Suchergebnisse

AbstractQuantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices.

Intro -- Preface -- Acknowledgements -- Contents -- Abbreviations -- Basic Notations -- 1 Introduction: Main Models and LLN Methodology -- 1.1 What Is a Markov Chain -- 1.2 Mean-Field Interacting Particle Systems -- 1.3 Migration via Binary Interaction -- 1.4 Introducing Principals -- 1.5 Pressure and Resistance Framework -- 1.6 Nash Equilibria and the Fixed Points of Kinetic Equations -- 1.7 Inspection and Corruption -- 1.8 Cybersecurity, Biological Attack-Defense, and Counterterrorism Modeling -- 1.9 Coalition Building, Merging and Splitting, Network Growth, Queues -- 1.10 The Minority Game, Its Strategic Enhancement, and the Sex Ratio Game -- 1.11 Threshold Collective Behavior -- 1.12 Project Selection, Investment Policies, and Optimal Allocation -- 1.13 Games on Networks and Positivity-Preserving Evolutions -- 1.14 Swarm Intelligence and Swarm-Based Robotics -- 1.15 Replicator Dynamics with Mean-Field Dependent Payoffs: Controlled … -- 1.16 Fluctuations -- 1.17 Brief Guide to the Content of This Book -- Part I Multi-agent Interaction and Nonlinear Markov Games -- 2 Best-Response Principals -- 2.1 Preliminaries: Markov Chains -- 2.2 Preliminaries: ODEs and First-Order PDEs -- 2.3 Mean-Field Interacting Particle Systems -- 2.4 Dynamic LLN: Smooth Coefficients -- 2.5 Dynamic LLN: Lipschitz Coefficients -- 2.6 Dynamic LLN with Major Players -- 2.7 Dynamic LLN with Distinguished (Tagged) Player -- 2.8 Stationary Points of Limiting Dynamics and Nash Equilibria -- 2.9 The Main Class of Examples: Inspection, Corruption, Security -- 2.10 Optimal Allocation -- 2.11 Stability of Stationary Points and Its Consequences -- 2.12 Stability for Two-State Models -- 2.13 Discontinuous Transition Rates -- 2.14 Nonexponential Waiting Times and Fractional LLN -- 3 Dynamic Control of Major Players -- 3.1 Multistep Decision-Making of the Principal.