Most of the cellular tra c (voice and data) is generated indoors. Due to attenuation from walls, quality of service (QoS) of di erent applications degrades for indoor tra c. Thus in order to provide QoS for such users the Macro base station (MBS) has to transmit at high power. This increases recurring costs to the service provider and contributes to green house emissions. Hence, Femtocells (FC) are considered as an option. Femto Access Points (FAP) are low cost, low powered, small base stations deployed indoors by customers. A substantial part of indoor tra c is diverted from the Macrocell (MC) through the FAP. Since the FCs also use the same channels as the MC, deployment of FCs causes interference to not only its neighbouring FCs but also to the users in the MC. Thus, we need better interference management techniques for this system. In this thesis, we consider a system with multiple Femtocells operating in a Macrocell. FCs and MC use same set of multiple channels and support multiple users. Each user may have a minimum rate requirement. To limit interference to the MC, there is a peak power constraint on each channel. In the rst part of the thesis, we consider sparsely deployed FCs where the interference between the FCs is negligible. For this we formulate the problem of channel allocation and power control in each FC. We develop computationally e cient, suboptimal algorithms to satisfy QoS of each user in the FC. If QoS of each user is not satis ed, we provide solutions which are fair to all the users. In the second part of the thesis, we consider the case of densely deployed FCs where we formulate the problem of channel allocation and power control in each Femtocell as a noncooperative Game. We develop e cient decentralized algorithms to obtain a Nash equilibrium (NE) at which QoS of each user is satis ed. We also obtain e cient decentralized algorithms to obtain fair NE when it may not be feasible to satisfy the QoS of all the users in the FC. Finally, we extend our algorithms to the case where there may be voice and data users in the system. In the third part of the thesis, we continue to study the problem setup in the second part, where we develop algorithms which can simultaneously consider the cases where QoS of users can be satis ed or not. We provide algorithms to compute Coarse Correlated Equilibrium (CCE), Pareto optimal points and Nash bargaining solutions. In the nal part of the thesis, we consider interference limit at the MBS and model FCs as sel sh nodes. The MBS protects itself via pricing subchannels per usage. We obtain a Stackelberg equilibrium (SE) by considering MBS as a leader and FCs as followers.

Real estate firms operating in the field of tenant representation may represent both tenants and landlords. As such their interests might be in conflict with those of the tenants they represent. A questionnaire was created and sent to 800 individuals. 78% of respondents indicated that they are aware of such conflicts. Meanwhile, only 40% of the respondents perceived such conflicts to be positive rather than negative. A game theoretical model showing how a tenant hires a tenant representative was created with three different scenarios, all using data from the questionnaire to mimic reality as closely as possible. In the first scenario, a tenant hires a representative at one occasion only. In the second scenario a tenant is considering to hire the representative for two sequential transactions. In the third scenario, a contract is introduced to ensure alignment of incentives between the representative and the tenant. The questionnaire and the modelling shows that if no incentivized contract is written the representative always benefits from engaging in conflicting business and, regardless of the conflict of interest and extra benefit to the representative, the tenant always benefits from hiring the representative. However, introducing a contract to align incentives it is shown that there exists a break even point at which the representative starts to benefit from not engaging in business with conflicting interests. Moreover, there exists little correlation between awareness and perceived effect, involvement, or premises size. / Real estate firms operating in the field of tenant representation may represent both tenants and landlords. As such their interests might be in conflict with those of the tenants they represent. A questionnaire was created and sent to 800 individuals. 78% of respondents indicated that they are aware of such conflicts. Meanwhile, only 40% of the respondents perceived such conflicts to be positive rather than negative. A game theoretical model showing how a tenant hires a tenant representative was created with three different scenarios, all using data from the questionnaire to mimic reality as closely as possible. In the first scenario, a tenant hires a representative at one occasion only. In the second scenario a tenant is considering to hire the representative for two sequential transactions. In the third scenario, a contract is introduced to ensure alignment of incentives between the representative and the tenant. The questionnaire and the modelling shows that if no incentivized contract is written the representative always benefits from engaging in conflicting business and, regardless of the conflict of interest and extra benefit to the representative, the tenant always benefits from hiring the representative. However, introducing a contract to align incentives it is shown that there exists a break even point at which the representative starts to benefit from not engaging in business with conflicting interests. Moreover, there exists little correlation between awareness and perceived effect, involvement, or premises size.