Cette thèse traite les jeux différentiels stochastiques de somme non nulle (JDSNN) dans le cadre de Markovien et de leurs liens avec les équations différentielles stochastiques rétrogrades (EDSR) multidimensionnelles. Nous étudions trois problèmes différents. Tout d'abord, nous considérons un JDSNN où le coefficient de dérive n'est pas borné, mais supposé uniquement à croissance linéaire. Ensuite certains cas particuliers de coefficients de diffusion non bornés sont aussi considérés. Nous montrons que le jeu admet un point d'équilibre de Nash via la preuve de l'existence de la solution de l'EDSR associée et lorsque la condition d'Isaacs généralisée est satisfaite. La nouveauté est que le générateur de l'EDSR, qui est multidimensionnelle, est de croissance linéaire stochastique par rapport au processus de volatilité. Le deuxième problème est aussi relatif au JDSNN mais les payoffs ont des fonctions d'utilité exponentielles. Les EDSRs associées à ce jeu sont de type multidimensionnelles et quadratiques en la volatilité. Nous montrons de nouveau l'existence d’un équilibre de Nash. Le dernier problème que nous traitons, est un jeu bang-bang qui conduit à des hamiltoniens discontinus. Dans ce cas, nous reformulons le théorème de vérification et nous montrons l’existence d’un équilibre de Nash qui est du type bang-bang, i.e., prenant ses valeurs sur le bord du domaine en fonction du signe de la dérivée de la fonction valeur ou du processus de volatilité. L'EDSR dans ce cas est un système multidimensionnel couplé, dont le générateur est discontinu par rapport au processus de volatilité. / This dissertation studies the multiple players nonzero-sum stochastic differential games (NZSDG) in the Markovian framework and their connections with multiple dimensional backward stochastic differential equations (BSDEs). There are three problems that we are focused on. Firstly, we consider a NZSDG where the drift coefficient is not bound but is of linear growth. Some particular cases of unbounded diffusion coefficient of the diffusion process are also considered. The existence of Nash equilibrium point is proved under the generalized Isaacs condition via the existence of the solution of the associated BSDE. The novelty is that the generator of the BSDE is multiple dimensional, continuous and of stochastic linear growth with respect to the volatility process. The second problem is of risk-sensitive type, i.e. the payoffs integrate utility exponential functions, and the drift of the diffusion is unbounded. The associated BSDE is of multi-dimension whose generator is quadratic on the volatility. Once again we show the existence of Nash equilibria via the solution of the BSDE. The last problem that we treat is a bang-bang game which leads to discontinuous Hamiltonians. We reformulate the verification theorem and we show the existence of a Nash point for the game which is of bang-bang type, i.e., it takes its values in the border of the domain according to the sign of the derivatives of the value function. The BSDE in this case is a coupled multi-dimensional system, whose generator is discontinuous on the volatility process.

Dans cette thèse, nous nous intéressons à la modélisation de mouvements de foules. Nous proposons un modèle microscopique basé sur la théorie des jeux. Chaque individu a une certaine vitesse souhaitée, celle qu'il adopterait en l'absence des autres. Une personne est influencée par certains de ses voisins, pratiquement ceux qu'elle voit devant elle. Une vitesse réelle est considérée comme possible si elle réalise un équilibre de Nash instantané: chaque individu fait son mieux par rapport à un objectif personnel (vitesse souhaitée), en tenant compte du comportement des voisins qui l'influencent. Nous abordons des questions relatives à la modélisation ainsi que les aspects théoriques du problème dans diverses situations, en particulier dans le cas où chaque individu est influencé par tous les autres, et le cas où les relations d'influence entre les individus présentent une structure hiérarchique. Un schéma numérique est développé pour résoudre le problème dans le second cas (modèle hiérarchique) et des simulations numériques sont proposées pour illustrer le comportement du modèle. Les résultats numériques sont confrontés avec des expériences réelles de mouvements de foules pour montrer la capacité du modèle à reproduire certains effets.Nous proposons une version macroscopique du modèle hiérarchique en utilisant les mêmes principes de modélisation au niveau macroscopique, et nous présentons une étude préliminaire des difficultés posées par cette approche.La dernière problématique qu'on aborde dans cette thèse est liée aux cadres flot gradient dans les espaces de Wasserstein aux niveaux continu et discret. Il est connu que l'équation de Fokker-Planck peut s'interpréter comme un flot gradient pour la distance de Wasserstein continue. Nous établissons un lien entre une discrétisation spatiale du type Volume Finis pour l'équation de Fokker-Planck sur une tesselation de Voronoï et les flots gradient sur le réseau sous-jacent, pour une distance de type Wasserstein récemment introduite sur l'espace de mesures portées par les sommets d'un réseaux. / We are interested in the modeling of crowd motion. We propose a microscopic model based on game theoretic principles. Each individual is supposed to have a desired