The Master Equation for Large Population Equilibriums

We use a simple N-player stochastic game with idiosyncratic and common noises to introduce the concept of Master Equation originally proposed by Lions in his lectures at the Coll\`ege de France. Controlling the limit N tends to the infinity of the explicit solution of the N-player game, we highlight the stochastic nature of the limit distributions of the states of the players due to the fact that the random environment does not average out in the limit, and we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic Partial Differential Equations (SPDEs). The first one is a forward stochastic Kolmogorov equation giving the evolution of the conditional distributions of the states of the players given the common noise. The second is a form of stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the optimization problem when the flow of conditional distributions is given. Being highly coupled, the system reads as an infinite dimensional Forward Backward Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its Markov property lead to the representation of the solution of the backward equation (i.e. the value function of the stochastic HJB equation) as a deterministic function of the solution of the forward Kolmogorov equation, function which is usually called the decoupling field of the FBSDE. The (infinite dimensional) PDE satisfied by this decoupling field is identified with the \textit{master equation}. We also show that this equation can be derived for other large populations equilibriums like those given by the optimal control of McKean-Vlasov stochastic differential equations. The paper is written more in the style of a review than a technical paper, and we spend more time and energy motivating and explaining the probabilistic interpretation of the Master Equation, than identifying the most general set of assumptions under which our claims are true

Balancing national objectives and settling rivalries: towards a new European asylum system?

We offer a holistic view (and formal representation) of the Common European Asylum System (CEAS), in which the national objectives of the states emerge in terms of trade-off between control and respect of the asylum seekers' human rights. Control of access implies spillover of the flows towards neighbouring states and rivalries between the states in a non-cooperative game. An asylum system is a set of rules designed to settle rivalries, balancing out the national objectives. Thus, on the basis of a Pareto (unanimity) criterion, one system is preferred to another if all the states are better able to achieve their national objectives. We examine the conditions under which a relocation system based on quotas may be preferable to the Dublin rules, and the reasons why closer harmonisation of standards can favour adoption of such a system. Finally, we comment on the possible outcomes when unanimity is not achieved

Translation invariant mean field games with common noise

This note highlights a special class of mean field games in which the coefficients satisfy a convolution-type structural condition. A mean field game of this type with common noise is related to a certain mean field game without common noise by a simple transformation, which permits a tractable construction of a solution of the problem with common noise from a solution of the problem without

Evaluating Competition Strategies for Generic Drug Industries Using Game Theory - A supplement to report “Scenario Planning as a Tool for Long Term Strategic Planning”

This report is a supplement to the project titled “Scenario Planning as a Tool for Long Term Strategic Planning - The Generics Drug Industry in the European Union.” The report will further evaluate the resultant scenarios and strategies build and recommended as part of the titled project mentioned above. Game theory perspectives will be used as a tool to analyse how economic agents (stakeholders of the pharmaceutical generic drugs industry) will react when what they do affects the actions of others. The report will evaluate hypothetical actions taken by generic drugs industry players and their outcomes/payoffs relative to the competitions. It will draw on strategic and extensive forms of games to identify how to act and how to think about your rival’s actions. What would be a more powerful in business strategy than this

Evaluating Competition Strategies for Generic Drug Industries Using Game Theory - A supplement to report “Scenario Planning as a Tool for Long Term Strategic Planning”

This report is a supplement to the project titled “Scenario Planning as a Tool for Long Term Strategic Planning - The Generics Drug Industry in the European Union.” The report will further evaluate the resultant scenarios and strategies build and recommended as part of the titled project mentioned above. Game theory perspectives will be used as a tool to analyse how economic agents (stakeholders of the pharmaceutical generic drugs industry) will react when what they do affects the actions of others. The report will evaluate hypothetical actions taken by generic drugs industry players and their outcomes/payoffs relative to the competitions. It will draw on strategic and extensive forms of games to identify how to act and how to think about your rival’s actions. What would be a more powerful in business strategy than this

On Proportionate and Truthful International Alliance Contributions: An Analysis of Incentive Compatible Cost Sharing Mechanisms to Burden Sharing