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: There are many examples of cooperation in action in society and in nature. In some cases cooperation leads to the increase of the overall welfare of those involved, and in other cases cooperation may be to the detriment of the larger society. The presence of cooperation seems natural if there is a direct bene t to individuals who choose to cooperate. However, in examples of cooperation this bene t is not always immediately obvious. The so called prisoner's dilemma is often used as an analogy to study cooperation and tease out the factors that lead to cooperation. In classical game theory, each player is assumed to be rational and hence typically seeks to select his strategy in such a way as to maximise his own expected pay-o . In the case of the classical prisoner's dilemma, this causes both players to defect. In evolutionary game theory, on the other hand, it is assumed that players have limited knowledge of the game and only bounded rationality. Games in evolutionary game theory are repeated in rounds and players are a orded the opportunity to adapt and learn as this repetition occurs. Past studies have revealed that cooperation may be a viable strategy if the prisoner's dilemma is placed in an evolutionary context, where the evolutionary tness of a strategy is directly related to the pay-o achieved by the player adopting the strategy. One of the mechanisms that promote the persistence of cooperation in the evolutionary prisoner's dilemma is structured interaction between players. A mathematical framework for representing the evolutionary prisoner's dilemma (ESPD) is developed in this thesis. The mathematical framework is used to undertake an analytical approach (i.e. avoiding the use of simulation) towards investigating the dynamics of the ESPD with a path, cycle, plane grid or toroidal grid as underlying graph. The objective of this investigation is to determine the likelihood of the emergence of persistent cooperation between players. The ESPD on a path or a cycle admits two fundamentally di erent parameter regions; large values of the temptation-to-defect parameter are not capable of inducing persistent cooperation, while small values of this parameter allow for the possibility of persistent cooperation. It is found that the likelihood of cooperation increases towards certainty as the order of the underlying graph increases if the underlying graph is a path or cycle. The state space of the ESPD with a plane or toroidal grid graph as underlying graph grows very quickly as a function of the graph order. The automorphism classes of game states are enumerated to determine exactly how fast the size of the state space of the game grows as a function of the order of the underlying graph. Finally, the dynamics of the ESPD is investigated for a grid graph as underlying graph (in cases where the state space is small enough) by means of constructing the corresponding state graphs of the ESPD. / AFRIKAANSE OPSOMMING: Daar is baie voorbeelde van samewerking in the gemeenskap en in die natuur. In sommige gevalle lei samewerking tot 'n toename in die algehele welvaart van die betrokkenes, terwyl samewerking in ander gevalle tot nadeel van die bre er gemeenskap mag wees. Die voorkoms van samewerking blyk natuurlik te wees indien daar 'n direkte voordeel vir die individue is wat kies om saam te werk. In voorbeelde van samewerking is s o 'n voordeel egter nie altyd voor-diehand- liggend nie. Die sogenaamde prisoniersdilemma word dikwels as voorbeeld in die studie van samewerking gebruik om die faktore wat na samewerking lei, te ontbloot. In klassieke speleteorie word daar aangeneem dat elke speler rasioneel is en dus poog om sy spelstrategie op s o 'n manier te kies dat sy eie verwagte uitbetaling gemaksimeer word. In die geval van die klassieke prisoniersdilemma veroorsaak dit dat beide spelers mekaar verraai. In evolusion^ere speleteorie, daarenteen, word daar slegs aangeneem dat elke speler oor beperkte kennis van die spel en begrensde rasionaliteit beskik. Spele in evolusion^ere speleteorie word in rondtes herhaal en spelers word die geleentheid gebied om gedurende hierdie herhalingsproses aan te pas en te leer. Vorige studies het getoon dat samewerking 'n lewensvatbare strategie is indien die prisoniersdilemma in 'n evolusion^ere konteks gespeel word, waar die evolusion^ere ksheid van 'n strategie direk afhang van die uitbetaling van 'n speler wat die strategie volg. Een van die meganismes wat volhoubare samewerking in die evolusion^ere prisoniersdilemma voortbring, is gestruktureerde interaksie tussen spelers. 'n Wiskundige raamwerk word vir die voorstelling van die evolusion^ere prisoniersdilemma in hierdie tesis ontwikkel. Hierdie wiskundige raamwerk word gebruik om 'n analitiese studie (met ander woorde sonder die gebruik van simulasie) van die dinamika van die prisoniersdilemma op 'n pad, siklus, rooster in die vlak, of rooster op die torus as onderliggende gra ek van stapel te stuur. Die doel van hierdie studie is om die waarskynlikheid vir die ontstaan van volhoubare samewerking tussen spelers te bepaal. Die prisoniersdilemma op 'n pad of siklus as onderliggende gra ek het twee fundamenteel verskillende parametergebiede tot gevolg; groot waardes van die versoeking-om-te-verraai parameter lei nie tot volhoubare samewerking nie, terwyl volhoubare samewerking wel vir klein waardes van hierdie parameter moontlik is. Daar word gevind dat die kans vir volhoubare samewerking toeneem tot sekerheid namate die orde van die onderliggende gra ek groei. Die toestandsruimte van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek groei baie vinnig as 'n funksie van die orde van die gra ek. Die outomor smeklasse van die speltoestande word getel met die doel om te bepaal presies hoe vinnig die toestandsruimte van die spel as 'n funksie van die orde van die onderliggende gra ek groei. Die dinamika van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek word laastens deur middel van konstruksies van die ooreenstemmende toestandsgra eke ondersoek (in gevalle waar die toestandsruimte klein genoeg is).