velocity, it is the one he would like to have in the absence of others. We consider that each individual is influenced by some of his neighbors, practically the ones that he sees. A possible actual velocity is an instantaneous Nash equilibrium: each individual does its best with respect to a personal objective (desired velocity), considering the behavior of the neighbors that influence him. We address theoretical and modeling issues in various situations, in particular when each individual is influenced by all the others, and in the case where the influence relations between individuals are hierarchical. We develop a numerical strategy to solve the problem in the second case (hierarchical model) and propose numerical simulations to illustrate the behavior of the model. We confront our numerical results with real experiments and prove the ability of the hierarchical model to reproduce some phenomena.We also propose to write a macroscopic counterpart of the hierarchical model by translating the same modeling principles to the macroscopic level and make the first steps towards writing such model.The last problem tackled in this thesis is related to gradient flow frameworks in the continuous and discrete Wasserstein spaces. It is known that the Fokker-Planck equation can be interpreted as a gradient flow for the continuous Wasserstein distance. We establish a link between some space discretization strategies of the Finite Volume type for the Fokker- Planck equation in general meshes (Voronoï tesselations) and gradient flows on the underlying networks of cells, in the framework of discrete Wasserstein-like distance on graphs recently introduced.

Multi-Agent Planning (MAP) is a topic of growing interest that deals with the problem of automated planning in domains where multiple agents plan and act together in a shared environment. In most cases, agents in MAP are cooperative (altruistic) and work together towards a collaborative solution. However, when rational self-interested agents are involved in a MAP task, the ultimate objective is to find a joint plan that accomplishes the agents' local tasks while satisfying their private interests. Among the MAP scenarios that involve self-interested agents, non-cooperative MAP refers to problems where non-strictly competitive agents feature common and conflicting interests. In this setting, conflicts arise when self-interested agents put their plans together and the resulting combination renders some of the plans non-executable, which implies a utility loss for the affected agents. Each participant wishes to execute its plan as it was conceived, but congestion issues and conflicts among the actions of the different plans compel agents to find a coordinated stable solution. Non-cooperative MAP tasks are tackled through non-cooperative games, which aim at finding a stable (equilibrium) joint plan that ensures the agents' plans are executable (by addressing planning conflicts) while accounting for their private interests as much as possible. Although this paradigm reflects many real-life problems, there is a lack of computational approaches to non-cooperative MAP in the literature. This PhD thesis pursues the application of non-cooperative games to solve non-cooperative MAP tasks that feature rational self-interested agents. Each agent calculates a plan that attains its individual planning task, and subsequently, the participants try to execute their plans in a shared environment. We tackle non-cooperative MAP from a twofold perspective. On the one hand, we focus on agents' satisfaction by studying desirable properties of stable solutions, such as optimality and fairness. On the other hand, we look for a combination of MAP and game-theoretic techniques capable of efficiently computing stable joint plans while minimizing the computational complexity of this combined task. Additionally, we consider planning conflicts and congestion issues in the agents' utility functions, which results in a more realistic approach. To the best of our knowledge, this PhD thesis opens up a new research line in non-cooperative MAP and establishes the basic principles to attain the problem of synthesizing stable joint plans for self-interested planning agents through the combination of game theory and automated planning. / La Planificación Multi-Agente (PMA) es un tema de creciente interés que trata el problema de la planificación automática en dominios donde múltiples agentes planifican y actúan en un entorno compartido. En la mayoría de casos, los agentes en PMA son cooperativos (altruistas) y trabajan juntos para obtener una solución colaborativa. Sin embargo, cuando los agentes involucrados en una tarea de PMA son racionales y auto-interesados, el objetivo último es obtener un plan conjunto que resuelva las tareas locales de los agentes y satisfaga sus intereses privados. De entre los distintos escenarios de PMA que involucran agentes auto-interesados, la PMA no cooperativa se centra en problemas que presentan un conjunto de agentes no estrictamente competitivos con intereses comunes y conflictivos. En este contexto, pueden surgir conflictos cuando los agentes ponen en común sus planes y la combinación resultante provoca que algunos de estos planes no sean ejecutables, lo que implica una pérdida de utilidad para los agentes afectados. Cada participante desea ejecutar su plan tal