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ABSTRACT Usually, real game problems encountered in our daily lives are so complicated that the existing methods are no longer sufficient to deal with them. This motivates us to investigate several kinds of differential game problems, which have not been considered or solved yet, including a pursuit-evasion game with n pursuers and one evader, a problem of guarding a territory with two guarders and two invaders, and a payoff-switching differential game. In this thesis, firstly the geometric method is used to consider the pursuit-evasion game with n pursuers and one evader. Two criteria used to find the solutions of the game in some cases are given. It will be shown that the one-on-one pursuit-evasion game is a special case of this game. Secondly, the problem of guarding a territory with two guarders and two invaders is considered both qualitatively and quantitatively. The investigation of this problem reveals a variety of situations never occurring in the case with one guarder and one invader. An interesting thing found in this investigation is that some invader may play the role as a pursuer for achieving a more favorable payoff in some cases. This will make the problem more complicated and more difficult to be solved. The payoff-switching differential game, first proposed by us, is a kind of differential game with incomplete information. The main difference between this problem and traditional differential games is that in a payoff-switching differential game, any one player at any time may have several choices of payoffs for the future. The optimality in such a problem becomes questionable. Some reasoning mechanisms based on different methods will be provided to determine a reasoning strategy for some player in a payoff-switching differential game. A practical payoff-switching differential game problem, i.e., the guarding three territories with one guarder against one invader, is presented to illustrate the situations of such a game problem. Many computer simulations of this example are given to show the performances of different reasoning strategies. The proposition of the payoff-switching differential game is an important breakthrough in dealing with some kinds of differential games with incomplete information.

京都大学 / 0048 / 新制・課程博士 / 博士(経済学) / 甲第18759号 / 経博第510号 / 新制||経||273(附属図書館) / 31710 / 京都大学大学院経済学研究科経済学専攻 / (主査)教授 柴田 章久, 教授 三野 和雄, 教授 矢野 誠 / 学位規則第4条第1項該当 / Doctor of Economics / Kyoto University / DGAM