In this work we present a general duality-theory framework for revenue maximization in additive Bayesian auctions involving multiple items and many bidders whose values for the goods follow arbitrary continuous joint distributions over some multi-dimensional real interval. Although the single-item case has been resolved in a very elegant way by the seminal work of Myerson [1981], optimal solutions involving more items still remain elusive. The framework extends linear programming duality and complementarity to constraints with partial derivatives. The dual system reveals the natural geometric nature of the problem and highlights its connection with the theory of bipartite graph matchings. We demonstrate the power of the framework by applying it to various special monopoly settings where a seller of multiple heterogeneous goods faces a buyer with independent item values drawn from various distributions of interest, to design both exact and approximately optimal selling mechanisms. Previous optimal solutions were only known for up to two and three goods, and a very limited range of distributional priors. The duality framework is used not only for proving optimality, but perhaps more importantly, for deriving the optimal mechanisms themselves. Some of our main results include: the proposal of a simple deterministic mechanism, which we call Straight-Jacket Auction (SJA) and is defined in a greedy, recursive way through natural geometric constraints, for many uniformly distributed goods, where exact optimality is proven for up to six items and general optimality is conjectured; a scheme of sufficient conditions for exact optimality for two-good settings and general independent distributions; a technique for upper-bounding the optimal revenue for arbitrarily many goods, with an application to uniform and exponential priors; and the proof that offering deterministically all items in a single full bundle is the optimal way of selling multiple exponentially i.i.d. items.

In this thesis we study decentralized dynamics for non-cooperative and cooperative games. The dynamics are behaviorally motivated and assume that very little information is available about other players' preferences, actions, or payoffs. For example, this is the case in markets where exchanges are frequent and the sheer size of the market hinders participants from learning about others' preferences. We consider learning dynamics that are based on trial-and-error and aspiration-based heuristics. Players occasionally try to increase their performance given their current payoffs. If successful they stick to the new action, otherwise they revert to their old action. We also study a dynamic model of social influence based on findings in sociology and psychology that people have a propensity to conform to others' behavior irrespective of the payoff consequences. We analyze the dynamics with a particular focus on two questions: How long does it take to reach equilibrium and what are the stability and welfare properties of the equilibria that the process selects? These questions are at the core of understanding which equilibrium concepts are robust in environments where players have little information about the game and the high rationality assumptions of standard game theory are not very realistic. Methodologically, this thesis builds on game theoretic techniques and prominent solution concepts such as the Nash equilibrium for non-cooperative games and the core for cooperative games, as well as refinement concepts like stochastic stability. The proofs rely on mathematical techniques from random walk theory and integer programming.

