In this paper we characterize the "price of anarchy", i.e., the inefficiency between user and system optimal solutions, when costs are non-separable, asymmetric and nonlinear, generalizing earlier work that has addressed "price of anarchy" under separable costs. The generalization models traffice equilibria, competitive multi-period pricing and competitive supply chains. The bounds established in the paper are tight and explicitly account for the degeee of asymmetry and nonlinearity of the cost function. We introduce and alternate proof method for providing bounds that uses ideas from semidenfinite optimization. Finally, in the context of nulti-period pricing our analysis establishes that user and system optimal soulutions coincide.

We study the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-hard, even when all arc latency functions are linear and there is a single source and sink. Still, one can prove that an optimal flow and an equilibrium flow share a desirable property in this situation: all flow-carrying paths have the same length; i.e., these solutions are "fair," which is in general not true for the optimal flow in networks with nonlinear latency functions. In addition, the maximum latency of the Nash equilibrium, which can be computed efficiently, is within a constant factor of that of an optimal solution. That is, the so-called price of anarchy is bounded. In contrast, we present a family of instances that shows that the price of anarchy is unbounded for instances with multiple sources and a single sink, even in networks with linear latencies. Finally, we show that an s-t-flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective, and it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latenc

Στην παρούσα διπλωματικής εργασία εφαρμόζουμε τις αρχές της Θεωρίας Παιγνίων, και συγκεκριμένα τις έννοιες των Ισορροπιών Nash και των Παι­γνίου Συμφόρησης, ώστε να αναλύσουμε την επίδραση που έχει στην καθο­λική απόδοση ενός δικτύου και γενικότερα ενός συστήματος διαμοιραζόμενων πόρων η εγωιστική και ανταγωνιστική συμπεριφορά των χρηστών του. Αρχικά ασχολούμαστε με το πρόβλημα της εγωιστικής δρομολόγησης φορ­τίων από μια κοινή πηγή προς έναν κοινό προορισμό σε ένα δίκτυο επικοι­νωνίας. Σε ένα τέτοιο περιβάλλον οι χρήστες επιλέγουν εγωιστικά τις στρα­τηγικές τους, οι οποίες στην περίπτωση μας αντιστοιχούν σε μονοπάτια από την πηγή προς τον προορισμό Όταν οι χρήστες δρομολογούν τα φορτία τους σύμφωνα με τις στρατηγικές που επιλέγουν, έρχονται αντιμέτωποι με μια κα­θυστέρηση που προκαλείται από τα φορτία όλων των χρηστών καθώς διαμοι­ράζονται τις ακμές. Κάθε χρήστης στοχεύει στην ελαχιστοποίηση του εγωι­στικού τον κόστους, που αντιστοιχεί σε αυτήν ακριβώς την καθυστέρηση, γεγονός που συνήθως έρχεται σε αντίθεση με το στόχο της βελτιστοποίησης της καθολικής απόδοσης του δικτύου. Η θεωρία των ισορροπιών Nash μας παρέχει μία σημαντική αρχή επίλυ­σης για τέτοιου είδους καταστάσεις: μια ισορροπία Nash, είναι μια κατάσταση του συστήματος τέτοια ώστε δεν υπάρχει κάποιος χρήστης που να μπορεί να μειώσει το εγωιστικό του κόστος αλλάζοντας μονομερώς τη στρατηγική του. Σε ένα τέτοιο δίκτυο λοιπόν περιμένουμε οι χρήστες να καταλήξουν σε μια ισορροπία Nash. Ωστόσο, ο υπολογισμός μιας τέτοιας ισορροπίας παραμέ­νει ένα πρόβλημα του οποίου η πολυπλοκότητα είναι, στη γενική περίπτωση, άγνωστη. Στα πλαίσια αυτής της διπλωματικής εργασίας δείχνουμε πειραματικά ότι ο υπολογισμός μιας αγνής ισορροπίας Nash σε ένα περιβάλλον εγωιστικής δρομολόγησης, όπου η καθυστέρηση σε κάθε ακμή ισούται με το φορτίο της. μπορεί να γίνει σε πολυωνυμικό χρόνο για μια μεγάλη ποικιλία δικτύων και κατανομών των φορτίων των χρηστών. Επιπλέον, προτείνουμε μια αρχική ανάθεση χρηστών σε μονοπάτια η οποία, όπως δείχνουν οι προσομοιώσεις μας, οδηγεί σε μια αξιοσημείωτη μείωση του συνολικού αριθμού των βημάτων που απαιτούνται ώστε να καταλήξουμε σε μια αγνή ισορροπία Nash. Επίσης αποδεικνύουμε την ύπαρξη αγνών ισορροπιών Nash και για την περίπτωση που η καθυστέρηση σε κάθε ακμή είναι εκθετική συνάρτηση του φορτίου της. Στη συνέχεια προτείνουμε και αναλύουμε ένα νέο μηχανισμό κόστους που θέτει τιμές για την ανταγωνιστική χρησιμοποίηση πόρων από ένα σύνολο χρη­στών. Το βασικό πλεονέκτημα αυτού του μηχανισμού είναι ότι οι πόροι θέ­τουν τα κόστη με έναν ισοδύναμο, δίκαιο τρόπο, και το πλέον σημαντικό είναι ότι κανένας πόρος δεν επωφελείται εις βάρος των χρηστών. Αυτός ο δίκαιος μηχανισμός κόστους επαγάγει ένα μη συνεργατικό παί­γνιο μεταξύ των χρηστών, για το οποίο αναλύουμε τις ισορροπίες Nash. Απο­δεικνύουμε ότι δεν υπάρχουν αγνές ισορροπίες Nash, εκτός από την περί­πτωση όπου όλα τα φορτία είναι ίσα, ενώ δείχνουμε ότι υπάρχει πάντα μία πλήρως μικτή ισορροπία Nash. Επίσης αναλύουμε για το παίγνιο αυτό το Κόστος της Αναρχίας, που εκφράζει την απόκλιση της απόδοσης του συ­στήματος στη χειρότερη ισορροπία Nash από τη βέλτιστη απόδοση. Αποδει­κνύουμε ότι το Κόστος της Αναρχίας στη χειρότερη περίπτωση είναι γραμ­μικό ως προς το πλήθος των χρηστών και ότι το φράγμα αυτό είναι αυστηρό. Ωστόσο προτείνουμε δύο τρόπους για να μετριάσουμε τη δυσάρεστη αυτή διαπίστωση. Καταρχήν, μελετάμε την περίπτωση όπου τα φορτία των χρηστών επιλέ­γονται από μία πολύ ευρεία οικογένεια φραγμένων κατανομών πιθανότητας. Ορίζουμε το Διαχεόμενο Κόστος της Αναρχίας το οποίο λαμβάνει υπόψη την κατανομή πιθανότητας των φορτίων και αποδεικνύουμε ότι Διαχεόμενο Κόστος της Αναρχίας φράσσεται εκ των άνω από μία μικρή σταθερά. Επιπλέον, προτείνουμε έναν υβριδικό μηχανισμό κόστους, ο οποίος επιτυγχάνει ένα ση­μαντικά μικρότερο Κόστος της Αναρχίας, ενώ το κέρδος κάθε πόρου παρα­μένει αμελητέο. / -

