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Abstract; Contents; 1 Introduction; 1.1 Motivation of Research; 1.2 List of Notations; 2 Game Theory and Multi-Agent Optimization; 2.1 Game Theory; 2.1.1 Introduction to Game Theory; 2.1.2 Nash Equilibrium; 2.1.3 Potential Games; 2.2 Potential Game Design in Multi-Agent Optimization; 2.2.1 Multi-Agent Systems Modeled by Means of Potential Games; 2.2.2 Learning Optimal States in Potential Games; 2.3 Distributed Optimization in Multi-Agent Systems; References; 3 Logit Dynamics in Potential Games with Memoryless Players; 3.1 Introduction
3.2 Memoryless Learning in Discrete Action Games as a Regular Perturbed Markov Chain3.2.1 Preliminaries: Regular Perturbed Markov Chains; 3.2.2 Convergence in Total Variation of General Memoryless Learning Algorithms; 3.3 Asynchronous Learning; 3.3.1 Log-Linear Learning in Discrete Action Games; 3.3.1.1 An Example: Log-Linear Learning for Consensus Problem; 3.3.2 Convergence to Potential Function Maximizers; 3.4 Synchronization in Memoryless Learning; 3.4.1 Additional Information is Needed; 3.4.2 Independent Log-Linear Learning in Discrete Action Games
3.4.3 Convergence to Potential Function Maximizers3.5 Convergence Rate Estimation and Finite Time Behavior; 3.5.1 Convergence Rate of Time-Inhomogeneous Log-Linear Learning; 3.5.2 Convergence Rate of Time-Inhomogeneous Independent Log-Linear Learning; 3.5.3 Simulation Results: Example of a SensorCoverage Problem; 3.5.3.1 Inhomogeneous Log-Linear Learning in Coverage Problem; 3.5.3.2 Inhomogeneous Independent Log-Linear Learning in Coverage Problem; 3.6 Learning in Continuous Action Games; 3.6.1 Log-Linear Learning in Continuous Action Games
4.3 Push-Sum Algorithm in Non-convex Distributed Optimization4.3.1 Problem Formulation: Push-Sum Algorithm and Assumptions; 4.3.2 Convergence to Critical Points; 4.3.3 Perturbed Procedure: Convergence to Local Minima; 4.3.4 Convergence Rate of the Perturbed Process; 4.3.5 Simulation Results: Illustrative Example and Congestion Routing Problem; 4.4 Communication-Based Memoryless Learningin Potential Games; 4.4.1 Simulation Results: Code Division Multiple Access Problem; 4.5 Payoff-Based Learning in Potential Games; 4.5.1 Convergence to a Local Maximum of the PotentialFunction
3.2 Memoryless Learning in Discrete Action Games as a Regular Perturbed Markov Chain3.2.1 Preliminaries: Regular Perturbed Markov Chains; 3.2.2 Convergence in Total Variation of General Memoryless Learning Algorithms; 3.3 Asynchronous Learning; 3.3.1 Log-Linear Learning in Discrete Action Games; 3.3.1.1 An Example: Log-Linear Learning for Consensus Problem; 3.3.2 Convergence to Potential Function Maximizers; 3.4 Synchronization in Memoryless Learning; 3.4.1 Additional Information is Needed; 3.4.2 Independent Log-Linear Learning in Discrete Action Games
3.4.3 Convergence to Potential Function Maximizers3.5 Convergence Rate Estimation and Finite Time Behavior; 3.5.1 Convergence Rate of Time-Inhomogeneous Log-Linear Learning; 3.5.2 Convergence Rate of Time-Inhomogeneous Independent Log-Linear Learning; 3.5.3 Simulation Results: Example of a SensorCoverage Problem; 3.5.3.1 Inhomogeneous Log-Linear Learning in Coverage Problem; 3.5.3.2 Inhomogeneous Independent Log-Linear Learning in Coverage Problem; 3.6 Learning in Continuous Action Games; 3.6.1 Log-Linear Learning in Continuous Action Games
4.3 Push-Sum Algorithm in Non-convex Distributed Optimization4.3.1 Problem Formulation: Push-Sum Algorithm and Assumptions; 4.3.2 Convergence to Critical Points; 4.3.3 Perturbed Procedure: Convergence to Local Minima; 4.3.4 Convergence Rate of the Perturbed Process; 4.3.5 Simulation Results: Illustrative Example and Congestion Routing Problem; 4.4 Communication-Based Memoryless Learningin Potential Games; 4.4.1 Simulation Results: Code Division Multiple Access Problem; 4.5 Payoff-Based Learning in Potential Games; 4.5.1 Convergence to a Local Maximum of the PotentialFunction