Esta dissertação aborda a teoria dos jogos diferenciais em sua estreita relação com a teoria das equações de Hamilton-Jacobi (HJ). Inicialmente, uma revisão da noção de solução em teoria dos jogos é empreendida. Discutem-se nesta ocasião as idéias de equilíbrio de Nash e alguns de seus refinamentos. Em seguida, tem lugar uma introdução à teoria dos jogos diferenciais, onde noções de solução como a função de valor de Isaacs e de Friedman são discutidas. É nesta altura do trabalho que fica evidente a conexão entre este conceito de solução e a teoria das equações de Hamilton-Jacobi. Por ocasião desta conexão, é explorada a noção de solução clássica e é exposta uma demonstração do fato de que se um jogo diferencial possuir uma função de valor pelo menos continuamente diferenciável, esta será uma solução da equação de Hamilton-Jacobi associada ao jogo. Este resultado faz uso do princípio da programação dinâmica, devido a Bellman, e cuja demonstração está presente no texto. No entanto, quando a função de valor do jogo é apenas contínua, então embora esta não seja uma solução clássica da equação HJ associada a jogo, vemos que ela será uma solução viscosa, ou solução no sentido da viscosidade - e a esta altura são discutidos os elementos e propriedades desta classe de soluções, um teorema de existência e unicidade e alguns exemplos. Por fim, retomamos o estudo dos jogos diferenciais à luz das soluções viscosas da equação de Hamilton-Jacobi e, assim, expomos uma demonstração de existência da função de valor e do princípio da programação dinâmica a partir das noções da viscosidade / This dissertation aims to address the topic of Differential Game Theory in its connection with the Hamilton-Jacobi (HJ) equations framework. Firstly we introduce the idea of solution for a game, through the discussion of Nash equilibria and its refinements. Secondly, the solution concept is then translated to the context of Differential Games and the idea of value function is introduced in its Isaacs\'s as well as Friedman\'s version. As the value function is discussed, its relationship with the Hamilton-Jacobi equations theory becomes self-evident. Due to such relation, we investigate the HJ equation from two distinct points of view. First of all, we discuss a statement according to which if a differential game has a continuously differentiable value function, then such function is a classical solution of the HJ equation associated to the game. This result strongly relies on Bellman\'s Dynamic Programming Principle - and this is the reason why we devote an entire chapter to this theme. Furthermore, HJ is still at our sight from the PDE point of view. Our motivation is simple: under some lack of regularity - a value function which is continuous, but not continuously differentiable - a game may still have a value function represented as a solution of the associated HJ equation. In this case such a solution will be called a solution in the viscosity sense. We then discuss the properties of viscosity solutions as well as provide an existence and uniqueness theorem. Finally we turn our attention back to the theory of games and - through the notion of viscosity - establish the existence and uniqueness of value functions for a differential game within viscosity solution theory.

The interaction of theoretical computer science with game theory and economics has resulted in the emergence of two very interesting research directions. First, it has provided a new model for algorithm design, which is to optimize in the presence of strategic behavior. Second, it has prompted us to consider the computational aspects of various solution concepts from game theory, economics and auction design which have traditionally been considered mainly in a non-constructive manner. In this thesis we present progress along both these directions. We first consider optimization problems that arise in the design of combinatorial auctions. We provide an online algorithm in the important case of budget-bounded utilities. This model is motivated by the recent development of the business of online auctions of search engine advertisements. Our algorithm achieves a factor of $1-1/e$, via a new linear programming based technique to determine optimal tradeoffs between bids and budgets. We also provide lower bounds in terms of hardness of approximation in more general submodular settings, via a PCP-based reduction. Second, we consider truth-revelation in auctions, and provide an equivalence theorem between two notions of strategy-proofness in randomized auctions of digital goods. Last, we consider the problem of computing an approximate Nash equilibrium in multi-player general-sum games, for which we provide the first subexponential time algorithm.