como fue concebido, pero las congestiones y conflictos que pueden surgir entre las acciones de los diferentes planes fuerzan a los agentes a obtener una solución estable y coordinada. Las tareas de PMA no cooperativa se abordan a través de juegos no cooperativos, cuyo objetivo es hallar un plan conjunto estable (equilibrio) que asegure que los planes de los agentes sean ejecutables (resolviendo los conflictos de planificación) al tiempo que los agentes satisfacen sus intereses privados en la medida de lo posible. Aunque este paradigma refleja muchos problemas de la vida real, existen pocos enfoques computacionales para PMA no cooperativa en la literatura. Esta tesis doctoral estudia el uso de juegos no cooperativos para resolver tareas de PMA no cooperativa con agentes racionales auto-interesados. Cada agente calcula un plan para su tarea de planificación y posteriormente, los participantes intentan ejecutar sus planes en un entorno compartido. Abordamos la PMA no cooperativa desde una doble perspectiva. Por una parte, nos centramos en la satisfacción de los agentes estudiando las propiedades deseables de soluciones estables, tales como la optimalidad y la justicia. Por otra parte, buscamos una combinación de PMA y técnicas de teoría de juegos capaz de calcular planes conjuntos estables de forma eficiente al tiempo que se minimiza la complejidad computacional de esta tarea combinada. Además, consideramos los conflictos de planificación y congestiones en las funciones de utilidad de los agentes, lo que resulta en un enfoque más realista. Bajo nuestro punto de vista, esta tesis doctoral abre una nueva línea de investigación en PMA no cooperativa y establece los principios básicos para resolver el problema de la generación de planes conjuntos estables para agentes de planificación auto-interesados mediante la combinación de teoría de juegos y planificación automática. / La Planificació Multi-Agent (PMA) és un tema de creixent interès que tracta el problema de la planificació automàtica en dominis on múltiples agents planifiquen i actuen en un entorn compartit. En la majoria de casos, els agents en PMA són cooperatius (altruistes) i treballen junts per obtenir una solució col·laborativa. No obstant això, quan els agents involucrats en una tasca de PMA són racionals i auto-interessats, l'objectiu últim és obtenir un pla conjunt que resolgui les tasques locals dels agents i satisfaci els seus interessos privats. D'entre els diferents escenaris de PMA que involucren agents auto-interessats, la PMA no cooperativa se centra en problemes que presenten un conjunt d'agents no estrictament competitius amb interessos comuns i conflictius. En aquest context, poden sorgir conflictes quan els agents posen en comú els seus plans i la combinació resultant provoca que alguns d'aquests plans no siguin executables, el que implica una pèrdua d'utilitat per als agents afectats. Cada participant vol executar el seu pla tal com va ser concebut, però les congestions i conflictes que poden sorgir entre les accions dels diferents plans forcen els agents a obtenir una solució estable i coordinada. Les tasques de PMA no cooperativa s'aborden a través de jocs no cooperatius, en els quals l'objectiu és trobar un pla conjunt estable (equilibri) que asseguri que els plans dels agents siguin executables (resolent els conflictes de planificació) alhora que els agents satisfan els seus interessos privats en la mesura del possible. Encara que aquest paradigma reflecteix molts problemes de la vida real, hi ha pocs enfocaments computacionals per PMA no cooperativa en la literatura. Aquesta tesi doctoral estudia l'ús de jocs no cooperatius per resoldre tasques de PMA no cooperativa amb agents racionals auto-interessats. Cada agent calcula un pla per a la seva tasca de planificació i posteriorment, els participants intenten executar els seus plans en un entorn compartit. Abordem la PMA no cooperativa des d'una doble perspectiva. D'una banda, ens centrem en la satisfacció dels agents estudiant les propietats desitjables de solucions estables, com ara la optimalitat i la justícia. D'altra banda, busquem una combinació de PMA i tècniques de teoria de jocs capaç de calcular plans conjunts estables de forma eficient alhora que es minimitza la complexitat computacional d'aquesta tasca combinada. A més, considerem els conflictes de planificació i congestions en les funcions d'utilitat dels agents, el que resulta en un enfocament més realista. Des del nostre punt de vista, aquesta tesi doctoral obre una nova línia d'investigació en PMA no cooperativa i estableix els principis bàsics per resoldre el problema de la generació de plans conjunts estables per a agents de planificació auto-interessats mitjançant la combinació de teoria de jocs i planificació automàtica. / Jordán Prunera, JM. (2017). Non-Cooperative Games for Self-Interested Planning Agents [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/90417 / TESIS

Tato práce se zabývá současnými trendy v aplikaci teorie her k tvorbě ekonomických modelů, které se následně využívají při ekonomickém rozhodování s podporou prostředků informatiky. Práce se zejména opírá o poznatky teorie statických a dynamických her a her s dokonalými a nedokonalými informacemi. Zkoumány jsou modely týkající se sdílení zdrojů, aukcí a managementu. Pro každý z popsaných modelů je prezentována konkrétní aplikace.