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This dissertation is concerned with three problems within the field of optimization for agent--based systems. Firstly, the inverse optimal control problem is investigated for the single-agent system. Given a dynamic process, the goal is to recover the quadratic cost function from the observation of optimal control sequences. Such estimation could then help us develop a better understanding of the physical system and reproduce a similar optimal controller in other applications. Next, problems of optimization over networked systems are considered. A novel differential game approach is proposed for the optimal intrinsic formation control of multi-agent systems. As for the credit scoring problem, an optimal filtering framework is utilized to recursively improve the scoring accuracy based on dynamic network information. In paper A, the problem of finite horizon inverse optimal control problem is investigated, where the linear quadratic (LQ) cost function is required to be estimated from the optimal feedback controller. Although the infinite-horizon inverse LQ problem is well-studied with numerous results, the finite-horizon case is still an open problem. To the best of our knowledge, we propose the first complete result of the necessary and sufficient condition for the existence of corresponding LQ cost functions. Under feasible cases, the analytic expression of the whole solution space is derived and the equivalence of weighting matrices is discussed. For infeasible problems, an infinite dimensional convex problem is formulated to obtain a best-fit approximate solution with minimal control residual, where the optimality condition is solved under a static quadratic programming framework to facilitate the computation. In paper B, the optimal formation control problem of a multi-agent system is studied. The foraging behavior of N agents is modeled as a finite-horizon non-cooperative differential game under local information, and its Nash equilibrium is studied. The collaborative swarming behaviour derived from non-cooperative individual actions also sheds new light on understanding such phenomenon in the nature. The proposed framework has a tutorial meaning since a systematic approach for formation control is proposed, where the desired formation can be obtained by only intrinsically adjusting individual costs and network topology. In contrast to most of the existing methodologies based on regulating formation errors to the pre-defined pattern, the proposed method does not need to involve any information of the desired pattern beforehand. We refer to this type of formation control as intrinsic formation control. Patterns of regular polygons, antipodal formations and Platonic solids can be achieved as Nash equilibria of the game while inter-agent collisions are naturally avoided. Paper C considers the credit scoring problem by incorporating dynamic network information, where the advantages of such incorporation are investigated in two scenarios. Firstly, when the scoring publishment is merely individual--dependent, an optimal Bayesian filter is designed for risk prediction, where network observations are utilized to provide a reference for the bank on future financial decisions. Furthermore, a recursive Bayes estimator is proposed to improve the accuracy of score publishment by incorporating the dynamic network topology as well. It is shown that under the proposed evolution framework, the designed estimator has a higher precision than all the efficient estimators, and the mean square errors are strictly smaller than the Cramér-Rao lower bound for clients within a certain range of scores. / I denna avhandling behandlas tre problem inom optimering för agentbaserade system. Inledningsvis undersöks problemet rörande invers optimal styrning för ett system med en agent. Målet är att, givet en dynamisk process, återskapa den kvadratiska kostnadsfunktionen från observationer av sekvenser av optimal styrning. En sådan uppskattning kan ge ökad förståelse av det underliggande fysikaliska systemet, samt vara behjälplig vid konstruktion av en liknande optimal regulator för andra tillämpningar. Vidare betraktas problem rörande optimering över nätverkssystem. Ett nytt angreppssätt, baserat på differentialspel, föreslås för optimal intrinsisk formationsstyrning av system med fler agenter. För kreditutvärderingsproblemet utnyttjas ett filtreringsramverk för att rekursivt förbättra kreditvärderingens noggrannhet baserat på dynamisk nätverksinformation. I artikel A undersöks problemet med invers optimal styrning med ändlig tidshorisont, där den linjärkvadratiska (LQ) kostnadsfunktionen måste uppskattas från den optimala återkopplingsregulatorn. Trots att det inversa LQ-problemet med oändlig tidshorisont är välstuderat och med flertalet resultat, är fallet med ändlig tidshorisont fortfarande ett öppet problem. Så vitt vi vet presenterar vi det första kompletta resultatet med både tillräckliga och nödvändiga villkor för existens av en motsvarande LQ-kostnadsfunktion. I fallet med lösbara problem härleds ett analytiskt uttryck för hela lösningsrummet och frågan om ekvivalens med viktmatriser behandlas. För de olösbara problemen formuleras ett oändligtdimensionellt konvext optimeringsproblem för att hitta den bästa approximativa lösningen med den minsta styrresidualen. För att underlätta beräkningarna löses optimalitetsvillkoren i ett ramverk för statisk kvadratisk programmering. I artikel B studeras problemet rörande optimal formationsstyrning av ett multiagentsystem. Agenternas svärmbeteende modelleras som ett icke-kooperativt differentialspel med ändlig tidshorisont och enbart lokal information. Vi studerar detta spels Nashjämvikt. Att, ur icke-kooperativa individuella handlingar, härleda ett kollaborativt svärmbeteende kastar nytt ljus på vår förståelse av sådana, i naturen förekommande, fenomen. Det föreslagna ramverket är vägledande i den meningen att det är ett systematiskt tillvägagångssätt för formationsstyrning, där den önskade formeringen kan erhållas genom att endast inbördes justera individuella kostnader samt nätverkstopologin. I motstat till de flesta befintliga metoder, vilka baseras på att reglera felet i formeringen relativt det fördefinierade mönstret, så behöver den föreslagna metoden inte på förhand ta hänsyn till det önskade mönstret. Vi kallar denna typ av formationsstyrning för intrinsisk formationsstyrning. Mönster så som regelbundna polygoner, antipodala formeringar och Platonska kroppar kan uppnås som Nashjämvikter i spelet, samtidigt som kollisioner mellan agenter undviks på ett naturligt sätt. Artikel C behandlar kreditutvärderingsproblemet genom att lägga till dynamisk nätverksinformation. Fördelarna med en sådan integrering undersöks i två scenarier. Då kreditvärdigheten enbart är individberoende utformas ett optimalt Bayesiskt filter för riskvärdering, där observationer från nätverket används för att tillhandahålla en referens för banken på framtida finansiella beslut. Vidare föreslås en rekursiv Bayesisk estimator (stickprovsvariabel) för att förbättra noggrannheten på den skattade kreditvärdigheten genom att integrera även den dynamiska nätverkstopologin. Inom den föreslagna ramverket för tidsutveckling kan vi visa att, för kunder inom ett visst intervall av värderingar, har den utformade estimatorn högre precision än alla effektiva estimatorer och medelkvadrafelet är strikt mindre än den nedre gränsen från Cramér-Raos olikhet. / <p>QC 20190603</p>