Distributed learning is the iterative process of decision-making in the presence of other decision-makers. In recent years, researchers across fields as disparate as engineering, biology, and economics have identified mathematically congruous problem formulations at the intersection of their disciplines. In particular, stochastic processes, game theory, and control theory have been brought to bare on certain very basic and universal questions. What sort of environments are conducive to distributed learning? Are there any generic algorithms offering non-trivial performance guarantees for a large class of models? The first half of this thesis makes contributions to two particular problems in distributed learning, self-assembly and language. Self-assembly refers to the emergence of high-level structures via the aggregate behavior of simpler building blocks. A number of algorithms have been suggested that are capable of generic self-assembly of graphs. That is, given a description of the objective they produce a policy with a corresponding performance guarantee. These guarantees have been in the form of deterministic convergence results. We introduce the notion of stochastic stability to the self-assembly problem. The stochastically stable states are the configurations the system spends almost all of its time in as a noise parameter is taken to zero. We show that in this framework simple procedures exist that are capable of self-assembly of any tree under stringent locality constraints. Our procedure gives an asymptotically maximum yield of target assemblies while obeying communication and reversibility constraints. We also present a slightly more sophisticated algorithm that guarantees maximum yields for any problem size. The latter algorithm utilizes a somewhat more presumptive notion of agents' internal states. While it is unknown whether an algorithm providing maximum yields subject to our constraints can depend only on the more parsimonious form of internal state, we are able to show that such an algorithm would not be able to possess a unique completing rule--- a useful feature for analysis. We then turn our attention to the problem of distributed learning of communication protocols, or, language. Recent results for signaling game models establish the non-negligible possibility of convergence, under distributed learning, to states of unbounded efficiency loss. We provide a tight lower bound on efficiency and discuss its implications. Moreover, motivated by the empirical phenomenon of linguistic drift, we study the signaling game under stochastic evolutionary dynamics. We again make use of stochastic stability analysis and show that the long-run distribution of states has support limited to the efficient communication systems. We find that this behavior is insensitive to the particular choice of evolutionary dynamic, a fact that is intuitively captured by the game's potential function corresponding to average fitness. Consequently, the model supports conclusions similar to those found in the literature on language competition. That is, we expect monomorphic language states to eventually predominate. Homophily has been identified as a feature that potentially stabilizes diverse linguistic communities. We find that incorporating homophily in our stochastic model gives mixed results. While the monomorphic prediction holds in the small noise limit, diversity can persist at higher noise levels or as a metastable phenomenon. The contributions of the second half of this thesis relate to more basic issues in distributed learning. In particular, we provide new results on the problem of distributed convergence to Nash equilibrium in finite games. A recently proposed class of games known as stable games have the attractive property of admitting global convergence to equilibria under many learning dynamics. We show that stable games can be formulated as passive input-output systems. This observation enables us to identify passivity of a learning dynamic as a sufficient condition for global convergence in stable games. Notably, dynamics satisfying our condition need not exhibit positive correlation between the payoffs and their directions of motion. We show that our condition is satisfied by the dynamics known to exhibit global convergence in stable games. We give a decision-theoretic interpretation for passive learning dynamics that mirrors the interpretation of stable games as strategic environments exhibiting self-defeating externalities. Moreover, we exploit the flexibility of the passivity condition to study the impact of applying various forecasting heuristics to the payoffs used in the learning process. Finally, we show how passivity can be used to identify strategic tendencies of the players that allow for convergence in the presence of information lags of arbitrary duration in some games.