No description available.

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

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.

São analisados nesta dissertação problemas em teoria dos jogos, com enfoque no efeito que perturbações acarretam em jogos. A análise suavizada (smoothed analysis) é utilizada para tal análise, e dois tipos de jogos são o foco principal desta dissertação, jogos bimatrizes e o problema de atribuição de tráfego (Traffic Assignment Problem.) O algoritmo de Lemke-Howson é um método utilizado amplamente para computar um equilíbrio Nash de jogos bimatrizes. Esse problema é PPAD-completo (Polynomial Parity Arguments on Directed graphs), e existem instâncias em que um tempo exponencial é necessário para terminar o algoritmo. Mesmo utilizando análise suavizada, esse problema permanece exponencial. Entretanto, nenhum estudo experimental foi realizado para demonstrar na prática como o algoritmo se comporta em casos com perturbação. Esta dissertação demonstra como as instâncias de pior caso conhecidas atualmente podem ser geradas e mostra que a performance do algoritmo nestas instâncias, quando perturbações são aplicadas, difere do comportamento esperado provado pela teoria. O Problema de Atribuição de Tráfego modela situações em uma rede viária onde usuários necessitam viajar de um nodo origem a um nodo destino. Esse problema pode ser modelado como um jogo, usando teoria dos jogos, onde um equilíbrio Nash acontece quando os usuários se comportam de forma egoísta. O custo total ótimo corresponde ao melhor fluxo de um ponto de vista global. Nesta dissertação, uma nova medida de perturbação é apresentada, o Preço da Anarquia Suavizado (Smoothed Price of Anarchy), baseada na análise suavizada de algoritmos, com o fim de analisar os efeitos da perturbação no Preço da Anarquia. Usando esta medida, são estudados os efeitos que perturbações têm no Preço da Anarquia para instâncias reais e teóricas para o Problema de Atribuição de Tráfego. É demonstrado que o Preço da Anarquia Suavizado se mantém na mesma ordem do Preço da Anarquia sem perturbações para funções de latência polinomiais. Finalmente, são estudadas instâncias de benchmark em relação à perturbação. / This thesis analyzes problems in game theory with respect to perturbation. It uses smoothed analysis to accomplish such task and focuses on two kind of games, bimatrix games and the traffic assignment problem. The Lemke-Howson algorithm is a widely used algorithm to compute a Nash equilibrium of a bimatrix game. This problem is PPAD-complete (Polynomial Parity Arguments on Directed graphs), and there exists an instance which takes exponential time (with any starting pivot.) It has been proven that even with a smoothed analysis it is still exponential. However, no experimental study has been done to verify and evaluate in practice how the algorithm behaves in such cases. This thesis shows in detail how the current known worst-case instances are generated and shows that the performance of the algorithm on these instances, when perturbed, differs from the expected behavior proven in theory. The Traffic Assignment Problem models a situation in a road network where users want to travel from an origin to a destination. It can be modeled as a game using game theory, with a Nash equilibrium happening when users behave selfishly and an optimal social welfare being the best possible flow from a global perspective. We provide a new measure, which we call the Smoothed Price of Anarchy, based on the smoothed analysis of algorithms in order to analyze the effects of perturbation on the Price of Anarchy. Using this measure, we analyze the effects that perturbation has on the Price of Anarchy for real and theoretical instances for the Traffic Assignment Problem. We demonstrate that the Smoothed Price of Anarchy remains in the same order as the original Price of Anarchy for polynomial latency functions. Finally, we study benchmark instances in relation to perturbation.

