública des d'un lloc aliè al servei TDX. Tampoc s'autoritza la presentació del seu contingut en una finestra o marc aliè a TDX (framing). Aquesta reserva de drets afecta tant als continguts de la tesi com als seus resums i índexs. ADVERTENCIA. El acceso a los contenidos de esta tesis doctoral y su utilización debe respetar los derechos de la persona autora. Puede ser utilizada para consulta o estudio personal, así como en actividades o materiales de investigación y docencia en los términos establecidos en el art. 32 del Texto Refundido de la Ley de Propiedad Intelectual (RDL 1/1996). Para otros usos se requiere la autorización previa y expresa de la persona autora. En cualquier caso, en la utilización de sus contenidos se deberá indicar de forma clara el nombre y apellidos de la persona autora y el título de la tesis doctoral. No se autoriza su reproducción u otras formas de explotación efectuadas con fines lucrativos ni su comunicación pública desde un sitio ajeno al servicio TDR. Tampoco se autoriza la presentación de su contenido en una ventana o marco ajeno a TDR (framing). Esta reserva de derechos afecta tanto al contenido de la tesis como a sus resúmenes e índices. WARNING. Access to the contents of this doctoral thesis and its use must respect the rights of the author. It can be used for reference or private study, as well as research and learning activities or materials in the terms established by the 32nd article of the Spanish Consolidated Copyright Act (RDL 1/1996). Express and previous authorization of the author is required for any other uses. In any case, when using its content, full name of the author and title of the thesis must be clearly indicated. Reproduction or other forms of for profit use or public communication from outside TDX service is not allowed. Presentation of its content in a window or frame external to TDX (framing) is not authorized either. These rights affect both the content of the thesis and its abstracts and indexes. The Complexity of Angel-Daemons and Game Isomorphism Alina Garc´ıa Chacon´ PhD Thesis advised by Joaqu´ım Gabarro´ Departament de Llenguatges i Sistemes Informatics` Universitat Politecnica` de Catalunya April 2012 The Complexity of Angel-Daemons and Game Isomorphism Alina Garc´ıa Chacon´ PhD Thesis advised by Joaqu´ım Gabarro´ Departament de Llenguatges i Sistemes Informatics` Universitat Politecnica` de Catalunya April 2012 To my dear son Dani “There is a driving force more powerful than steam, electricity and nuclear power: the will.” Albert Einstein Abstract The analysis of the computational aspects of strategic situations is a basic field in Com- puter Sciences. Two main topics related to strategic games have been developed. First, introduction and analysis of a class of games (so called angel/daemon games) designed to asses web applications, have been considered. Second, the problem of isomorphism between strategic games has been analysed. Both parts have been separately considered. Angel-Daemon Games A service is a computational method that is made available for general use through a wide area network. The performance of web-services may fluctuate; at times of stress the performance of some services may be degraded (in extreme cases, to the point of failure). In this thesis uncertainty profiles and Angel-Daemon games are used to analyse service- based behaviours in situations where probabilistic reasoning may not be appropriate. In such a game, an angel player acts on a bounded number of “angelic” services in a beneficial way while a daemon player acts on a bounded number of “daemonic” services in a negative way. Examples are used to illustrate how game theory can be used to analyse service-based scenarios in a realistic way that lies between over-optimism and over-pessimism. The resilience of an orchestration to service failure has been analysed - here angels and daemons are used to model services which can fail when placed under stress. The Nash equilibria of a corresponding Angel-Daemon game may be used to assign a “robustness” value to an orchestration. Finally, the complexity of equilibria problems for Angel-Daemon games has been analysed. It turns out that Angel-Daemon games are, at the best of our knowledge, the first natural example of zero-sum succinct games. The fact that deciding the existence of a pure Nash equilibrium or a dominant strategy p for a given player is Σ2 -complete has been proven. Furthermore, computing the value of an Angel-Daemon game is EXP-complete. Thus, matching the already known