Η παρούσα διπλωματική εργασία πραγματεύεται την θεωρία παιγνίων και το πώς αυτή εφαρμόζεται στην οικονομική επιστήμη. Συγκεκριμένα, στόχος μας είναι να απαντήσουμε στο ερώτημα: «Πως αποφασίζονται οι τελικές στρατηγικές που θα επικρατήσουν σε ένα παίγνιο με την πάροδο του χρόνου;». Η εργασία είναι χωρισμένη σε δύο μέρη. Αρχικά αναφερόμαστε στην κλασσική θεωρία παιγνίων και αναλύουμε τα βασικά της στοιχεία και στη συνέχεια περνάμε στην ανάλυση της εξελικτικής θεωρίας παιγνίων. Στο 1ο μέρος της παρούσας εργασίας, λοιπόν, αναφέρουμε τα όσα είναι σχετικά με την κλασσικά θεωρία παιγνίων. Συγκεκριμένα, στο πρώτο κεφάλαιο γίνεται μία σύντομη ιστορική αναδρομή της θεωρίας αυτής και στο δεύτερο κεφάλαιο την ορίζουμε ως την επίσημη μελέτη που εξετάζει την ορθολογικότητα σε ένα επιχειρηματικό περιβάλλον και παρουσιάζουμε τα βασικά στοιχεία ενός παιγνίου. Αναφέρουμε τα δύο επίπεδα περιγραφής των παιγνίων, δηλαδή τα παίγνια συνεργασίας και μη-συνεργασίας, καθώς και τους δύο τρόπους αναπαράστασής τους που είναι η στρατηγική ή αλλιώς κανονική μορφή (μήτρες) και η εκτεταμένη ή αλλιώς αναλυτική μορφή (δέντρα παιγνίων). Στο τρίτο κεφάλαιο ορίζονται οι κυρίαρχες στρατηγικές και η αντίστοιχη ισορροπία κυρίαρχης στρατηγικής και στο τέταρτο κεφάλαιο ορίζεται η Ισορροπία Nash, η οποία αποτελεί τη στάνταρ έννοια της ισορροπίας στα οικονομικά. Στα δύο αυτά κεφάλαια (3 και 4) υπάρχουν παραδείγματα εφαρμογής που στοχεύουν στην καλύτερη κατανόηση, και αναλύεται και το Δίλημμα του Φυλακισμένου που αποτελεί το πιο κλασσικό παράδειγμα στη θεωρία παιγνίων. Στην περίπτωση, τώρα, όπου δεν υπάρχει Ισορροπία Nash (κάτι το οποίο συμβαίνει σε παίγνια στρατηγικής μορφής) το παίγνιο λύνεται με τη βοήθεια των μικτών στρατηγικών οι οποίες αναλύονται στο πέμπτο κεφάλαιο. Συνεχίζουμε με το έκτο κεφάλαιο, όπου παρουσιάζονται τα εκτεταμένα παίγνια πλήρους πληροφόρησης και αναλύεται η μέθοδος της προς τα πίσω επαγωγής (αναδίπλωση). Στο έβδομο κεφάλαιο παρουσιάζονται τα παίγνια ελλιπούς πληροφόρησης και στο όγδοο κεφάλαιο αναφέρονται τα παίγνια μηδενικού αθροίσματος (π.χ. σκάκι) και το πώς μπορούν να χρησιμοποιηθούν μαζί με τους τυχαιοποιημένους αλγόριθμους για την ανάλυση προβλημάτων στον απευθείας σύνδεσης υπολογισμό. Το 1ο μέρος κλείνει με ένα παράδειγμα εφαρμογής της θεωρίας παιγνίων, τις δημοπρασίες. Τι γίνεται όμως όταν ένα παίγνιο επαναλαμβάνεται και παίζεται περισσότερες από μία φορές; Το ερώτημα αυτό έρχεται να μας το απαντήσει η εξελικτική θεωρία παιγνίων στο 2ο μέρος της παρούσας διπλωματικής εργασίας. Στα δύο πρώτα κεφάλαια, του μέρους αυτού, ορίζονται τα εξελικτικά παίγνια, γίνεται αναφορά για το που μπορούν να βρουν εφαρμογή καθώς και στους λόγους που δεν είναι ακόμη γνωστές οι οικονομικές εφαρμογές τους. Το τρίτο και το πέμπτο κεφάλαιο αποτελούν τα πιο σημαντικά κεφάλαιο του 2ου μέρους. Στο τρίτο κεφάλαιο παρουσιάζεται αναλυτικά το μοντέλο των εξελικτικών παιγνίων και τα στοιχεία που το αποτελούν (αναμενόμενες ανταμοιβές, πληθυσμός, καταστάσεις). Περιγράφεται το στάδιο παιγνίου το οποίο ορίζεται από μία συνάρτηση καταλληλότητας και δίνεται έμφαση στις δύο γραμμικές προδιαγραφές που έχουν οι συναρτήσεις αυτές. Στη συνέχεια, αναλύεται πλήρως το πιο αντιπροσωπευτικό παράδειγμα της εξελικτικής θεωρίας παιγνίων, το παίγνιο Hawk-Dove, που αποτελεί ένα γενικό μοντέλο καταστάσεων με επιθετικές και αμυντικές αγορές. Το παίγνιο αυτό έχει δύο ειδών παίκτες, αυτοί που επιλέγουν να είναι επιθετικοί (Hawk) και αυτοί που επιλέγουν να είναι αμυντικοί (Dove), και ερευνάται το ποιο είδος παικτών θα επικρατήσει τελικά. Μέσα από την διαφορική εξίσωση που αναλύεται στο πέμπτο κεφάλαιο, στις δυναμικές, φαίνεται πως το αποτέλεσμα εξαρτάται από τρεις παραμέτρους: από τον αρχικό πληθυσμό, από την πιθανότητα να παιχτεί η καθεμία στρατηγική και από τον πίνακα με τις ανταμοιβές των παικτών. Έτσι απαντάται το