São analisados nesta dissertação problemas em teoria dos jogos, com enfoque no efeito que perturbações acarretam em jogos. A análise suavizada (smoothed analysis) é utilizada para tal análise, e dois tipos de jogos são o foco principal desta dissertação, jogos bimatrizes e o problema de atribuição de tráfego (Traffic Assignment Problem.) O algoritmo de Lemke-Howson é um método utilizado amplamente para computar um equilíbrio Nash de jogos bimatrizes. Esse problema é PPAD-completo (Polynomial Parity Arguments on Directed graphs), e existem instâncias em que um tempo exponencial é necessário para terminar o algoritmo. Mesmo utilizando análise suavizada, esse problema permanece exponencial. Entretanto, nenhum estudo experimental foi realizado para demonstrar na prática como o algoritmo se comporta em casos com perturbação. Esta dissertação demonstra como as instâncias de pior caso conhecidas atualmente podem ser geradas e mostra que a performance do algoritmo nestas instâncias, quando perturbações são aplicadas, difere do comportamento esperado provado pela teoria. O Problema de Atribuição de Tráfego modela situações em uma rede viária onde usuários necessitam viajar de um nodo origem a um nodo destino. Esse problema pode ser modelado como um jogo, usando teoria dos jogos, onde um equilíbrio Nash acontece quando os usuários se comportam de forma egoísta. O custo total ótimo corresponde ao melhor fluxo de um ponto de vista global. Nesta dissertação, uma nova medida de perturbação é apresentada, o Preço da Anarquia Suavizado (Smoothed Price of Anarchy), baseada na análise suavizada de algoritmos, com o fim de analisar os efeitos da perturbação no Preço da Anarquia. Usando esta medida, são estudados os efeitos que perturbações têm no Preço da Anarquia para instâncias reais e teóricas para o Problema de Atribuição de Tráfego. É demonstrado que o Preço da Anarquia Suavizado se mantém na mesma ordem do Preço da Anarquia sem perturbações para funções de latência polinomiais. Finalmente, são estudadas instâncias de benchmark em relação à perturbação. / This thesis analyzes problems in game theory with respect to perturbation. It uses smoothed analysis to accomplish such task and focuses on two kind of games, bimatrix games and the traffic assignment problem. The Lemke-Howson algorithm is a widely used algorithm to compute a Nash equilibrium of a bimatrix game. This problem is PPAD-complete (Polynomial Parity Arguments on Directed graphs), and there exists an instance which takes exponential time (with any starting pivot.) It has been proven that even with a smoothed analysis it is still exponential. However, no experimental study has been done to verify and evaluate in practice how the algorithm behaves in such cases. This thesis shows in detail how the current known worst-case instances are generated and shows that the performance of the algorithm on these instances, when perturbed, differs from the expected behavior proven in theory. The Traffic Assignment Problem models a situation in a road network where users want to travel from an origin to a destination. It can be modeled as a game using game theory, with a Nash equilibrium happening when users behave selfishly and an optimal social welfare being the best possible flow from a global perspective. We provide a new measure, which we call the Smoothed Price of Anarchy, based on the smoothed analysis of algorithms in order to analyze the effects of perturbation on the Price of Anarchy. Using this measure, we analyze the effects that perturbation has on the Price of Anarchy for real and theoretical instances for the Traffic Assignment Problem. We demonstrate that the Smoothed Price of Anarchy remains in the same order as the original Price of Anarchy for polynomial latency functions. Finally, we study benchmark instances in relation to perturbation.