Burden sharing within an international alliance is a contentious topic, especially in the current geopolitical environment, that in practice is generally imposed by a central authority\u27s perception of its members\u27 abilities to contribute. Instead, we propose a cost sharing mechanism such that burden shares are allocated to nations based on their honest declarations of the alliance\u27s worth. Specifically, we develop a set of multiobjective nonlinear optimization problem formulations that respectively impose Bayesian Incentive Compatible (BIC), Strategyproof (SP), and Group Strategyproof (GSP) mechanisms based on probabilistic inspection efforts and deception penalties that are budget balanced and in the core. Any feasible solution to these problems corresponds to a single stage Bayesian stochastic game wherein a collectively honest declaration is a Bayes-Nash equilibrium, a Nash Equilibrium in dominant strategies, or a collusion resistant Nash equilibrium, respectively, but the optimal solution considers the alliance\u27s central authority preferences. Each formulation is shown to be a nonconvex optimization problem. The solution quality and computational effort required for three heuristic algorithms as well as the BARON global solver are analyzed to determine the superlative solution methodology for each problem. The Pareto fronts associated with each multiobjective optimization problem are examined to determine the tradeoff between inspection frequency and penalty severity required to obtain truthfulness under stronger assumptions. Memory limitations are examined to ascertain the size of alliances for which the proposed methodology can be utilized. Finally, a full block design experiment considering the clustering of available alliance valuations and the member nations\u27 probability distributions therein is executed on an intermediate-sized alliance motivated by the South American alliance UNASUR

A Note on Anti-Berge Equilibrium for Bimatrix Game

Game theory plays an important role in applied mathematics, economics and decision theory. There are many works devoted to game theory. Most of them deals with a Nash equilibrium. A global search algorithm for finding a Nash equilibrium was proposed in [13]. Also, the extraproximal and extragradient algorithms for the Nash equilibrium have been discussed in [3]. Berge equilibrium is a model of cooperation in social dilemmas, including the Prisoner’s Dilemma games [15]. The Berge equilibrium concept was introduced by the French mathematician Claude Berge [5] for coalition games. The first research works of Berge equilibrium were conducted by Vaisman and Zhukovskiy [18; 19]. A method for constructing a Berge equilibrium which is Pareto-maximal with respect to all other Berge equilibriums has been examined in Zhukovskiy [10]. Also, the equilibrium was studied in [16] from a view point of differential games. Abalo and Kostreva [1; 2] proved the existence theorems for pure-strategy Berge equilibrium in strategic-form games of differential games. Nessah [11] and Larbani, Tazdait [12] provided with a new existence theorem. Applications of Berge equilibrium in social science have been discussed in [6; 17]. Also, the work [7] deals with an application of Berge equilibrium in economics. Connection of Nash and Berge equilibriums has been shown in [17]. Most recently, the Berge equilibrium was examined in Enkhbat and Batbileg [14] for Bimatrix game with its nonconvex optimization reduction. In this paper, inspired by Nash and Berge equilibriums, we introduce a new notion of equilibrium so-called Anti-Berge equilibrium. The main goal of this paper is to examine Anti-Berge equilibrium for bimatrix game. The work is organized as follows. Section 2 is devoted to the existence of Anti-Berge equilibrium in a bimatrix game for mixed strategies. In Section 3, an optimization formulation of Anti-Berge equilibrium has been formulated