In this dissertation, we consider differential games for multi-agent systems under distributed information where every agent is only able to acquire information about the others according to a directed information graph of local communication/sensor networks. Such games arise naturally from many applications including mobile robot coordination, power system optimization, multiplayer pursuit-evasion games, etc. Since the admissible strategy of each agent has to conform to the information graph constraint, the conventional game strategy design approaches based upon Riccati equation(s) are not applicable because all the agents are required to have the information of the entire system. Accordingly, the game strategy design under distributed information is commonly known to be challenging. Toward this end, we propose novel open-loop and feedback game strategy design approaches for Nash equilibrium and noninferior solutions with a focus on linear quadratic differential games. For the open-loop design, approximate Nash/noninferior game strategies are proposed by integrating distributed state estimation into the open-loop global-information Nash/noninferior strategies such that, without global information, the distributed game strategies can be made arbitrarily close to and asymptotically converge over time to the global-information strategies. For the feedback design, we propose the best achievable performance indices based approach under which the distributed strategies form a Nash equilibrium or noninferior solution with respect to a set of performance indices that are the closest to the original indices. This approach overcomes two issues in the classical optimal output feedback approach: the simultaneous optimization and initial state dependence. The proposed open-loop and feedback design approaches are applied to an unmanned aerial vehicle formation control problem and a multi-pursuer single-evader differential game problem, respectively. Simulation results of several scenarios are presented for illustration.

Une grande partie de la ressource halieutique est, à ce jour, surexploitée, voire menacée d’extinction. Il est donc nécessaire de gérer cette ressource de manière à assurer la viabilité du secteur économique de la pêche. L’objectif de ce travail est d’étendre et de compléter les modèles existant dans la littérature économique sur le sujet qui visent à analyser les effets de différentes stratégies de préservation de la ressource sur le profit du secteur de la pêche. Ces travaux souffrent en effet de deux limites principales.D’une part, ils mettent principalement en avant les caractéristiques biologiques de la ressource dans l’espace et leurs travaux se limitent généralement à la détermination du niveau de pêche qui assure le rendement soutenable maximal. Cependant, d’un point de vue économique, ces travaux sont limités puisqu’ils ne prennent pas en compte les interactions stratégiques entre agents.D’autre part, concernant les politiques de gestion de la ressource halieutique dans l’espace, peu de travaux intègre les stratégies individuelles des agents pour évaluer les impacts de ces règlementations sur l’état des stocks. De plus, en ce qui concerne les aires marines protégées, les travaux existant se concentrent sur les impacts de ce type de mesures sur les stocks de ressource et leur approche ne permet pas d’évaluer l’impact de la taille d’une aire marine protégée.Il est donc important d’élargir l’analyse dans deux directions. Tout d’abord, Nous allons tenter d’améliorer les travaux existants en y intégrant les stratégies des pêcheurs dans le cadre d’un secteur oligopolistique, (structure de marché représentative du secteur de la pêche), tout en prenant en compte les caractéristiques biologiques en termes de dispersion pour savoir dans quelle mesure ces différents aspects influencent les perspectives de préservation.Enfin, nous nous intéresserons, dans le même cadre d’analyse, aux politiques de gestion des ressources halieutiques pour évaluer leur efficacité. / Much of the fishery resource is, to date, overexploited even endangered. It is therefore necessary to manage this resource to ensure the sustainability of the fishery’s economic sector. The aim of this work is to extend and complete the existing models in the literature on the topic which aim for analyze the effects of different strategies for resource conservation benefit of the fishery sector. Indeed, these studies suffer from two main limitations.On the one hand, they put forward mainly the biological characteristics of the resource in space and their works are generally limited to the determination of the fishing level which provides the maximum sustainable yield. However, from an economic point of view, these works are limited, as they do not take into account the strategic behavior between agents.On the other hand, concerning the management policies of fishery resources in space, a few studies integrate agents’ individual strategies to evaluate the impacts of these regulations on the stocks state. In addition, with regard to marine protected areas, existing works focus on the impact of such measures on resources stock and their approach does not allow evaluating the impact of the size of a marine reserve.It is therefore important to extend the analysis in two directions. Firstly, we will try to improve existing works on integrating fishermen strategies in an oligopolistic sector (market structure representing the fishery sector), while taking into account the biological characteristics in terms of dispersion, to investigate in which extent these aspects influence the preservation prospects.Then, we focus, in the same analytical framework, the fishery resources management policies to evaluate their effectiveness.

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.

在經濟活動發展迅速的情形下，產品的競爭與多樣化使得廠商在價格之外，也須將行銷策略納入考量。研究行銷通路中成員互動的理論模型從靜態模型開始，到用動態微分賽局的模型來研究廠商間的互動，過去的研究文獻得到了兩點結論：第一，通路合作為Pareto最適。第二，在通路無法合作的情況下，利用合作機制的建立，能使均衡結果產生Pareto改善。 然而，過去的文獻卻未說明將通路合作放入合作廣告的模型中，通路合作是否仍為Pareto最適。因此，本研究沿用Jørgensen et al.(2003)合作廣告模型的設定，將通路合作的情況放入模型中，比較通路合作、零售商遠視、零售商短視與合作廣告四種情形均衡時的行銷策略及廠商的利潤，並探討通路合作在合作廣告的模型中是否仍為Pareto最適。

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.