αρχικό μας ερώτημα και προκύπτει η στρατηγική που τελικά θα επικρατήσει, που ονομάζεται εξελικτική στρατηγική (evolutionary stable strategy-ESS). Στο τέταρτο κεφάλαιο ορίζεται η Ισορροπία Nash (I.N.), η Εξελικτική Σταθερή Στρατηγική (ESS) και η Εξελικτική Ισορροπία (E.E.) και στο έκτο κεφάλαιο αναφέρουμε την τοπική κατάταξη συστημάτων με χαμηλές διαστάσεις και συγκεκριμένα τα γραμμικά παίγνια μίας-διάστασης, τα συστήματα δύο μεταβλητών και άλλα συστήματα δύο-διαστάσεων και μη-γραμμικά. Κλείνοντας το 2ο μέρος και γενικά την παρούσα εργασία, παρουσιάζουμε τρία παραδείγματα στα οποία φαίνεται η εφαρμοσιμότητα των όσων αναφέραμε. Συγκεκριμένα αναλύονται τρία γνωστά παίγνια τα οποία χρησιμοποιήθηκαν από την πολιτική και παραλληλίστηκαν με καταστάσεις που είχαν να αντιμετωπίσουν εκείνη τη στιγμή. / This thesis deals with the evolutionary game theory and how it applies to economics. First of all, it is necessary to refer the original game theory and to analyze the key elements and then move to the analysis of evolutionary game theory. In the first part of this study, therefore, we indicate what is on game theory. Specifically, in the first chapter is a brief history of game theory and the second chapter defined game theory as a formal study examining the rationality in a business environment and presents the basics elements of a game. Also, at this chapter i reffer to the description of games, namely, games of cooperation and non-cooperation, and the two ways of representing their strategy, the normal form (matrix) and the extensive form (game tree). The third chapter sets out the dominant strategy and the corresponding dominant strategy equilibrium and in the fourth chapter we define the Nash Equilimbrium, which is the standard notion of equilibrium in economics. In these two chapters (third and fourth) there are examples of the application for better understanding, and we analyze the prisoner's dilemma, which is the most classic example of game theory. If there is no Nash Equilimbrium (this could happen at narmal strategy games) the game is solved by mixed strategies, which are analyzed in the fifth chapter. Continuing, at the sixth chapter we can see the extensive games with perfect information and we analyze the method of backward induction. In the seventh chapter, we can see the extensive games with imperfect information and the eighth chapter refers to the zero-sum games and how they can be used together with randomized algorithms for the analysis of problems on-line calculation. Finally, the first part closes with an example application of game theory, the auctions. The question is “What happens when a game is played more than once?” The answer comes from the second part of this thesis in which we analyse the evolutionary game theory. In the first two chapters of this part we define evolutionary games, we refere where evolutionary games might be applicable and why economic application aren’t common already. The third and fourth chapter are the most important chapters of the second part. At the third chapter we present the model of the evolutionary game and its elements (expected payoffs, population, states). We describe the stage game which is defined by a fitness function and we emphasize at its two linear specifications. Then we make a full analysis one of the most representative example of evolutionary game theory, the Hawk-Dove game. This game has two types of players, aggressive (Hawk) and defensive (Dove), which reflects the situation where there is a competitive and an uncompetitive business, and the point is to find which of the two types will eventually prevail. Based on a differential equation, we conclude that the result depends on three parameters: the initial population, the probability with which each strategy is played and the payoff matrix. All this leads in a strategy which is known as evolutionary stable (ESS). In chapter five, we define the Nash Equilibrium, the Evolutionary Stable Strategy (ESS) and Evolutionary Equilibrium (EE) and in chapter six we analyze the local classification of low dimensions systems. To make clear the applicability of all those we mention at this thesis, we are closing with three examples. More specific we analyze three well-known games which were used by the political and paralleled with situations they had to face with.