Social Shaping for Multi-Agent Systems

Multi-agent systems have gained attention due to advances in automation, technology, and AI. In these systems, intelligent agents collaborate through networks to achieve goals. Despite successes, multi-agent systems pose social challenges. Problems include agents finding resource prices unacceptable due to efficient allocation, interactions being cooperative/competitive, leading to varying outcomes, and sensitive data being at risk due to sharing. Problems are: 1. Price Acceptance; 2. Agent Cooperation and Competition; 3. Privacy Risks. For Price Acceptance, we address decentralized resource allocation systems as markets. We solve price acceptance in static systems with quadratic utility functions by defining allowed quadratic ranges. For dynamic systems, we present dynamic competitive equilibrium computation and propose a horizon strategy for smoothing dynamic pricing. Concerning Agent Cooperation and Competition, we study the well-known Regional Integrated Climate-Economy model (RICE). It's a dynamic game. We analyze cooperative and competitive solutions, showing impact on negotiations and consensus for regional climate action. Regarding Privacy Risks, we infer network structures from linear-quadratic game best-response dynamics to reveal agent vulnerabilities. We prove network identifiability tied to controllability conditions. A stable, sparse system identification algorithm learns network structures despite noise. Lastly, we contribute privacy-aware algorithms. We address network games where agents aggregate under differential privacy. Extending to network games, we propose a Laplace linear-quadratic functional perturbation algorithm. A tutorial example demonstrates meeting privacy needs through tuning. In summary, this thesis solves social challenges in multi-agent systems: Price Acceptance, Agent Cooperation and Competition, and Privacy Risks

Integrated management of chemical processes in a competitive environment

El objetivo general de esta Tesis es mejorar el proceso de la toma de decisiones en la gestión de cadenas de suministro, tomando en cuenta principalmente dos diferencias: ser competitivo considerando las decisiones propias de la cadena de suministro, y ser competitivo dentro de un entorno global. La estructura de ésta tesis se divide en 4 partes principales: La Parte I consiste en una introducción general de los temas cubiertos en esta Tesis (Capítulo 1). Una revisión de la literatura, que nos permite identificar las problemáticas asociadas al proceso de toma de decisiones (Capítulo 2). El Capítulo 3 presenta una introducción de las técnicas y métodos de optimización utilizados para resolver los problemas propuestos en esta Tesis. La Parte II se enfoca en la integración de los niveles de decisión, buscando mejorar la toma de decisiones de la propia cadena de suministro. El Capítulo 4 presenta una formulación matemática que integra las decisiones de síntesis de procesos y las decisiones operacionales. Además, este capítulo presenta un modelo integrado para la toma de decisiones operacionales incluyendo las características del control de procesos. El Capítulo 5 muestra la integración de las decisiones del nivel táctico y el operacional, dicha propuesta está basada en el conocimiento adquirido capturando la información relacionada al nivel operacional. Una vez obtenida esta información se incluye en la toma de decisiones a nivel táctico. Finalmente en el capítulo 6 se desarrolla un modelo simplificado para integrar múltiples cadenas de suministro. El modelo propuesto incluye la información detallada de las entidades presentes en una cadena de suministro (suministradores, plantas de producción, distribuidores y mercados) introduciéndola en un modelo matemático para su coordinación. La Parte III propone la integración explicita de múltiples cadenas de suministro que tienen que enfrentar numerosas situaciones propias de un mercado global. Asimismo, esta parte presenta una nueva herramienta de optimización basada en el uso integrado de métodos de programación matemática y conceptos relacionados a la Teoría de Juegos. En el Capítulo 7 analiza múltiples cadenas de suministro que cooperan o compiten por la demanda global del mercado. El Capítulo 8 incluye una comparación entre el problema resuelto en el Capítulo anterior y un modelo estocástico, los resultados obtenidos nos permiten situar el comportamiento de los competidores como fuente exógena de la incertidumbre típicamente asociada la demanda del mercado. Además, los resultados de ambos Capítulos muestran una mejora sustancial en el coste total de las cadenas de suministro asociada al hecho de cooperar para atender de forma conjunta la demanda disponible. Es por esto, que el Capítulo 9 presenta una nueva herramienta de negociación, basada en la resolución del mismo problema (Capítulo 7) bajo un análisis multiobjetivo. Finalmente, la parte IV presenta las conclusiones finales y una descripción general del trabajo futuro.This Thesis aims to enhance the decision making process in the SCM, remarking the difference between optimizing the SC to be competitive by its own, and to be competitive in a global market in cooperative and competitive environments. The structure of this work has been divided in four main parts: Part I: consists in a general introduction of the main topics covered in this manuscript (Chapter I); a review of the State of the Art that allows us to identify new open issues in the PSE (Chapter 2). Finally, Chapter 3 introduces the main optimization techniques and methods used in this contribution. Part II focuses on the integration of decision making levels in order to improve the decision making of a single SC: Chapter 4 presents a novel formulation to integrate synthesis and scheduling decision making models, additionally, this chapter also shows an integrated operational and control decision making model for distributed generations systems (EGS). Chapter 5 shows the integration of tactical and operational decision making levels. In this chapter a knowledge based approach has been developed capturing the information related to the operational decision making level. Then, this information has been included in the tactical decision making model. In Chapter 6 a simplified approach for integrated SCs is developed, the detailed information of the typical production‐distribution SC echelons has been introduced in a coordinated SC model. Part III proposes the explicit integration of several SC’s decision making in order to face several real market situations. As well, a novel formulation is developed using an MILP model and Game Theory (GT) as a decision making tool. Chapter 7 includes the tactical and operational analysis of several SC’s cooperating or competing for the global market demand. Moreover, Chapter 8 includes a comparison, based on the previous results (MILP‐GT optimization tool) and a two stage stochastic optimization model. Results from both Chapters show how cooperating for the global demand represent an improvement of the overall total cost. Consequently, Chapter 9 presents a bargaining tool obtained by the Multiobjective (MO) resolution of the model presented in Chapter 7. Finally, final conclusions and further work have been provided in Part IV.Postprint (published version