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This dissertation focuses on studying various network formation games in Economics. We explore a different model in each chapter to capture various aspects of networks. Chapter 1provides an overview of this dissertation. Chapter 2 studies the possible Nash equilibrium configurations in a model of signed network formation as proposed by Hiller (2017). We specify the Nash equilibria in the case of heterogeneous agents. We find 3 possible Nash equilibrium configurations: Utopia network, positive assortative matching, and disassortative matching. We derive the specific conditions under which they arise in a Nash equilibrium. In Chapter 3, we study a generalized model of signed network formation game where the players can choose not only positive and negative links but also neutral links. We check whether the results of the signed network formation model in the literature still hold in our generalized framework using the notion of pairwise Nash equilibrium. Chapter 4 studies inequality in a weighted network formation model using the notion of Nash equilibrium. As a factor of inequality, there are two types of players: Rich players and poor players. We show that both rich and poor players designate other rich players as their best friends. As a result, We present that nested split graphs are drawn from survey data because researchers tend to ask respondents to list only a few friends. / Doctor of Philosophy / This dissertation focuses on studying various network formation games in Economics. We explore a different model in each chapter to capture various aspects of networks. Chapter 1 provides an overview of this dissertation. Chapter 2 studies the possible singed network configurations in equilibrium. In the signed network, players can choose a positive (+) relationship or a negative (-) relationship toward each other player. We study the case that the players are heterogeneous. We find 3 possible categories of networks in equilibrium: Utopia network, positive assortative matching, and disassortative matching. We derive the specific conditions under which they arise in equilibrium. In Chapter 3, we study a generalized model of signed network formation game where the players can choose not only positive and negative links but also neutral links. We check whether the results of the signed network formation model in the literature still hold in our generalized framework. Chapter 4 studies inequality in a weighted network formation model using the notion of Nash equilibrium. In this weighted network model, each player can choose the level of relationship. As a factor of inequality, there are two types of players: rich players and poor players. We show that both rich and poor players choose other rich players as their best friends. As a result, we present that nested split graphs are drawn from survey data because these social network data are censored due to the limit of the number of responses.

Cyber attacks against companies are becoming more common as technology advances and digitalization is increasing exponentially. All Swedish insurance companies that sell cyber insurance encounter the same problem, there is not enough data to do good actuarial work. In order for the pricing procedure to improve and general knowledge of cyber insurance to increase, it has been proposed that insurance companies should share their data with each other. The goal of the thesis is to do mathematical calculations to explore data quality consequences of such a sharing regime. This thesis is based on some important assumptions and three scenarios. The most important assumptions are that there are two insurance companies forced to share all their data with each other and that they can reduce the uncertainty about their own product by investing in better data quality. In the first scenario, we assume a game between two players where they can choose how much to invest in reducing the uncertainty. In the second scenario, we assume that there is not a game, but the two insurance companies are forced to equal investments and thus have the same knowledge of their products. In the third scenario, we assume that the players are risk averse, that is, they are not willing to take high risk. The results will show how much, if any, the insurance companies should invest in the different scenarios to maximize their profits (if risk neutral) or utility (if risk averse). The results of this thesis show that in the first and second scenario, the optimal profit is reached when the insurance companies do not invest anything. In the third scenario though, the optimal investment is greater than zero, given that the companies are enough risk averse.

Braess' paradox is a counter-intuitive phenomenon which can occur in congesting networks. It refers to those cases where the introduction of a new link in the network results in the total travel time on the network increasing. The dissertation starts by introducing the traffic assignment problem and the concept of equilibrium in traffic assignment. The concept of equilibrium is based on Wardrop's first principle that all travellers will attempt to minimize their own travel time regardless of the effect on others. A literature review includes details of a number of papers that have been published investigating theoretical aspects of the paradox. There is also a brief description of Game Theory and the Nash Equilibrium. It has been shown that the equilibrium assignment is an example of Nash Equilibrium. The majority of work that has been published deals with networks where the delay functions that are used to compute the travel times on the links of the network do not include explicit representation of the capacity of the links. In this dissertation a network that is similar in form to the one first presented by Braess was constructed with the difference being that the well-known BPR function was used in the delay functions. This network was used to show that a number of findings that had been presented previously using simpler functions also applied to this network. It was shown that when it occurs, Braess' paradox only occurs over a range of values at relatively low levels of congestion. Real-world networks were then investigated and it was found that similar results occurred to those found in the simpler test networks that are often used in discussions of the paradox. Two methodologies of eliminating the paradox were investigated and the results are presented. / Decision Sciences / M.Sc.