complexity results of the corresponding problems for the generic families of succinctly represented games with exponential number of actions. Game Isomorphism The question of whether two multi-player strategic games are equivalent and the com- putational complexity of deciding such a property has been addressed. Three notions of isomorphisms, strong, weak and local have been considered. Each one of these iso- morphisms preserves a different structure of the game. Strong isomorphism is defined to preserve the utility functions and Nash equilibria. Weak isomorphism preserves only the player preference relations and thus pure Nash equilibria. Local isomorphism preserves preferences defined only on “close” neighbourhood of strategy profiles. The problem of the computational complexity of game isomorphism, which depends on the level of succinctness of the description of the input games but it is independent of the isomorphism to consider, has been shown. Utilities in games can be given suc- cinctly by Turing machines, boolean circuits or boolean formulas, or explicitly by tables. Actions can be given also explicitly or succinctly. When the games are given in general form, an explicit description of actions and a succinct description of utilities have been assumed. It is has been established that the game isomorphism problem for general form games is equivalent to the circuit isomorphism when utilities are described by Turing Ma- chines; and to the boolean formula isomorphism problem when utilities are described by formulas. When the game is given in explicit form, it is has been proven that the game isomorphism problem is equivalent to the graph isomorphism problem. Finally, an equivalence classes of small games and their graphical representation have been also examined. Resum L’analisi` dels aspectes computacionals de situacions estrategiques` es´ un camp basic` de les Ciencies` de la Computacio.´ En aquesta tesi s’han desenvolupat dos aspects fonamentals relacionats amb els jocs estrategics.` Primer, s’ha considerat la introduccio´ i l’analisi` d’una classe de jocs anomenada jocs d’angel-dimoni,` dissenyada per avaluar aplicacions web. Segon, s’ha analitzat el problema de l’isomorfisme entre jocs estrategics.` Ambdues parts han estat considerades de manera independent. Jocs d’Angel-Dimoni` Un servei es´ un metode` de calcul` que es fa disponible a traves´ de la xarxa per ser usat de manera general. L’acompliment dels serveis web pot fluctuar. Fins i tot, en moments d’estres` el funcionament d’alguns serveis es pot degradar, fins al punt de fallar en casos extrems. Per tant, els perfils d’incertesa i els jocs d’Angel-Dimoni` s’utilitzen en aquesta tesi per analitzar conductes basades en els serveis, en situacions on el raonament proba- bil´ıstic pot no ser apropiat. En aquests jocs, el jugador angel` actua dins d’una serie` limitada de serveis angeli- cals d’una manera beneficiosa. No obstant aixo,` el jugador dimoni actua dins de la serie` limitada de serveis demon´ıacs d’una manera negativa. Alguns dels exemples mostrats serveixen per il.lustrar com la teoria de jocs pot utilitzar-se en l’analisi` d’escenaris basats en serveis d’una manera realista, a mig cam´ı entre l’optimisme i el pessimisme. Addicionalment s’ha analitzat la resistencia` d’orquestracions a la manca de serveis. Aqu´ı, els angels` i els dimonis s’empren per a modelar els serveis que poden fallar en condicions de pressio.´ Els equilibris de Nash del joc corresponent d’Angel-Dimoni` es poden utilitzar per assignar cert valor de “solidesa” a una orquestracio.´ Per ultim,´ s’ha analitzat la complexitat de problemes d’equilibri en jocs d’Angel-Dimoni` . Aquesta classe de jocs d’Angel-Dimoni` constitueixe el primer exemple natural de jocs succints de suma zero, i aquest es´ un dels resultats d’aquesta tesi. A mes,´ s’ha pogut provar el fet que la decisio´ de l’existencia´ d’un equilibri de Nash p pur o d’una estrategia` dominant per a un jugador donat es´ Σ2 completa. Tambe,´ s’ha comprovat que el calcul` del valor d’un joc d’Angel-Dimoni` es´ EXP-complet, la qual cosa coincideix amb els coneguts resultats de complexitat dels problemes corresponents de les fam´ılies generiques` de jocs representats de manera succinta, amb un nombre exponencial d’accions. Isomorfisme de Jocs En aquesta tesi tambe´ s’ha abordat la questi¨ o´ de si dos jocs de multiples´ jugadors son´ equivalents aix´ı com la complexitat computacional de decidir aquest tipus de propietat.
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