Governing multi-actor decision processes in Dutch industrial area redevelopment

In the first part of the thesis, a literature review is presented. In this literature review, industrial areas in the Netherlands are discussed, leading to the conclusion that industrial areas are important for realizing sustainable economic growth in the Netherlands. Industrial areas play an important role in accommodating employment, in stimulating local and regional economies, and in creating a high value added. Furthermore, I conclude that process features have a significant influence on the outcome of industrial area (re)development projects. Subsequently, the most important problem aspects of the current industrial area planning approach are discussed, together with several causes of these problems. It is argued that most of these problematic failures can be traced back to one main problem: the rapid obsolescence of the existing stock of industrial areas. The dimensions of the Dutch industrial area redevelopment task affirm this. This creates a large necessity for redevelopment. However, based upon the disappointing figures on yearly realized redevelopment projects and on the low spatial yields of actually realized redevelopment projects, it is concluded that the execution of industrial area redevelopment projects stagnates. When starting up a redevelopment project in the current increasingly complex and rapidly changing environment, interdependent negotiation processes within and among organizations appear to be problematic, consuming substantial time and effort. Focus within this research lies therefore on studying, supporting and accommodating the consensus-building process within redevelopment projects. The point of departure in this research is the postulate that the main cause of the occurring stagnation in industrial area redevelopment is the absence of a well-functioning process governance system. Several authors support this statement; they posit that the regional arena is the appropriate level for executing such governance. Because of a lack of insight into effective ways to implement a governance system, and because of the presumed advantages related to the acceleration of industrial area redevelopment processes when gaining this insight, the general research objective is as follows: ‘To explore ways to effectively support the governance of involved stakeholders’ choice behavior, in order to stimulate the current decision-making processes in industrial area redevelopment projects’. Thus, governance – and especially meta-governance – is a promising approach for application to complex industrial area redevelopment projects. Several best-practice industrial area redevelopment projects reveal that centrally governing such alliances contributes to project success. The aim of meta-governance within industrial area redevelopment projects is to establish cooperation between relevant parties, in order to realize a number of functions and purposes from a public, social importance, through the establishment of one central governing agency, responsible for the management of the decision-making process. In Dutch industrial area redevelopment, regional development companies seem most appropriate for executing this central governance role. Assuming that meta-governance can be a solution to the occurring problems in the Dutch industrial area redevelopment market, it is essential to analyze the consensusfinding processes, as well as causes of tension and conflict, in order to theoretically support governing agencies in managing decision-making processes. Therefore, the specific goal of the research is to better understand how individual and interactive decision-making of the most important actors in industrial area redevelopment processes can be modeled, in order to analyze and predict the occurrence of cooperation or conflict, and how this decision-making can be influenced by a regional governing agency. A better understanding of these processes is a key requirement for the development of a decision support tool for this regional governing agency, in order to support the acceleration of industrial area redevelopment projects. A formal model of the collaborative decision process has not been developed for this domain, incorporating a governance approach. Therefore, several available techniques for analyzing both individual and interactive decision-making are explored in the second part of the thesis. From this, it is concluded that the discrete choice approach seems applicable for modeling individual choice behavior of actors. Furthermore, the application of game theory seems very interesting for modeling interactive and interdependent choice behavior. In order to make a game-theoretic model that is suitable for studying strategic interactions in industrial area redevelopment, a relatively new approach is advocated in which game theory is combined with a multiattribute