In this thesis we analyze the performance of communication networks using game theoretic approaches. The thesis is in two parts. The first part studies the performance of Ad hoc, cellular and transportation networks taking into consideration spatial effects. The second part studies economic issues in the communication networks related to the 'net neutrality' regulations. Here we study price competition and revenue sharing mechanisms between the network service providers. In the first part, we use Medium Access Control (MAC) game and Jamming game models to study the performance of a Mobile Ad hoc NETwork (MANET), and routing games to study the performance of a transportation network. In the cellular networks, we study the effect of reducing the spatial density of base stations on the amount of radiation to human body (green networking). We use tools from stochastic geometry to model spatial characters. We begin with the study of MAC game in a MANET where the nodes are noncooperative, and their locations form a Poisson point process. The nodes use Aloha protocol to access the channel, and choose their Medium Access Probability (MAP) such that it optimizes their utility. The utility of each node is defined as weighted difference between a performance metric (expected goodput or expected delay) and transmission costs. We first consider a scenario in which nodes can be priced for channel access. We show that by using a simple (linear) pricing mechanism and setting the price 'appropriately', the selfish behavior of the nodes can be used to achieve the same performance as social optima at equilibrium. In the case where channel access is free, we consider transmission energy costs and analyze the Price of Anarchy (PoA). For the game with goodput based utility, we show that the PoA is infinite at the price that achieves the global optimal goodput. We then study the performance of MANET in the presence of a Jammer while all the nodes cooperate by using a MAP that is assigned by a network an Operator. The objective of the Jammer is to degrade the spatial performance of the MANET by causing interference, whereas the objective of the Operator is to set a MAP to optimize it. We model the interaction between the Jammer and the Operator taking into account the transmission energy costs. We transform the resulting non zero sum game into a zero sum game by constructing an anti-potential, and show that for a given Jammer's transmission cost, if the nodes transmit at a power higher than certain threshold, then the Jammer has no incentive to jam, i.e., the nodes can operate as if there is no Jammer. Next, we consider cellular networks. We study energy saving by switching off a fraction of the base stations (BSs). This saving comes at some cost: the coverage is reduced, and the uplink transmission power of mobiles may increase. This implies exposure of the human body to stronger electromagnetic fields. We quantify this through the deactivation of base stations under the assumptions that the random location of base stations and mobiles form a Poisson processes. We consider the case of sparse network where the interference is negligible, and the case where it is non negligible. We observe that when the mobiles have no power constraints, unlike in the case of negligible interference, switching off base stations reduces the uplink power. Finally, we study a dynamic routing game by adding temporal dimension to the classical routing problem. We consider a scenario where each user have to ship a fixed demand on a shared link. The users can delay shipping their demand, but need to ship within T days, and each time they delay, a delay cost is incurred. We study the effect of delay costs on the user strategies by translating the time dimension into space dimension. Considering both atomic and non-atomic game models and polynomial congestion costs, we show that there exists a threshold time T T before which all users ship their demand at equilibrium, and during this period total demand on the link is decreasing in time. Further, we extend the analysis to the case when the demand of each user arrives randomly. In the second part of the thesis, we study economics aspects in communication networks. Representatives of several Internet access providers (ISPs) have expressed their wish to see a substantial change in the pricing policies in the Internet. In particular, they would like to see content providers (CPs) pay for use of the network, given the large amount of resources they use. This would be in clear violation of the "network neutrality" principle that had characterized the development of the wireline Internet. Our first goal here is to propose and study possible ways of implementing payments in a nonneutral network. We introduce a model that includes the users' behavior, the utilities of the ISP and of the CPs, and the monetary flow that involves the users, an ISP and CPs, and CP's revenues from advertisements. We consider various game models and obtain the resulting equilibrium prices. We show that if a regulator decides the amount of money that CPs pays to the ISP, rather it is decided by the ISP or the CPs, then it results in a favorable situation for all the players. Thus, we demonstrate the necessary for regulation of the monetary transactions between the ISP and the CPs in the nonneutral regime. The mechanism we propose for monetary interaction is based on the Nash bargaining solution. One of the central issues in the debate on network neutrality has been whether one should allow or prevent preferential treatment by an ISP. This raised the question of whether to allow an ISP to have exclusive agreement with a content provider (CP). We study the impact of exclusive contracts between a CP and an ISP in a nonneutral network. We consider a simple case of collusion where a CP and an ISP aim to maximize their sum of revenues, and show that such a collusion may not always be beneficial. We derive an explicit condition in terms of the advertisement revenues of the CPs which tells when a collusion is profitable to the colluding entities. Finally, we consider discrimination in the opposite direction, i.e., CP discriminating the ISPs. We derive models for studying the impact that the CP can have on the utilities of the ISPs by favoring one of them through exclusively revealing its private information on the demand. For a special case of linear demand function, we show that such preferential treatment always benefits the ISPs. However, if the CP charges the ISPs for providing the private information, then the ISP obtaining preferential treatment may not always gain. We then propose mechanisms based on weighted proportional fairness for deciding the payments between the CP and the ISP and compare them by proposing a new metric termed Price of Partial Bargaining.