São analisados nesta dissertação problemas em teoria dos jogos, com enfoque no efeito que perturbações acarretam em jogos. A análise suavizada (smoothed analysis) é utilizada para tal análise, e dois tipos de jogos são o foco principal desta dissertação, jogos bimatrizes e o problema de atribuição de tráfego (Traffic Assignment Problem.) O algoritmo de Lemke-Howson é um método utilizado amplamente para computar um equilíbrio Nash de jogos bimatrizes. Esse problema é PPAD-completo (Polynomial Parity Arguments on Directed graphs), e existem instâncias em que um tempo exponencial é necessário para terminar o algoritmo. Mesmo utilizando análise suavizada, esse problema permanece exponencial. Entretanto, nenhum estudo experimental foi realizado para demonstrar na prática como o algoritmo se comporta em casos com perturbação. Esta dissertação demonstra como as instâncias de pior caso conhecidas atualmente podem ser geradas e mostra que a performance do algoritmo nestas instâncias, quando perturbações são aplicadas, difere do comportamento esperado provado pela teoria. O Problema de Atribuição de Tráfego modela situações em uma rede viária onde usuários necessitam viajar de um nodo origem a um nodo destino. Esse problema pode ser modelado como um jogo, usando teoria dos jogos, onde um equilíbrio Nash acontece quando os usuários se comportam de forma egoísta. O custo total ótimo corresponde ao melhor fluxo de um ponto de vista global. Nesta dissertação, uma nova medida de perturbação é apresentada, o Preço da Anarquia Suavizado (Smoothed Price of Anarchy), baseada na análise suavizada de algoritmos, com o fim de analisar os efeitos da perturbação no Preço da Anarquia. Usando esta medida, são estudados os efeitos que perturbações têm no Preço da Anarquia para instâncias reais e teóricas para o Problema de Atribuição de Tráfego. É demonstrado que o Preço da Anarquia Suavizado se mantém na mesma ordem do Preço da Anarquia sem perturbações para funções de latência polinomiais. Finalmente, são estudadas instâncias de benchmark em relação à perturbação. / This thesis analyzes problems in game theory with respect to perturbation. It uses smoothed analysis to accomplish such task and focuses on two kind of games, bimatrix games and the traffic assignment problem. The Lemke-Howson algorithm is a widely used algorithm to compute a Nash equilibrium of a bimatrix game. This problem is PPAD-complete (Polynomial Parity Arguments on Directed graphs), and there exists an instance which takes exponential time (with any starting pivot.) It has been proven that even with a smoothed analysis it is still exponential. However, no experimental study has been done to verify and evaluate in practice how the algorithm behaves in such cases. This thesis shows in detail how the current known worst-case instances are generated and shows that the performance of the algorithm on these instances, when perturbed, differs from the expected behavior proven in theory. The Traffic Assignment Problem models a situation in a road network where users want to travel from an origin to a destination. It can be modeled as a game using game theory, with a Nash equilibrium happening when users behave selfishly and an optimal social welfare being the best possible flow from a global perspective. We provide a new measure, which we call the Smoothed Price of Anarchy, based on the smoothed analysis of algorithms in order to analyze the effects of perturbation on the Price of Anarchy. Using this measure, we analyze the effects that perturbation has on the Price of Anarchy for real and theoretical instances for the Traffic Assignment Problem. We demonstrate that the Smoothed Price of Anarchy remains in the same order as the original Price of Anarchy for polynomial latency functions. Finally, we study benchmark instances in relation to perturbation.

Orientador: Flávio Keidi Miyazawa / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-23T01:59:45Z (GMT). No. of bitstreams: 1 Pereira_ViniciusdeNovaesGuimaraes_M.pdf: 1343382 bytes, checksum: e44e4ecf8abf29e4b44af22979e1269b (MD5) Previous issue date: 2013 / Resumo: Uma das fontes de receita mais lucrativas da internet são os anúncios para sites de busca. O crescimento deste mercado bilionário foi, em média, 20% ao ano nos últimos anos. Como o público alvo e variedade de anunciantes deste mercado são grandes e diversificados, um pequeno aumento da eficiência deste mecanismo representa um grande aumento de receita para os sites. Neste trabalho discutimos a evolução dos mecanismos usados neste mercado, identificando as razões destas mudanças. Avaliamos os mecanismos usados atualmente, modelando-o de formas diferentes e calculando o seu preço da anarquia / Abstract: Sponsored search auction is one of the most profitable sources of revenue on the internet. The growth of this market was, on average, 20% per year over the past years. Since the target audience and advertiser variety are big and diverse, a small increase in efficiency in this mechanism can bring a huge increase in the sites profits. In this work we discuss the evolution of the mechanisms used in this market, identifying the reasons of these changes. We evaluate the currently used mechanism, modeling in different ways and calculating the price of anarchy / Mestrado / Ciência da Computação / Mestre em Ciência da Computação