trade-off technique. Eventually, the application of game theory leads to an insight in the occurrence of conflicts, and in the causes of these conflicts. The 2x2 game is regarded as most appropriate for application in this research because this game type has been used very often in conflict modeling and conflict management, and it suits the real world negotiation processes in which two players are involved, each roughly having two strategies. Within 2x2 games, three tools are most utilizable for solving conflicts: (1) changing the information of the involved players; (2) changing the payoffs of the players; and (3) changing the rules of the game, focusing on the sequence of decisionmaking and the possible allowance of communication in the game. Because the research focuses on static 2x2 games of complete information, it is concluded that gaining insight in the second tool is most feasible and interesting. In the third part, the results are represented. Firstly, the individual choice behavior of involved actors is modeled, thereby giving a better and more systematical insight in stakeholders’ preferences when accepting or rejecting a development plan, in the (dis)similarities between both stakeholder groups’ preferences in making that choice, and in the most important points of interest when composing a development plan proposal. Resulting data analysis showed that the plan attributes ‘technical quality’ and ‘cost coverage’ are important for both stakeholder groups when choosing a plan proposal. A high level of these attributes in a plan results in a high probability of being chosen, while a low level results in a low choice probability. Besides this, companies find the attribute ‘development speed’ very important when choosing a plan, and municipalities value ‘architectural quality’ highly. Furthermore, municipalities proved to be less demanding in accepting industrial area redevelopment plan proposals. Secondly, the interaction between involved stakeholders is modeled as an interdependent process, using a relative novel approach in which conjoint analysis and game theory are combined, in order to explain the occurrence of cooperation or conflict within Dutch industrial areas redevelopment negotiations. Data analysis reveals that there is one major source of conflicts; stakeholders choosing not to cooperate based upon the presented negotiation setting. A more in-depth analysis of negotiation settings ending up in mutual cooperation demonstrated that the appraisal of both stakeholders for the proposed development plan is the most influential factor, together with an eventual absolute difference between both players’ appraisals. This leads to the conclusion that the content of proposed plans is very important in such negotiations; factors like power and risks play a secondary role. Thirdly, a model is created that supports the decision-making of a central governing agency. This model is based upon the results of the individual and interactive choice models, giving recommendations on how to put meta-governance into practice in industrial area redevelopment. The model consists of three major steps: (1) giving assistance in assessing the initial state of the negotiation; (2) calculating whether the possible conflict occurrence can be prevented by marginally changing the payoffs of both players; and (3) indicating how the equalizing of appraisals can be put into practice. This final step gives insight in the contribution of specific changes in plan proposals to solving the conflicts that are discovered in the first part. After testing the model, it is concluded that altering payoffs in games in order to avoid conflicts is very effective in these games. Furthermore, these payoffs can often be altered through the use of minimal resources. In cases in which the municipality values the proposed plan lower than the company – raising the levels of the attributes technical quality, architectural quality, and value development results most often in an equaling of both players’ appraisals. Furthermore, the attribute architectural quality functions most often as the only solution. Reciprocally, in cases in which the company has a lower plan proposal appraisal, raising the levels of the attributes development speed, technical quality, and cost coverage most often results in an equal plan proposal appraisal. In general, tools are already available for executing interventions on above-mentioned attributes. Thus, focus should be on actual execution of the governance task, not on adding tools to the existing instrumental palette. Concluding, a model is created with which it is possible to give recommendations concerning the decision-making of a central governing agency in different possible industrial area redevelopment negotiations. It entails a new, structured way of solving conflicts, which is empirically testable, and delivers some real world recommendations